Last class we discussed:
• Ways of knowing reality (science is
chief means for psychology)
• Classification of Research Studies
– (1) Design (experiment / correlational
/ descriptive)
– (2) Setting (field / lab)
– (3) Data-collection method (selfreport / observation)
By the end of today’s class, you will be able to:
•
•
•
•
•
Define statistics and its two branches
Understand the four scales of measurement
Deal with measures of central tendancy
Calculate standard deviation
Discuss the properties of a distribution curve
Statistics
•is the branch of mathematics that deals
with the collection, organization, and
analysis of numerical data.
•Even if your data doesn’t begin as a set
of numbers, you can quantify it (turn it
into numbers) and use statistical analysis.
What are statistics?
• Numerical representations of large amounts of
data
• Two Branches
– Descriptive – tell us about the numbers
– Inferential – tell us what the numbers mean
The work of the statistician is no longer
confined to gathering and tabulating data
(descriptive statistics), but is chiefly a
process of interpreting the information
(inferential statistics).
How do we obtain statistics?
• Through Quantification (#)
• All behaviour needs to be quantified to
be analysed statistically
• How do we quantify behaviour?
– Though scales of measurement
What are Scales of Measurement?
Measurement is the assignment of numbers to
objects or events in a systematic fashion.
Four levels of measurement are
commonly distinguished:
• Nominal
• Ordinal
• Interval
• Ratio
• “NOIR”
Nominal
• Based on Categories
• No quantitative information is conveyed
and no ordering of the items is implied.
– Gender
– Religion
– Ethnicity
Nominal scale
• Classifies data
according to a
category only.
• Measurement:
Frequency
distributions. How
often do certain
items or responses
occur?
Ordinal
• Ordered in the sense that higher
numbers represent higher values.
• However, the intervals between the
numbers are not necessarily equal.
• e.g., “Strongly Disagree, Disagree,
Neither Agree nor Disagree, Agree,
Strongly Agree”
Ordinal scale
• Classifies data according to rank.
• The difference between mild and average hotness
may not represent the same difference as the
difference between a rating of hot and very hot.
• There is no "true" zero point for ordinal scales since
the zero point is chosen arbitrarily.
Interval
• One unit on the scale represents the
same magnitude on the trait or
characteristic being measured across
the whole range of the scale.
• For example, if anxiety were measured
on an interval scale, then a difference
between a score of 10 and a score of
11 would represent the same
difference in anxiety as would a
difference between a score of 50 and a
score of 51.
Interval scale
• Does not have true zero point (it is arbitrary).
• On the anxiety scale: score of 30 NOT twice as
anxious as score of 15. We know they are MORE
anxious, but that’s about where it ends.
• Similarly, a score of zero does not mean that they
have zero anxiety.
• True interval measurement is rare to nonexistent in
the behavioral sciences. Most measurement scales in
psychology are ordinal.
Ratio
• All of the previous + absolute zero
- e.g. weight and height
– This scale has an absolute zero.
• Temperature of 300 Kelvin is twice as
high as a temperature of 150 Kelvin.
• Time can have an absolute zero (no
time).
Ratio scale
• Similar to interval scales, but has a true
zero point.
• Racer C’s speed (below) was twice as fast
as Racer A’s speed.
Levels of Measurement
Mutually
Exclusive
Rank
Order
Equal
Intervals
Nominal

Ordinal


Interval



Ratio



Absolute
Zero

Note. Mutually Exclusive: Every point on every
measurement scale is separate from every other
point. You can’t be red and blue at the same time.
Concept Review
• Which rating scale is based on categories?
– Nominal
• Which rating scale has an absolute zero?
– Ratio
• Most psychological scales, such as the CESD, which measures depression, use which
rating scale?
– Ordinal
• How do we interpret data?
– Inferential Statistics
2 main branches of these
statistics:
Inferential / Descriptive
Inferential Statistics
•
Draw a sample from larger
population
– The sample is smaller than the population
and should be representative of the
population.
– Use inferential statistics to draw
inferences about the population based on
the sample.
Statistics prescribes ways to take
measurements of smaller groups
of people and infer the findings
true for all people matching the
defined criteria, such as every
Canadian adult between ages 25
and 34. The smaller group is our
sample, and we infer from the
sample to the population.
Descriptive Statistics
• Aims to describe sets of data values
– Ex: a group of individual test scores.
• Each data value is a single measurement of some
attribute being observed.
• The term data set refers to all data values considered
in a set of statistical calculations.
• Descriptive statistics summarize sets of information.
Descriptive Stats cont’d
• Examples
– Central Tendency (mean, median,
mode)
– Variability
– Shape of Distributions
– Relations (Correlation)
• We’ll discuss each of these in turn
What are measures
of central tendency?
• Each is a score that represents
the typical performance of the
sample
• Mode
• Mean
• Median
Mode
•Answers the question, "Which number or object
appears the most often in our data set?“
•Can have more than one mode.
•Example: Array of test scores 2,3,4,5,6,6,7,8,9,9 
Mode is 6 & 9. We say it’s bimodal.
• Looking ahead: median and mean are a
little more slippery (but not much more).
• It’s from the median and mean that we get
the term “central tendency,” measures of
the "center" (average) of the scores in a
given data set.
Mean
sum of the individual scores divided by
the total number of scores.
i.e. the average
Median
…is the score that’s right in the middle of your data set.
1/2 of the data set’s values less than or equal to the
median; 1/2 of the numbers in the data set will have
values equal to or greater than the median.
To find the median of a finite list of numbers:
Arrange all observations from lowest value to highest
value and pick the middle one. If there are an even
number of observations, you can take the mean of the
two middle values.
Central Tendency Example
• Hours watching The Bachelor per day for 10 people
Subject
– 1.
– 2.
– 3.
– 4.
– 5.
– 6.
– 7.
– 8.
– 9.
– 10.
Hours
2
3
2
7
3
5
2
2
1
3
• Take a minute to calculate mean, median, and mode
Central Tendency Example
• Hours watching The Bachelor per day for 10 people:
• Subject
–
–
–
–
–
–
–
–
–
–
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Hours
2
3
2
7
3
5
2
2
1
3
Mean = 3 (add
all scores and
divide by 10)
Median = 2.5
Mode = 2
Something important to keep in
mind when writing papers:
A conclusion is simply the place where you
got tired of thinking.
Variability
Consider Two Streets
• Conservative Street
• Wild Alley
• 45, 46, 47, 44, 43
• 25, 35, 45, 55, 65
• Mean = ?
• Mean = ?
Consider Two Streets
• Conservative Street
• Wild Alley
• 45, 46, 47, 44, 43
• 25, 35, 45, 55, 65
• Mean = 45
• Median =
• Mean = 45
• Median =
How are these streets different?
Consider Two Streets
• Conservative Street
• Wild Alley
• 45, 46, 47, 44, 43
• 25, 35, 45, 55, 65
• Mean = 45
• Median = 45
• Mean = 45
• Median = 45
The means are the same. The medians are the same.
How are these streets different?
Variability
What is Variability?
• Extent of dispersion around
the mean.
• Are the scores all pretty close
together or more spread out?
Variance
• An index of variability (i.e. one
way to express variability).
Variance Formula
s2
2
(X

X
)

-1

N
(Take the difference between each score and the
mean, squaring each of these deviations, and
then calculating the mean of the squared
deviations. Finally, you divide by the number of
scores minus one.)
The problem with variance
Has an ATTITUDE
It is an index in the wrong units! Everything
was SQUARED to get the variance.
We need something else…
H
m
m
m
(ideas?)
Standard Deviation
• Square root of the squared
deviations about the mean.
• i.e. square root of variance
(Wasn’t that disturbingly simple?)
Standard Deviation Formula
• Definitional Formula
 (X  X)
2
s
N
-1
• The standard deviation is equal to the
square root of the sum of the squared
deviations divided by the total number of
scores
Practice Run: Calculate SD and Variance
• Hours watching tv per day for 10 people
•
•
•
•
•
•
•
•
•
•
•
Subject
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Hours
2
3
2
7
3
5
2
2
1
3
(Hours – mean)
• Calculate SD and variance for the data.
squared
• First step:
–raw score minus mean.
–then square the deviations.
–Sum these together.
• Next:
–Divide by n-1 to get variance.
• Then:
–Take the square root of this for SD.
Calculate SD and Variance
• Hours watching tv per day for 10 people
Subject
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Hours
2
3
2
7
3
5
2
2
1
3
(Hours – mean)
2–3
3–3
2–3
7- 3
3- 3
5- 3
2- 3
2- 3
1- 3
3- 3
squared
1
0
1
16
0
4
1
1
4
0
• Sum = 28
• N–1=9
• 28/9 = 3.11
Square Root of 3.11 = 1.76
When you collect data…
• Assumption – normality. We assume that many
things in life take the form of a “normal curve” when
graphed, with most scores occurring close to the
mean, and some scores more spread out.
• Assumption sometimes violated
• Example: Give a middle school-level math test to 6year-olds and to 18-year-olds. Result will not be a
normal curve; rather, you will see two clumps: high
scores (18-yr-olds) and low scores (6-yr-olds).
Properties of a distribution curve
• Skewness
– Happens when the mean, median, and mode
aren’t the same.
- symmetry of a curve
- determined by the tail in the curve
Positively skewed
• Tail on the right side of the distribution
• Most scores cluster on the left side of the distribution
• If this was a math test, most people had a low score
and just a few performed very well.
Negatively skewed
• Tail on the left side of the distribution
• Most scores cluster on the right side of the
distribution
• How did people perform on the math test this time?
Why is Skew a Problem?
• Can’t use certain formula (t-tests,
correlation)
– Assumption of normality is important
• Mathematically, we can manipulate the
data to try to normalize distribution
• Or, use different formula
Kurtosis
• Peakedness of the curve compared to the
normal curve.
• Are the scores grouped more toward the
middle (the mean) or more toward the tails
of the distribution
Leptokurtic
• Pointed shape
• Kurtosis >3
• Most scores cluster at the middle of the
distribution
• Simultaneously “peaked” centre and
“fat” tails
Platykurtic
• Flat shape
• Kurtosis <3
• Many scores spread toward the tails of
the distribution
• Simultaneously less peaked and thinner
tails
Mesokurtic
• Normal curve distribution
• Kurtosis = 0 (no skew)
Normal Distribution
• Bell shape
• Mean, median, and
mode are equal
•Located at centre
of distribution
Area of the Normal Curve
• 34% is one SD above or below the mean
• 68% within one SD above and below
• 47.5% of scores is between 0 and 2 SD
• 95% within two SD above and below