Summary Introduction to Statistics including examples and pictures
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Statistics
Questions worth doing again: p.96 Q5, p.129 Q17, p.191 Q25, p.222 Q17, p.265 Q15, alle
opdrachten week 9
 best way to make sense of great loads of data
= a set of mathematical procedures for organizing, summarizing, and interpreting
information
Chapter 1
Descriptive statistics
Statistical procedures used to summarize, organize, and simplify data
- About describing the data
- Through summary statistics (organized in a table or computing the average)
- The wealthiest 1% own 50% of the goods
Inferential statistics
Consist of techniques that allow us to study samples and then make generalizations about
the populations from which they were selected
- To make an inference from something to something else
 make a5513n inference from a sample to the population
(= Select a few from your population and test these, then make an inference from
these few to the population)
Parameter: a value (usually numerical) that describes a population, it is usually derived from
measurements of the individuals in the population
Statistic: a value (usually numerical) that describes a sample, it is usually derived from
measurements of the individuals in the sample
Data differs
- Height (in cm)
- Annual income (EUR)
- Smoker vs. non-smoker  no value/ number
- Pet (dog, cat, hamster, bunny)  no value/ number
- How much you support Trump (from -5 to +5)  scale
Research methods
Correlational method: two different variables are observed to determine whether there is a
relationship between them
- Relationship between wake up time and academic performance in students
- Scores are often of categorical, not numerical, value
- Shows a correlation, but never a cause-and-effect relationship
Statistics used for comparing two (or more) groups of scores: examining descriptive statistics
that summarize and describe the scores in each group, and we use inferential statistics to
determine whether differences between the groups can be generalized to the entire
population
- Calculating averages and comparing them
- Computing proportions and comparing them
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Experimental method: one variable is manipulated (independent variable) while another
variable is observed and measured (dependent variable). To establish a cause-and-effect
relationship between the two variables, an experiment attempts to control all other
variables to prevent them from influencing the results
- Participant variables (= age, gender, intelligence) and environmental variables (=
lightning, time of day, weather) must not differ between groups
 random assignment, matching (each group has equal males and females), and
holding variables constant (only using boys)
- Using two groups, control group and experimental group
Control group gets no treatment, or neutral/placebo treatment
Experimental group gets the experimental treatment
Non-experimental methods
- Nonequivalent groups design: groups are already divided; the researcher has no
ability to control
- Pre-post design: the researcher has no control over other variables that change with
time, like the light and weather
- Quasi-independent variable: the independent variable that is used to create the
different groups of scores. It cannot be manipulated by the researcher
Dimensions of data
Constructs vs operationalizations
Construct: internal attributes or characteristics that cannot be directly observed but are
useful for describing and explaining behavior
- Interest in relationship between construct A and B
 needs operationalizations
- Interest in relationship between intelligence and uni performance
 operationalization intelligence: IQ test
 operationalization uni performance: Average grade
Discrete vs continuous variables
Discrete variables:
Variables that only consist of a limited number of categories
- Gender, eye color, native language, how many pets/ siblings
 there cannot be a value between two neighboring categories
 there cannot be a value between 1 and 2 pets
Continuous variables:
Variables that can take all values between two points
- Income, height, weight, speed
- Height can be 1,76 m, 176 cm, or 1,751356344 m
 it is very rare to obtain identical measurements for two different individuals, there
is an infinite number of possible values
 a continuous variable is actually an interval, when two people claim to weigh 150
pounds, they do not weigh exactly the same but are both around 150 pounds. Thus
150 is not a specific point on the scale but an interval. To differentiate 150 from 149
there need to be specific boundaries (real limits) and are placed exactly halfway
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between adjacent scores  X=150 is between 149,5 (lower real limit) and 150,5
(upper real limit)
The nominal scale
Named categories (dog, cat, hamster)
- No quantitative distinction between them  you cannot say ‘a dog is more than a
cat’/ room 109 is not 9 bigger than room 100
- No true zero
The ordinal scale
Consists of a set of categories that are organized in an ordered sequence
Ranked named categories (1st, 2nd, 3rd)
- No equal distance between ranks (1st place was much faster than 2nd and 3rd)
- No true zero
The interval scale
Consists of ordered categories of equally sized intervals between values
 Each unit has the same size
- Temperature (from 21C to 26C, or from 1C to 6C)
 both have the same difference
- No true zero (arbitrarily chosen, can have the value 0)
0 degrees does not mean that there is no temperature, it can even go lower
The ratio scale
Consists of equally sized intervals between values
 each unit has the same size
- There is an absolute zero!!!
- Distance: a distance of 0 means your bike has not moved
- 0 gallons means that your tank is empty
Statistical notations
X = raw scores  when having two scores Y is also indicated as raw scores
N = number of scores in a population
n = number of scores in a sample
 = summation  10,6,4,7 X=27 and N=4
- The  is always followed by a symbol or mathematical expression, which identifies
exactly which values are to be added
Order of mathematical operations
1. Any calculation within parentheses
2. Squaring (or raising other exponents)
3. Multiplying and/or dividing
4. Summation
5. Any other addition and/or subtraction
Chapter 2
Representing data
Frequency distribution
Organized tabulation of the number of individuals located in each category on the scale of
measurement
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-
Places the disorganized scores in order from highest to lowest, grouping together
individuals who all have the same score
How many pets do you have?
- Ask 10 people the number of pets that currently live in their main
household
- Construct: number of pets
Operationalization: number of pets that currently live in a
person’s main household
Add structure
- Count how often each option occurs
 how many people have 0, 1, 2… pets
 Frequencies of values
- by adding up the frequencies, you obtain the total number of
individuals  f=N
The structured table is then called a frequency distribution table
Grouped frequency distributions
Bundle some value ranges together = class intervals
- 10 intervals is the general guide
- The width should be a relatively simple number (2,5,10,20)
- Low (0-25000)
Middle (25001-50000)
Upper middle (50001-75000)
High (75000+)
- Frequency distributions for continuous variables
 because we cannot put them together in normal frequency
distributions
Higher sample number
Pet example: instead of n=10  n=10000
- Frequency goes up
- The large dataset differs much from the small data set
 proportion: p = f/n
- All proportions add up to 1!!!
 percentage: p = f/n * 100
- All percentages add up to 100%
Frequency distribution graphs
Graphs for interval or ratio data
Histograms
- Numerical scores along the x-axis and bars above each X
value in which the height corresponds to the frequency for
that category on the y-axis.
- Ability to compare n=10 to n=10000 when put into
proportions
- No spaces between the bars
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Polygons
- Numerical scores along the x-axis, then a dot is centered above each score so that
the vertical position of the dot corresponds to the frequency for the category, then a
continuous line is drawn from dot to dot
- With class intervals the dot is drawn directly above the midpoint of the class interval
In a class interval of 20-29, the dot is drawn above 24,5
Graphs for nominal or ordinal data
Bar graph
- the same as a histogram, except that spaces are left between adjacent bars
 separate distinct categories and the categories are not all the same size (the bars
are)
Graphs for population distributions
- it is impossible to obtain an exact count of the number of people in a large
population. A frequency distribution can be constructed when they have relative
frequencies and smooth curves
relative frequencies
you don’t know the exact score but based on past census data and general trends you know
certain numbers are close (like how many males and females there are) so you draw the
outnumbering category slightly above the latter category
smooth curves
you are not connecting a series of dots, but you are showing the relative changes that occur
from one score to the next
- normal curve
the shape of a frequency distribution
frequency distribution graphs all have the same three characteristics
1. shape
2. central tendency (= where the center of the distribution is located)
3. variability (= the degree to which the scores are spread over a wide range or are
clustered together)
symmetrical distribution: possibility to draw a vertical line through the middle so that one
side of the distribution is a mirror image of the other
skewed distribution: the scores tend to pile up toward one end of the scale and taper off
gradually at the other end
tail of the distribution: the section where the scores taper off toward one end of a
distribution
positively/negatively skewed: positively = tail on the right
side because the tail points toward the positive end of
the x-axis, negatively = tail on the left side
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Locating data points
We may want to find where a value lies relative to the whole data
- Are 3 pets a lot or not?
- (percentile) rank: rank of a score that is defines as the percentage of individuals in
the distribution with scores at or below a particular value
 locate points based on the frequency distribution
Percentiles
Score that is identified
1. Sort the frequency table (X, f, prop, perc)
2. We calculate a cumulative percentage
c%= (cf/N) x100
Option 0 and 1 cover 60,48% of the data
3. We locate our data point of interest (having 3 pets)
3 pets correspond to a cumulative percentage of
95,17%
‘3 pets’ has a percentile rank of 95,17%
‘3 pet’s are the 95th percentile
Interpolation
A method of finding intermediate values
1. A single interval is measured on two separate scales, the endpoints of the interval are
known for each scale
2. You are given an intermediate value on one of the scales, the problem is to find the
corresponding intermediate value on the other scale
3. The interpolation process requires 4 steps
a. Find the width of the interval on both scales
b. Locate the position of the intermediate value in the interval, this position
corresponds to a fraction of the whole interval
Fraction = distance from the top of the interval/ interval width
c. Use the same fraction to determine the corresponding position on the other
scale, first use the fraction to determine the distance from the top of the interval
Distance = (fraction) x (width)
d. Use the distance from the top to determine the position on the other scale
Question 1: find the percentile rank for X = 9
Table:
X
C%
10-14
75%
5-9
60%
0-4
10%
Step 1: find the width of the interval on both scales, this is the top of interval 5-9  9,5 and
the top of interval 0-4  4.5
9 falls between these intervals
top
9.5
60%
9
?
bottom
4.5
10%
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Step 2: Locate the position of the intermediate value in the interval, this position
corresponds to a fraction of the whole interval
Fraction = distance from the top of the interval/ interval width
This is the top (9.5) – the value (9) = 0.5
To get the fraction you do 0.5 (the distance from top of the interval) / 5 (total interval width
 9.5-4.5 = 5) = 0.1
Step 3: Use the same fraction to determine the corresponding position on the other scale,
first use the fraction to determine the distance from the top of the interval
Distance = (fraction) x (width)
The distance between both intervals is 50 (60% - 10%)
To calculate the distance from 9 to 9.5 but in percentages  0.1 x 50 = 5
Step 4: Use the distance from the top to determine the position on the other scale
60% - 5 = 55 %
 so the percentage of X=9 is 55%
Stem and leaf display
A way to represent data in a different way than the
frequency table
1. Take all first numbers that are unique (1,2,3,4,5…)
2. Take the second numbers and place them behind the
line
3. You can flip it so you can see it as bars (you see that
most answers are in the 30’s)
Chapter 3
Central tendency
A statistical measure to determine a single score that defines the center of a distribution.
The goal of central tendency is to find the single score that is most typical or most
representative of the entire group (average or typical individual)
- Characterizes what is typical for a large population and in doing so makes large
amounts of data more digestible
It is difficult to have one way of defining a center in different distributions because they all
look different
 solution: the mean, median and the mode
The mean
The arithmetic average  computed by adding all the scores in the distribution and dividing
them by the number of scores
- The mean of a population is defined by the μ  μ = ΣX/N
The mean of a sample is defined by M  M = ΣX / n
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Alternative definitions
- The mean as the amount each individual receives when the total is divided equally
among all the individuals
- The mean as a balance point for the distribution. When you have 7 points below the
mean and 7 points above the mean, the mean is the balance point.
The weighted mean
When combining two sets of scores you need to
find the overall mean for the combined group,
you need:
1. The overall sum of the scores for the
combined group
2. The total number of scores in the
combined group (n)
 M = ΣX1 + ΣX2 / n1 + n2
Characteristics of the mean
Every score in the distribution contributes to the value of the mean
- Changing the value of any score will change the mean
- Adding or removing a score will change the mean, unless the score is located exactly
at the mean
-
If a constant value is added/removed to every score in a distribution, the same
constant will be added/removed from the mean
If every score in a distribution is multiplied/divided by a constant value, the mean will
change in the same way
The median
The goal is to locate the midpoint of the distribution
If the scores in a distribution are listed in order from smallest to largest, the median is the
midpoint of the list. It is the point on the measurement scale below which 50% of the scores
in the distribution are located
- No special symbols or notation, just median
To calculate the median, you need to put all scores from smallest to largest, you start with
the smallest and count the scores as you move up the list. The median is the first point you
reach that is greater than 50% of the scores in the distribution
- With this set of scores: 3, 5, 8, 10, 11  the median is 8
- With this set of scores: 3, 5, 8, 10  the median is 5 + 8 / 2 = 6.5
Finding the precise median for a continuous variable
A continuous variable consists of categories that can be split into an infinite number of
fractional parts. It is not the same as the discrete variables shown above. When
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encountering a set of scores like 1,2,3,4,4,4,4,6 the median of continuous variables cannot
be 4.  X=4 corresponds to an interval from 3.5-4.5
to find the precise median, we first observe that the distribution contains 8 scores,
represented by 8 boxed in the graph. to make it equal that both sides of the median contain
4 boxes you need to split the 4  ¼ belongs with 1,2,3 and ¾ belongs with 6
- Fraction = number needed to reach 50% / number in the interval
Fraction = 1/4 = 0.25
 median is 3,75
 You can also use interpolation by trying to find a score of 50%
The mode
In a frequency distribution, the mode is the score or category that has the greatest
frequency, it is used to determine the typical or most frequent value of any scale of
measurement
- No special symbols or notation, just median
- Can be used when using category’s like restaurants
In a questionnaire 42 out of 100 students chose Luigi’s to be their favorite restaurant,
this was the most frequently chosen restaurant  the mode is Luigi’s
- In a distribution there can only be one mean, and one median, but there can be
multiple modes  when multiple scores have the same frequency
You call this a bimodal or multimodal distribution
- The mode is technically the score with the highest frequency, but when two scores
are close to each other and are both peaks they are both modes  bimodal
distribution with a major and minor mode
Selecting a measure of central tendency
The goal is to find the single value that best represents the entire distribution  using the
mean is most common
When to use the median
- Extreme scores or skewed distributions
Unable to use the mean because one or two extreme values can have a large
influence and cause the mean to be displaced  these outliers can be spotted by
taking context into account
- Undetermined values (when participants never finish the experiment)
Unable to use the mean because one value is unknown
- Open-ended distributions = when there is no limit (number of pizza’s eaten with
largest category ‘5 or more’)
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Unable to use the mean because you do not have an exact total score
- Ordinal scale, it allows you to determine direction but do not allow you to determine
distance  median is the best way, this shows half of the scores below and half of
the scores above the median
When to use the mode
- Nominal scales: the categories that make up a nominal scale are differentiated only
by name, they do not measure quantity
Unable to use the mean or median
- Discrete variables: exist only in whole, indivisible categories
Unable to use the mean because it would calculate categories that do not exist (1,6
children and 5,3 rooms instead of when using the mode 2 children and 5 rooms)
- Describing shape: it does not need calculation so it can be used as an indication of
the mean or median
Using graphs to present means and medians
A graph allows several means (or medians) to be shown simultaneously so it is possible to
make quick comparisons between groups or treatment conditions
- X-axis: groups and treatment conditions
- Y-axis: values for the dependent variable
- The means (or medians) are displayed using a line graph, histogram or bar graph
Mean, median and modes in distributions
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Chapter 4
Variability: provides a quantitative measure of the differences between scores in a
distribution and describes the degree to which the scores are spread out or clustered
together
The purpose is to obtain and objective measure of how the scores are spread out in a
distribution  describe the data
A good measure of variability has:
1. It describes a distribution; it tells whether the scores are clustered together or are
spread out over a large distance  defined in terms of distance
2. It measured how well an individual score represents an entire distribution it
provides information about how much error to expect if you are using a sample to
represent a population
 to measure variability you have the range, standard deviation and variance
The range
The distance covered by the scores in a distribution
- Calculated by measuring the difference between the largest score and the smallest
score  Range = Xmax – Xmin
- Values 1-5 have a range of 4
Values ¼, ½, ¾ have a range of 1  between 0 and 1
- When the measurements are continuous values the range is defined by the upper
real limit for the largest score, and the lower real limit for the smallest score
Values 1-5 have a range of 5,5 – 0,5 = 5
- when the scores are all whole numbers the range is  Range = Xmax – Xmin + 1
the range is rarely used because it does not consider all scores in the distribution, outliers
make a whole other range than no outliers  unreliable
Standard deviation and variance
Most commonly used and most important variability measure. It used the mean of the
distribution as a reference point and measures variability by considering the distance
between each score and the mean
Step 0: calculate the mean
Step 1: determine the deviation/distance from the mean, for each individual score
- deviation score = X – μ
with a μ = 50 and a score X = 53  deviation is 3
- when adding up all deviation scores in a distribution you should arrive at 0, because
the total of distances above the mean are equal to the total of
distances below the mean
- the mean of the deviations is also always 0
Step 2: because the mean of the deviation is always 0, you need to get
rid of all the – /+ signs  square each deviation score, add them up
and calculate the mean
- compute the mean squared deviation  variance
- variance = Σ (X – μ)2 / N
Step 3: to get rid of the squared deviation you need to take the root
- standard deviation: the square root of the variance and provides a measure of the
standard, or average distance from the mean  expressed by σ
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-
standard deviation = √ variance
Standard deviation and variance in a population
The concepts of standard deviation are the same for both samples and populations, however
the calculations slightly differ
Variance = mean squared deviation = sum of squared deviations (= SS)/ number of scores
To compute the SS (= sum of squares) you need two formulas
1. The definition formula: SS = Σ (X – μ)2
first, find each deviation score, then square each
deviations core and then add the squared
deviations
2. The computational formula: SS = ΣX2 – (ΣX)2 / N
First, square each score and then add the values,
then find the sum of the scores and square this
total, divide this by N, and finally subtract the
second part from the first
- Used when the mean is not a whole number, the deviations all contain decimals or
fractions, and the calculations become difficult
Population variance:
Represented by the symbol σ2 and equals the mean squared distance from the mean,
population variance is obtained by dividing the sum of squared by N
- Variance (σ2) = SS/N
Population standard deviation
Represented by the symbol σ and equals the square root of the population variance
- Standard deviation (σ) = √ SS/√ N
Standard deviation and variance in a sample
A few extreme scores in a population can make the variability quite large, however the
chance that these scores are included in a sample are very small, which makes the sample
variability relatively small  biased estimate of population variability
Calculations to fix this problem
- Calculating the sum of squared deviations is the same, except the notations
Definition formula: SS = Σ (X – μ)2  SS = Σ (X – M)2
Computational formula: SS = ΣX2 – (ΣX)2 / N  SS = ΣX2 – (ΣX)2 / n
- Calculating the sample variance and standard deviation is different
Sample variance
Represented by the symbol s2 and equals the mean squared distance from the mean, sample
variance is obtained by dividing the sum of squared by n – 1
- Variance (s2) = SS / n – 1
Sample standard deviation
represented by the symbol s and equal the square root of the sample variance
- Standard deviation (s) = √ SS/ √ n-1
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Degrees of freedom
For a sample of n scores the degrees of freedom (df) for the sample variance are defined as
df = n – 1. The degrees of freedom determine the number of scores in the sample that are
independent and free to vary.
Sample variance as an unbiased statistic
Unbiased: a sample statistic is unbiased if the average value of the statistic is equal to the
population parameter (the average value of the statistic is obtained from all the possible
samples for a specific sample size, n)
- divided by n-1
Biased: a sample statistic is biased if the average value of the statistic either underestimates
or overestimates the corresponding population parameter  correct this by using the sum
of squares
- divided by n
Transformations of scale
Occasionally a set of scores is transformed by adding a constant to each score or by
multiplying each score by a constant value. The easiest way to determine the effect of a
transformation is to remember that the standard deviation is a measure of distance. If you
select any two scores and see what happens to the distance between them, you will also find
out what happens to the standard deviation
1. Adding a constant to each score does not change the standard deviation, the
distance between two scores will stay the same. It does change the mean, which will
also increase or decrease by the same number
2. Multiplying each score by a constant causes the standard deviation to be multiplied
by the same constant, the distance between the scores will also multiply by the same
amount, as does the mean
Chapter 5
The individual scores within a distribution
z-score / standard score: identify and describe the exact location of each score in a
distribution
- A score by itself does not necessarily provide much information about its position
within a distribution
- Raw scores: original, unchanged scores that are the direct result of measurement
The process of transforming X values into z-scores serves the purpose of:
1. Each z-score tells the exact location of the original X value within the distribution
2. The z-scores form a standardized distribution that can be directly compared to other
distributions that also have been transformed into z-scores
The z-score
Location within a distribution
The z score accomplishes to describe the exact location by transforming each X value into a
signed number (+ or -)  the sign tells whether the score is located above or below the
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mean, and the number tells the distance between the score and the mean in terms of the
number of standard deviations
- In a distribution with μ=100, σ=15, in a score of X=130  z= +2,00
- Formula: z = (X – μ) / σ
 X – μ is a deviation score, it measures the distance in points between X and μ and
indicates whether X is located above or below the mean
Determining raw scores from a z-score
In general, it is easier to use the definition of a z-score, rather than a formula, when you are
changing z-scores into X values. But you can use a formula:
- X = μ + (z x σ)
- μ = 60, σ = 8, which value of X corresponds to z = -1,50
determine the distance corresponding 1,5 standard deviations  1,5 x 8 = 12
then find the value of X located below the mean by 12 points  60 – 12 = 48
Standardized distribution
a composition of scores that have been transformed to create predetermined values for μ
and σ, standardized distributions are used to make dissimilar distributions comparable
You can transform every value of X to its corresponding z-score, this new distribution of
z-scores has characteristics that make the z-score transformation a very useful tool:
1. Shape: the distribution of z-scores will have exactly the same shape as the original
distribution of scores
2. The mean: the z-score distribution will always have a mean of zero, which makes the
mean a convenient reference point
3. The standard deviation: the distribution of z-scores will always have a standard
deviation of 1, the numerical value of a z-score is exactly the same as the number of
standard deviations from the mean
Many people find z-score distributions not nice because they contain negative values and
decimals, so it is common to standardize a distribution by transforming the scores into a new
distribution with a predetermined mean and standard deviation that are whole round
numbers  create a new distribution that has simple values for the mean and standard
deviation, but does not change any location within the distribution = standardized scores
This is a two-step process:
1. The original raw scores are transformed into z-scores
2. The z-scores are transformed into new X values so that the specific μ and σ are
attained
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Z scores are most commonly used in a population, but the same principles can be used to
identify individual locations within a sample. You can use the exact same method, as long as
you use the sample mean and the sample standard deviation
Researchers use z-scores when comparing results of a sample to the original population, to
determine if the individuals in the sample are noticeably different you need the z-score to
decide whether a sample is noticeably different
Chapter 6
Probability in inferential statistics:
- You start with developing probability as a bridge from populations to samples, this
involves identifying the types of samples that probably would be obtained from a
specific population. Then you select from which population you should sample to it is
as representative as possible
Probability: a fraction of a proportion of all the possible outcomes  probability of A =
number of outcomes classified as A / total number of possible outcomes
- The chance of getting a king in a card deck  4/52 (4 out of 52)
- Symbol: p
- Identifies probability as a fraction or a proportion
- Probability values range from 0—1
- When calculating multiple probabilities, you need to multiply them or put one to the
power of how many predictions.
P(correct) x P(correct) x etc.  P(correct)10
Joint probability
Two events occurring together is less probable than each event happening individually:
P(A) = 0.6
P(B) = 0.7
P(C)  A + B  0.6 x 0.7 = 0.42
Conditional probability
The probability of how likely something is, given that something else is true
P (X I A)  probability of something, given A
(simple) random sample: each individual in the population has an equal chance of being
selected
Independent random sample: each individual has an equal chance of being selected and the
probability of being selected stays constant from one selection to the next if more than one
individual is selected
 used the most, is a required component in statistical applications
Requirements:
- There is no bias in the selection process, you select randomly in your population
- Sample with replacement: to keep the probabilities from changing from one selection
to the next, it is necessary to return each individual to the population before you
make the next selection
Normal distribution
Bell shaped, but not all bell-shaped distribution is a normal distribution
Function: gives you the probability density  formula:
Standard normal: mean of 0 and standard deviation of 1
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Using probability: what is the probability of randomly selecting an individual from a
population with a score greater than 700?
1. All scores are the population distribution, from which the mean is 500, so the score
700 is right to the mean. We are interested in scores greater than 700 to we need to
be at the right of the 700 mark.
2. To identify the exact position, you need to compute the z-score  z = (X – μ) / σ
 (700 – 500) / 100 = 2.00
The score 700 is exactly 2 standard deviations
above the mean
3. What is the probability?  use the standard
percentages of a normal distribution
P(z > + 2.00) = 2.28%
The unit normal table
A complete listing of z-scores and proportions (instead of a graph, which shows just a few
z-score values)
1. The body always corresponds to the larger part
of the distribution, no matter on which side it is.
The tail is always the smaller section
2. The normal distribution is symmetrical, so the
proportions on the right-hand side are exactly
the same as the proportions on the left-hand
side. The table does not list negative z-score
values
3. The z-score values change signs, but the
proportions are always positive
In calculations you should always sketch a distribution,
locate the mean with a vertical line and shade in the portion you are trying to determine,
this will help you avoid making errors
Most of the time you need to find probabilities (/ proportions, they mean the same) for
specific X values, so the first step you need to do is transform the X values into z-scores, and
then you can use the unit normal table to look up the proportions corresponding to the zscore values
 examples of calculations on pages 173 – 177
The binomial distribution
Binomial data: categorical variables, only a few options
A variable that is measured on a scale consisting of exactly two categories, the resulting data
are called binomial. This occurs when a variable naturally exists with only two categories
(male and female; head or tails)
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The researcher often knows the probabilities associated with each of the two categories  a
coin, p(heads) = p(tails) = ½
Notation:
1. The two categories are identified as A and B
2. The probabilities associated with each category are identifies as
p(A) = p = the probability of A
p(B) = q = the probability of B
 p + q = 1.00
3. The number of individuals or observations in a sample is identified as n
4. The variable X refers to the number of times category A occurs in the sample
The binomial distribution tends to approximate a normal distribution, particularly when n is
large. Then it has the following parameters:
- Mean: μ = pn
- Standard deviation: σ = √ npq
- Z-score: z = (X – μ) / σ  z = (X – pn) / √ npq
Binomial values are discrete (coin tosses, 2 or 3, not 2,5), normal values are continuous
Chapter 7
A z-score value near zero indicates a central, representative sample; a z-value beyond +2.00
or -2.00 indicates an extreme sample
The difficulty with working with samples is that a sample provides an incomplete picture of
the population. A sample almost certainly misses some segments of the population that are
not included.
Sampling error: the natural discrepancy, or amount of error, between a sample statistic and
its corresponding population parameter
The distribution of sample means
The collection of sample means for all the possible random samples of a particular size (n)
that can be obtained from a population
 two separate samples will be different even though they are taken from the same
population
- Sampling distribution: a distribution obtained by selecting all the possible samples of
a specific size from a population
- you can obtain the distribution of sample means by selecting a random sample of a
specific size (n) from a population, calculate the sample mean, and place the sample
mean in the distribution. Do this with every random sample
 you can change the specific size (n) of a sample, and how many times you take a
sample of that size. When you increase both the distribution will start to look more
and more like a normal distribution and the sampling error will reduce
characteristics:
- Samples are not expected to be perfect, but they are representative of the
population, so they should be relatively close to the population mean
- The pile of sample means should tend to form a normal-shaped distribution
- The larger the sample size, the closer the sample means should be to the population
mean
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Central limit theorem
Way of determining exactly what the distribution of sample means looks like without taking
hundreds of thousands of samples
For any population with the mean μ and the standard deviation σ, the distribution of sample
means for a sample size n will have a mean of μ and a standard deviation of σ/ √ n and will
approach a normal distribution as n approaches infinity
- It describes the distribution of sample means for any population
- The distribution of sample means approaches a normal distribution very rapidly,
when the number of scores in each sample is relatively large and when the
population is a normal distribution
- The rule of thumb is that n needs to be at least 30 and then it will rapidly approach
the normal distribution
The mean
The mean of the distribution of sample means is always identical to the population mean, it
is called the expected value of M. it is identified by the symbol μM or μ
The standard error
The standard deviation for the distribution of sample means, it is identified by the symbol σM
and is called the standard error of M
1. It describes the distribution of sample means, provides a measure of how much
difference is expected from one sample to another
2. It measures how well an individual sample mean represents the entire distribution. It
provides a measure of how much distance is reasonable to expect between a sample
mean and the overall mean for the distribution of sample means
The magnitude of the standard error is determined by:
1. The size of the sample
Law of large numbers: the larger the sample size, the more probable it is that the
sample mean will be close to the population mean
2. The standard deviation of the population from which the sample is selected
Bigger samples have smaller error, and smaller samples have bigger error
- When n=1, σM = σ (standard error = standard deviation)
- Standard error (SE) = σM = σ/ √ n
- The population variance and the population standard deviation are directly related,
and it is easy to substitute variance into the equation of standard error
 standard error (SE) = σM = σ/ √ n = √ σ2/n
Probability
The primary use of the distribution of sample means is to find the probability associated with
any specific sample. Because the distribution of sample means represents the entire set of
possible samples, you can use proportions to determine probabilities
1. Determine the standard error of the population by σM = σ/ √ n
2. Determine the z-score and the mean and use it to locate the exact point given
3. Then you can look up the probability in the normal unit table
We are finding a location within the distribution of sample means: by finding the location for
sample mean M rather than a score X, and the standard deviation for the distribution of
sample means is the standard error
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 the z formula for the z-score of sample means is z = M- μ / σM
Standard error vs sampling error
When you are working with a sample mean, you must use the standard error
Sampling error: a sample typically will not provide a perfectly accurate representation of its
population, typically there is some error between a statistic in a sample and the population
Standard error: most sample means are close to the population mean these samples provide
a fairly accurate representation of the population. Some means are far from the population
mean and are not representative. For each sample you can measure the error (or distance)
between the sample mean and the population mean
- When sample size increases, the standard error gets smaller and the sample means
tend to get closer to the population mean
- With increasing n, we decrease the standard error SE, and thereby reduce the
sampling error
The standard error measures how much discrepancy you should expect, between a sample
statistic and a population parameter. Statistical inference involves using sample statistics to
make a general conclusion about a population parameter, thus standard error plays a crucial
role in inferential statistics
Chapter 8
Hypothesis testing
a statistical procedure that allows researchers to use sample data to draw inferences about
the population of interest, by using sample data to evaluate a hypothesis about a population
- It is one of the most commonly used procedures in inferential statistics
- The details of a hypothesis test change from one situation to another, but the general
process remains constant
The process:
1. State a hypothesis about a population
American adults gain an average of μ = 7 pounds a year
2. Predict the characteristics the sample should have
The sample should have a mean around 7 pounds
3. Obtain a random sample from the population
4. Compare the obtained sample data with the prediction that was made from the
hypothesis
The unknown population
The researcher begins with a known population (the group before the treatment), which a
certain standard deviation and a mean. The purpose of the study is to determine what
happens to the population after the treatment. With any effect you can add or subtract a
certain number from each individual score ( this changes the mean, but not the shape of
the distribution or the standard deviation)
- To do this research you use a sample as your population because the research wants
to generalize it, but it is impossible to test a whole population
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The four steps of a hypothesis test
Step 1: state a hypothesis about a population
The null hypothesis
Known as H0, states that in the general population there is no change, no difference, or no
relationship. In the context of an experiment, H0 predicts that the independent variable
(treatment) has no effect on the dependent variable (scores) for the population
The alternative hypothesis
Known as H1 (or HA), states that there is a change, a difference, or a relationship for the
general population. In the context of an experiment, H1 predicts that the independent
variable (treatment) does have an effect on the dependent variable
 data will evaluate the credibility of the hypotheses
According to the null hypothesis, the distribution of sample means is divided into two
sections:
1. Sample means that are likely to be obtained if H0 is true: that is, sample means that
are close to the null hypothesis = high probability
2. Sample means that are very unlikely to be obtained if H0 is true: that is, sample
means that are very different from the null hypothesis = low probability
 null hypothesis significance testing: the null hypothesis is likely or unlikely
Step 2: predict the characteristics the sample should have/ set the criteria for a decision
Alpha level
To find the boundaries that separate the high-probability
samples from the low-probability samples, we must define
what exactly is meant by high and low probability  level
of significance / alpha level
The alpha (α) value is a small probability that is used to
identify the low-probability samples
- Alpha = .05 (5%), alpha = .10 (10%)  separate the
most unlikely 5% of the sample means, from the
most likely 95%
- The extremely unlikely values make up the critical
region
- The extremely unlikely values can also be called the
convincing evidence that the treatment has an effect
To find the critical region you use the unit normal table. The boundaries separate the
extreme 5% from the middle 95%. The 5% is split up in two tails, so there is exactly 2.5% in
each tail  in the unit normal table you look up .0250 in column C and you find the z-score
boundary
Step 3: collect data and compute sample statistics
The collection of data happens after the hypothesis is set, so the researcher makes an
honest, objective evaluation of the data
1. The raw data is summarized, for example with the sample mean.
2. Then the data is compared with the hypothesis. This is done by computing a z-score
exactly where the sample mean is located relative to the hypothesized population
mean from H0  z = M – μ / σM
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The value of the sample mean M is obtained from the sample data, the value of μ is
obtained from the null hypothesis
Step 4: make a decision by comparing the obtained sample data with the prediction that
was made from the hypothesis
The researcher uses the z-score value obtained in step 3 to make a decision about the null
hypothesis according to the criteria established in step 2, there are two possible outcomes:
1. The sample data are located in the critical region, the sample is not consistent with
H0 and the null hypothesis is rejected
The z-score is 2.25, this falls within the boundary of 1,96 (alpha level .05)
2. The sample data re not in the critical region, the sample mean is reasonably close to
the population mean specified in the null hypothesis, so we fail to reject the null
hypothesis, the treatment has no effect
The z-score is 0.75, this does not fall within the critical region (1.96)
The z-score
The z-score statistic used in the hypothesis test is a test statistic: the sample data are
converted into a single, specific statistic that is used to test the hypothesis
- The z-score as a recipe: you follow the instructions and use the right ingredients and
you produce a z-score
- The z-score as a ratio: it indicates a large discrepancy between sample data and the
null hypothesis
Errors in hypothesis testing
Type 1 errors
Occurs when a researcher rejects a null hypothesis that is actually true. In a typical research
situation, a type 1 error means the researcher concludes that a treatment does have an
effect when in fact it has no effect
- Occurs when a researcher unknowingly obtains an extreme, nonrepresentative
sample
- The alpha level determines the probability of a type 1 error, it determines the
probability of obtaining sample data in the critical region even though the null
hypothesis is true
Type 2 errors
Occurs when a researcher fails to reject a null hypothesis that is really false. In a typical
research situation, a type 2 error means that the hypothesis test failed to detect a real
treatment effect
- Occurs when the sample mean is not in the critical region, even though the
treatment has an effect on the sample, this often happens when the effect of the
treatment is relatively small
- The probability depends on a variety of factors and therefore is a function, it is
represented by the Greek letter β
Selecting an alpha level
To avoid type 1 errors, setting an alpha level is very important. The largest value is .05, but to
be more careful many researchers use an alpha level of .01 or .001 to reduce the risk of a
false report
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Though, a lower alpha level means less risk of a type 1 error, but it also means that
the hypothesis test demands more evidence from the research results
Statistical significance
A result is said to be (statistically) significant if it is very unlikely to occur when the null
hypothesis is true. That is, the result is sufficient to reject the null hypothesis, thus a
treatment has a significant effect
- The p (probability) of obtaining a sample mean in the critical region is extremely
small (less than .05) if there is no treatment effect
 the p-value must be smaller than .05 to be considered statistically significant
Factors that influence a hypothesis test
The variability of scores
High variability can make it very difficult to see any clear patterns in the results from a
research study. In a hypothesis test, higher variability can reduce the chances of finding a
significant treatment effect
- Increasing the variability of the scores produces a larger standard error and a smaller
value for the z-score
Number of scores in the sample
Increasing the number of scores in the sample produces a smaller standard error and a
larger value for the z-score. If all factors held constant, the larger the sample size is, the
greater the likelihood of finding a significant treatment effect
Other factors that can influence the hypothesis test are considered to be constant, these
are:
- Random sampling
- Independent observation
- The value of σ is unchanged by the treatment
- Normal sampling distribution (in order to use the unit normal table)
One tailed vs two tailed hypothesis tests
What is discussed before is in a two-tailed hypothesis test  the critical region is divided
between the two tails of the distribution
One-tailed test
Also called a directional hypothesis test, where the statistical
hypotheses (H0 and H1) specify an increase of a decrease in the
population mean. That is, they make a statement about the
direction of the effect
Alcohol is expected to slow reaction times
Step 1: state the statistical hypothesis
H0: alcohol does not slow reaction times  μ </= the
population mean
H1: alcohol does slow reaction times  μ > the population mean
Step 2: define the critical region
If the prediction is that the treatment will produce an increase in scores, then the critical
region is located entirely in the right-hand tail of the distribution
- You are allowed to use an alpha level of .05, for just one tail of the distribution
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Chapter 9 (9.1, 9.2, 9.4)
The problem with using a z-score is that the z-score formula requires more information than
is usually available, especially the value of the population standard deviation
- Circularity problem: we make an inference about a population based on a population
parameter, if you do not know the population parameter you cannot calculate the zscore
solution: the t-statistic = estimating the standard deviation
You can estimate the standard error: estimated standard error
- the estimated standard error (sM) is used as an estimate of the real standard error σM
when the value of σ is unknown. It is computed from the sample variance or sample
standard deviation and provides an estimate of the standard distance between a
sample mean M and the population mean μ
- variance = s2 = SS/ (n – 1) = SS / df
- df = degrees of freedom (sample of 400 has 399 as the degrees of freedom)
 the estimated value is computed from sample data (s) rather than from the actual
population parameter (σ)
- SM = s/√ n, or
SM = √ s2/n
Estimated standard error = √ sample variance / sample size
 we use the variance instead of the standard deviation because it is unbiased, and the tstatistic needs the variance instead of the standard deviation
The t-statistic
Used to test hypotheses about an unknown population mean (μ) when the value of σ is
unknown. The formula for the t statistic has the same structure as the z-score formula,
except that the t statistic used the estimated standard error in the denominator
- t = (M – μ) / sM

t = (M – μ) /√ s2/n
- to determine how well a t statistic approximates a z-score, you must determine how
well the sample variance approximates the population variance  degrees of
freedom = df = n – 1
The t-distribution
a complete set of t values computed for every possible random sample for a specific sample
size (n) or a specific degree of freedom (df). The t distribution approximates the shape of a
normal distribution
- if df gets large, the t distribution gets closer in shape to a normal z-score distribution
- it has an estimated standard error so it is flatter  you can’t put all the weight in the
middle because it’s an estimation
- you do not have one t-distribution but the t-family because the distribution depends
on the sample size used
The t-distribution table
Given the degrees of freedom and the tail proportions (alpha level), locate the critical tstatistic
- The two rows at the top of the table show proportions of the t distribution contained
in either one or two tails, depending on which row is used
- The first column lists the degrees of freedom for the t statistic
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The numbers in the body are the t values that mark the boundary between the tails
and the rest of the t distribution
 when your df value is not listed in the table (53), you loop up the critical t value for the
numbers surrounding it (40 and 60) and you use the smaller t-value!
Hypothesis tests with the t-statistic
Using the same procedure as with the z-score, the hypothesis test would result in a formula:
t = (sample mean, from the data) – (population mean, hypothesized from H 0) / estimated
standard error, computed from the sample data
- It works the same as with the z-score, if we obtain a large score then we can say that
the data is not consistent with the H0 hypothesis.
- You calculate the critical region by taking the degree of freedom: n-1
To calculate the t statistic, you need 3 steps:
1. Calculate the sample variance (the population variance is unknown, and you must
use the sample value)
 s2 = SS / (n – 1)
2. Use the sample variance and the sample size to compute the estimated standard
error
 sM = √ s2 / n
3. Compute the t statistic
 t = (M – μ) / sM
Assumptions of the t Test
- The values in the sample must consist of independent observations (no relationship)
- The population sampled must be normal
The t-statistic one tailed hypothesis test works the same as the z-score one tailed hypothesis
test
Chapter 10
General categories of two sets of data
1. independent measures research design / between subjects design: a research design
that uses a separate group of participants for each treatment condition (comparing
men and women)
2. Repeated measures research design / within subjects design: One group of people
before and after treatment
When you have two different groups in a study you also need to notate the data differently.
The number of scores in sample 1 would be n1 and the number of scores in sample two
would be n2
Independent-measures hypotheses
The goal is to evaluate the mean difference between two populations. The mean for the first
population is μ1 and for the second population is μ2  mean difference is μ1 – μ2
The null hypothesis means that there is no difference  H0: μ1 – μ2 = 0
The alternative hypothesis means that there is a difference, so H 1: μ1 – μ2 = not 0
Independent measures t-statistic
the basic structure of the t-statistic is the same for both the independent-measures and the
single-sample hypothesis test
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- t = (M – μ) / sM
the independent measures t is basically a two sample t that doubles all the elements of the
single sample t formulas
- t = ((M1 – M2) – (μ1 – μ2) / S(M1 – M2)  t = sample mean difference / estimated standard
error
The estimated standard error
Can be interpreted two ways:
1. it measures the standard distance between (M1 – M2) and (μ1 – μ2)
2. it measures the standard, or average size of (M1 – M2) if the null hypothesis is true.
That is, it measures how much difference is reasonable to expect between the two
sample means
calculation:
1. each of the two sample means represents its own population mean, but in each case
there is some error
M1 approximates μ1 with some error  same with M2
2. the amount of error associated with each sample mean is measured by the estimated
standard error of M, the estimated standard error for each sample mean is computes
as:
For M1, SM = √ s12 / n1  same with M2
3. for the independent-measures t statistic we want to know the total amount of error
involved in using two sample means to approximate two population means. To do
this, we will find the error from each sample separately and then add the two errors
together 
This formula only works when two sample sizes are exactly the same size. When two sample
variances are not equally good, they should not be treated equally  the variance obtained
from a large sample is a more accurate estimate of σ2 than the variance obtained from a
small sample
Pooled variance: method for correcting the bias in the standard error by combining the two
sample variances into a single value. This is obtained by averaging the two sample variances
using a procedure that allows the bigger sample to carry more weight in determining the
final value
- With one sample the sample variance is s2 = SS/df
- The pooled variance is:
 when in doubt always used the pooled variance
With this information you can now calculate an unbiased standard error for a sample mean
difference, this formula will be:
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Hypothesis test with the independent-measures t-statistic
For a two-tailed test
1. State the hypotheses and select the alpha level
H0: μ1 – μ2 = 0
H1: μ1 – μ2 ≠ 0
α = .05
2. Determine the critical region by calculating the degrees of freedom . It is an
independent measures design, so the calculation is:
df = df1 + df2
 look up boundaries in t-distribution table
3. Obtain the data and compute the test statistic
First, find the pooled variance
Second, use the pooled variance to compute the estimated standard error
Third, compute the t-statistic
4. Make a decision
Assumptions underlying the independent-measures t-formula
There are three assumptions that should be satisfied before you use the independentmeasures t-formula for hypothesis testing
1. The observations within each sample must be independent
2. The two populations from which the samples are selected must be normal
3. The two populations from which the samples are selected must have equal variances
- If the two sample variances are estimating different population variances, then the
average is meaningless
- If two population variances are equal, then the two sample variances should be very
similar, when the two sample variances are close, then the homogeneity assumption
is satisfied. When one variance is three or four times larger than the other, then
there is concern
Chapter 11
Repeated measures / within-subjects design
The dependent variable is measured two or more times for each individual in a single
sample, the same group is used in all of the treatment conditions
- Measured before and after treatment
 no risk for participant in one group are very different than the other group
Matched-subjects design
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Each individual in one sample is matched with an individual in the other sample. The
matching is done so that the two individuals are
equivalent with respect to a specific variable that
the researcher would like to control
- Study on learning, make sure people of all
different IQ scores are in one group. Match
one person with 120 with another person
with 120
The t-statistic for repeated-measured design
It is essentially the same as the single-sample t statistic,
but instead of using raw scores you use difference
scores
Because every person is measured twice you need to
calculate the difference score:
D = X2 – X1
 shows the direction of change for each person
Hypothesis:
H0: μD = 0  there is no difference between the time-points
H1: μD ≠ 0
For repeated-measures the t-statistic is almost the same as a single-sample t-statistic:

-
The estimated standard error is exactly the same as in a single-sample t-statistic
-
For the mean you calculate the mean by dividing the added difference scores by the
number of participants
The standard deviation is calculated by
dividing the squared difference scores by
the number of participants  it is the
standard deviation of the difference, not
the difference in standard deviations
-
Effect size
Whenever a treatment effect is found to be
statistically significant, it is recommended that
you also report a measure of the absolute
magnitude of the effect  Cohen’s d and r2
Cohen’s d
A standardized measure of the mean difference between treatments
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For a repeated measures study, the population mean and standard deviation are unknown,
so you use the sample mean difference divided by the sample mean deviation
The percentage of variance accounted for = r2
Computed using the obtained t value and the df value 
The answer is the percentage of the variance in the scores
that is explained by the effect
Differences repeated measures and independent measures
1. Number of subjects: a repeated-measures design requires fewer subjects
2. Study changes over time: the repeated-measures design is especially well suited for
studying learning, development, or other changes that take place over time
3. Individual differences: a repeated-measures design eliminates individual differences
 the disadvantage of repeated measures is:
- Order effects, so you can solve it by counterbalancing: participants are randomly
divided into groups and they do both treatments but switched. Group one does first
A, and then B. Group two does first B, and then A
- Participant loss (in longitudinal designs)
- Sometimes they are practically difficult
Chapter 8.5
There have been some concerns for hypothesis testing
1. The focus of a hypothesis test is on the data rather than the hypothesis. When the
null hypothesis is rejected, we are making a statement about the sample data, not
about the null hypothesis
2. Demonstrating a significant treatment effect does not necessarily indicate a
substantial treatment effect. Statistical significance does not provide any real
information about the absolute size of a treatment effect
 the test is making a relative comparison: the size of the treatment effect is being
evaluated relative to the standard error (a very small standard error can still be large
enough to be significant; a significant effect does not mean a big effect)
Effect size
Intends to provide a measurement of the absolute magnitude of a treatment effect,
independent of the size of the sample(s) being used
Cohen’s d:
 does not include the sample size, so it is not influenced by the number of scores in a
sample
Chapter 9.3
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In most situations the population values are not known, so you must substitute the
corresponding sample values in their place, it is the same with Cohen’s d; which becomes,
An estimated d of 1, indicates that the size of the treatment effect is equivalent to one
standard deviation
Confidence intervals
An interval, or range of values centered around a sample statistic. The logic behind a
confidence interval is that a sample statistic, such as a sample mean, should be relatively
near to the corresponding population parameter
- Normally we report findings using point estimates (M=5, SD=3)
- An interval accepts that there is uncertainty around an estimated value  instead of
saying M = 5, but say that the mean lies in an interval from A to B
- An interval has a certain width and is defined by a lower and upper boundary
 95% of the 95% confidence interval will contain the true value of M
Constructing a confidence interval
1. The observation that every sample mean has a corresponding t value defined by the
equation
2. You know M and SM, but t and μ are unknown, however you can estimate t. when you
know the sample has n scores, your df is n-1. If you want to be 80% confident that
the sample mean corresponds to a t value in a certain interval you can use a two
tailed proportion of 0.20 combined.
3. You then plug the estimated t value into the equation, and you can calculate the
value of μ.
The calculation is:
 it’s two values
Example:
Factors affecting width of a confidence interval
1. When changing the level of confidence, the width of the interval changes. To have
more confidence, the width must be increased. And when you have a smaller more
precise interval, you must give up confidence
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2. The width of the interval also changes if you change sample size, a bigger sample size
has a smaller interval
Chapter 8.6
Revision errors
Type 1 error
- False positive, we conclude that there is an effect when there isn’t
 keep the error low by an alpha level, with an alpha level of .01 1% of the values under
the null lie in that area, thus in 1% of the cases, we will incorrectly conclude that there is
an effect
Type 2 error
- Missed effects, we conclude that there is no difference, but in reality there is one
- This error term is β
- Keep these low as well  you can’t set them all low, one is always a compromise of
the other
- The higher your β (type 2 error), the lower you statistical power
 if we make α stricter (decreasing), we increase β, so we decrease statistical power (1 – β)
 if we make 1 – β higher (increasing), we decrease β, so we increase type 1 error α
Solutions (increasing statistical power)
- Increase sample size  statistical power increases (less overlap between
distributions)
- Increase the effect size of interest
- Increase α
Statistical power
Instead of measuring effect size, you can measure the power of the statistical test
 the power is the probability that the test will correctly reject a false null hypothesis. That
is, power is the probability that the test will identify a treatment effect if one really exists
- Whether a treatment has an effect has one of two outcomes, reject or accept the null
hypothesis. This should together add up to 1, failing to reject H0 when there is an
effect is a type 2 error, with a probability of p= β  second outcome is 1 – β
- Power of a hypothesis test is equal to 1 – β
- When the power of the test is 70% (1- β) then the probability of a type 2 error is 30%
(β)
- The power is directly related to the sample size, one reason to compute power is to
determine what sample size is necessary to achieve a reasonable probability for a
successful research study.
When comparing two normal distributions, one without treatment (μ=80) and one with
treatment (μ=88). You take an alpha level of .05 in a two-sided hypothesis test. You will see
that the distribution with treatment lies almost
entirely in the critical region, which means that
the treatment is close to a 100% power.
Calculation
Locate the exact boundary for the critical region
- SD x z-score  2 x 1.96 = 3.92
Add the distance to the sample mean
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 80 + 3.92 = 83,92
Determine what proportion of the treated samples are greater than M=83,92, calculate the
z- score
Look up the z-score in the unit normal table (-2.04) which corresponds to p=0.9793
 if the treatment has an 8-point effect, 97.93% of all the possible sample means will be in
the critical region and we will reject the null hypothesis
Chapter 15
Correlation
= a numerical value that describes and measures three characteristics of the relationship
between X and Y
1. The direction of the relationship: the sign of the correlation, positive of negative,
describes the direction of the relationship
 positive correlation: the variables tend to change in the same direction
 negative correlation: the variables tend to change in the opposite direction
2. Form of the relationship: linear form = points tend to cluster around a straight line
3. Strength or consistency: how the points fall on/around the line. A perfect correlation
is marked as 1.00, a correlation with no consistency is marked as 0.00
Why correlations are used
1. Prediction
2. Validity: checking if a test gives the result it was expected to, the psychologist could
measure the correlation between the new test and the other measures to
demonstrate that the new test is valid
3. Reliability: demonstrating that a test produces stable, consistent measurements
4. Theory verification: the prediction of a theory could be tested by determining the
correlation between two variables
Correlation and…
Causation
 there may be a causal relationship between two variables, but the existence of a
correlation does not prove it
Restricted range
A sample only shows a limited range of scores, a calculation with these scores could be
totally different from a calculation with a full range of scores
 To be safe you should not generalize any correlation beyond the range of data
represented
Outliers
Extreme data points, most easily noticed in a scatter plot
The coefficient of determination (r2)
= it measures the proportion of variability in one variable that can be determined from the
relationship with the other variable
- Used to measure the size of strength of the correlation
Pearson correlation (r)
= measures the degree and direction of the linear relationship between two variables
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- The correlation coefficient ranges from -1.00 to +1.00
Correlation coefficient
- Population notation: ρ
- Sample notation: r
- Effect size: R2  expresses absolute magnitude of the effect (by squaring r)
- Values approaching +/- 1.00 indicate a strong relationship, values around 0 indicate a
weak relationship
Sum of products (SP)
To calculate the Pearson correlation, you need the sum of products (SP)
This is calculated by:
or
- MX is the mean for X scores
- MY is the mean for the Ys
Steps:
1. Find the X deviation and the Y deviation for each individual
2. Find the product of the deviations for each individual
3. Add the products
Pearson correlation
- SP is used to measure the degree to which X and Y vary together (covariability)
- Both SS of X and Y are used to measure how they vary separately
Calculation:
Z-scores
Each X and Y score can be transformed into a z-score, so you can express the Pearson
correlation in terms of z-scores
Hypothesis test
 null hypothesis: there is no correlation, which is zero
 alternative hypothesis: there is a correlation, which is nonzero
T-statistic:
 sample value = calculated r
 population value = ρ (often 0)
Standard error:
Complete t statistic:
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Degrees of freedom: n – 2
Chapter 17
Associations between variables
Most calculations are done with numerical data, but often data are not numerical  could
be problematic
For example: experiment of extra lessons vs no extra lessons and pass vs fail
 2 x 2 table
 is there an association between the factor extra lessons (yes – no) and
the result (pass – fail)
 unable to use all calculations used before
The chi-square test for goodness of fit
= Uses sample data to test hypotheses about the shape or proportions of a population
distribution. The test determines how well the obtained sample proportions fit the
population proportions specified by the null hypothesis
- Present the scale of measurement as a series of boxes, with each box corresponding
to a separate category on the scale
- Symbol: χ2
Hypotheses
Null hypothesis: specifies the proportion (or percentage) of the population in each category
 50% of lawyers are men and 50% are women
1. No preference, equal proportions: no difference among the categories, the
population is equally divided among the categories
 alternative hypothesis would say that the population distribution has a different
shape
2. No difference from a known population: the proportions for one population are not
different from the proportions known to exist for another population
 alternative hypothesis states that the population proportions are not equal to the
values specified by the null hypothesis
Data
No need to calculate, just select a sample of n individuals
and count how many there are in each category
 the resulting values are the observed frequencies (fo) = the number of individuals from
the sample who are classified in a particular category, each individual is counted in one and
only one category
When you have a certain population in your null hypothesis that is distributed in a certain
way in percentages (group 1 is 25%, group 2 is 50% and group 3 is 25%), to find the exact
frequency expected for each category you multiply the sample size (n) by the proportion (or
percentage) from the null hypothesis
 these values are the expected frequencies (fe) = for each category the frequency value
that is predicted from the proportions in the null hypothesis and the sample size (n). the
expected frequencies define an idea, hypothetical sample distribution that would be
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obtained if the sample proportions were in perfect agreement with the proportions specified
in the null hypothesis
Chi square statistic
Purpose is to determine whether the sample data support or refute a hypothesis about the
population
Steps:
1.
2.
3.
4.
Find the difference between fo (the data) and fe (the hypothesis) for each category
Square the difference  makes all values positive
Divide the squared difference by fe
Sum the values from all the categories
Chi square distribution
the set of chi-square values for all the possible random samples when H 0 is true,
characteristics are:
1. The formula involves squaring, so all values are zero or larger  hypothesis is
positively skewed
2. When H0 is true, you expect the data (fo ) to be close to the hypothesis (fe) values 
we expect chi-square values to be small when H0 is
true
Degrees of freedom
Df = C – 1
 C is the amount of categories
Critical region
 chi square distribution table
Chi-square for goodness fit hypothesis test example
Data: n=50
Step 1: state the hypotheses and alpha level
H0: there is no preference, so all 4 categories are equal  each one has 25%
H1: one or more of the orientations is preferred over the others
α = .05
Step 2: locate the critical region
Degrees of freedom is C – 1  4 – 1 = 3
For df = 3 and alpha is .05, the critical value is 7.81
Step 3: calculate the chi square statistic
Compute the expected frequencies from H0
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 null hypothesis says that each category has 25%, the
sample is n=50
 fe = pn = ¼ x 50 = 12.5
Calculate the value of the chi square statistic
Step 4: make a decision
The chi square value lies in the critical region, so H0 is rejected  the four categories are not
equal
Chi square test for independence
= Uses the frequency data from a sample to evaluate the relationship between two variables
in a population. Each individual in the sample is classified on both of the two variables,
creating a two-dimensional frequency distribution matrix. The frequency distribution for the
sample is then used to test hypotheses about the corresponding frequency distribution in
the population
- Example:
Hypotheses
1: H0: for the general population, there is no relationship between A and B
 The value obtained for one variable is not related to the value for the second variable
2: H0: In the population, the proportions in the distribution of A are not different from the
proportions in the distributions of B, the two distributions have the same shape
 the data consists of two (or more) separate samples that are being used to test
differences between two (or more) populations
Data
Same logic as the chi-square test for goodness of fit.
1. Sample is selected, and each individual is classified or categorized
 two variables, so every individual is classified on both variables
 the frequencies in the sample distribution are the observed frequencies (fo)
2. To find the expected frequencies (fe) you determine the overall distribution of A, and
then apply this distribution to both categories of B
Using the example used above:
The total sample consists of 200 people, the proportion selecting red is 100 out of 200  50%
Using this you get:
- 50% prefer red, 10% prefer yellow, 20% prefer green, 20% prefer blue
According to the null hypothesis, both personality groups should have the same proportions
for color preferences. When calculating the introverts and red you do: 50% out of 50 = 25
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When you do this everywhere you get:
Instead of extensive calculations you
have a formula for the
expected frequencies:
-
Fc is the frequency total for the column
Fr is the frequency total for the row
Example: the red introvert  100 x 50 / 200 = 25
Statistic and degrees of freedom
The chi-square statistic is exactly the same:
The degrees of freedom is different, because you’re not only dealing with categories or
columns  df = (R – 1) x (C – 1)  R is row, C is column
Chi-square for independence hypothesis test example
Data: n = 200
Step 1: hypothesis and level of significance
Version 1:
H0: in the general population, there is no relationship between parents’ rule for alcohol use
and the development of alcohol related problems
H1: there is a relationship
Version 2:
H0: in the general population, the distribution of alcohol-related problems has the same
proportions for teenagers whose parents permit drinking and for those whose parents do
not
H1: there are different proportions
Alpha level: .05
Step 2: degrees of freedom and critical region
Degrees of freedom: (R – 1) x (C – 1)  (2 – 1) x (2 – 1) = 1
 critical value of 3.84
Step 3: determine the expected frequencies and calculate the chi-square statistic
- 160 out of 200 is 80%
40 out of 200 is 20%
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 null hypothesis states these are the same for both groups
For the teens who were allowed to drink:
- No: 64  80% out of 80
- Yes: 16  20% out of 80
For the teens who were not allowed to drink:
- No: 96  80% out of 120
- Yes: 24  20% out of 120
Chi-square statistic:
Step 4: make a decision
Reject the null hypothesis
Effect sizes of the Chi square test
Aim: “corrected” indication of the effect
Phi Coefficient
the square root of the chi-square value divided by the sample size
Cramer’s V
 beyond 2 by 2 tables
Square root of chi-square divided by n, multiplied with the degrees of freedom with an
asterix  the smaller of (R – 1) and (C – 1)
Chapter 18
Binomial test
= uses sample data to evaluate hypotheses about the values of p and q for a population
consisting of binomial data
- Binomial data: data classified in two distinct categories, in which each observation is
classified in one of these two categories, and the sample data consist of the
frequency or number of individuals in each category
Hypotheses
H0 falls in one of two categories
1. Just chance: the null hypothesis states that the two outcomes A and B occur in the
population with the proportions that would be predicted simply by chance
 50%  p = ½
2. No chance or no difference: when you know the proportions for one population and
want to determine if the same proportions apply to a different population
Z-score formula
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When pn and qn (both categories x sample size) are both equal or greater than 10, the
binomial distribution resembles a normal distribution
- The standard deviation is:
- The mean is p x n
the z-score calculation is:
 X/n is the proportion of individuals in the sample who are classified in
category A
 P is the hypothesized value for the proportion of individuals in the
population who are classified into category A

is the standard error of the sampling distribution of X/n and
provides a measure of the standard distance between the sample
statistic (X/n) and the population parameter
Direct formula
 when the data is not at least 10
Example
10 multiple choice questions  each answer has 4 options (25%)
Goal is to have 5 correct by guessing
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