Inferential Statistics:
Frequency Distributions
&
Z-Scores
Please refrain from typing, surfing or printing during our conversation! 
Outline of Today’s Discussion
1.
Frequency Distributions: Overview
2.
The Z-Score
3.
Z-scores & Percentile Rank
Part 1
Frequency Distributions:
An Overview
The Research Cycle
Real
World
Abstraction
Generalization
Research
Conclusions
Research
Representation
Methodology
***
Data Analysis
Frequency Distributions
Z-Scores / Percentiles
Evaluating Hypotheses
Research
Results
Frequency Distributions: Overview
1. Many naturally occurring phenomena are well
described by a bell-shaped curve.
2. The bell-shaped curve is also called the
Normal Distribution, or the Gaussian
Distribution (after its founder, Gauss).
http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss
Frequency Distributions: Overview
Frequency Distributions: Overview
1. The width of the normal distribution is an
indication of the standard deviation.
2. “Fat” distributions have big standard
deviations.
3. “Thin” distributions have small standard
deviations…
Frequency Distributions: Overview
Which distribution has the largest standard deviation?
Frequency Distributions: Overview
Which distribution would have the smallest error bars
if we were to plot the means and error bars on a bar-chart?
Gaussian “Width” & Error Bars
Dependent
Variable
Ordinate
starts
at zero
Motion Sensitivity (
d' )
0.6
0.5
Plus 1 S.D.
0.4
0.3
0.2
Minus 1 S.D.
0.1
0
Independent
Variable
Valid Cue
No Cue
Invalid Cue
Cue Condition
Frequency Distributions: Overview
1. Now let’s further develop some intuitions about
variability (variances, and standard deviations).
2. Each of the following four lists of numbers has a
mean equaling 5.
3. Just by looking at the lists, indicate which list has the
greatest standard deviation, and which one has the
smallest standard deviation….
Frequency Distributions: Overview
A
B
C
D
5
5
5
5
5
5
5
5
5
5
9
1
9
1
9
1
1
9
9
1
6
5
6
5
4
5
7
4
3
5
7
5
3
5
5
3
7
5
7
3
Frequency Distributions: Overview
1. Normal distributions are important in science
because they allow for standardizing data
across experiments and disciplines.
2. IQ scores, SAT scores, and height are just a
few of the variables that are well described by
the bell curve (i.e., “normally distributed”)…
Frequency Distributions: Overview
We’ll return to z-scores later,
this is just a peak to “prime” you.
Frequency Distributions: Overview
1. Not all distributions are symmetric.
2. Distributions that are not symmetric are said
to be skewed.
3. Positively Skewed Distributions have “tails” to
the right (the positive side of the axis).
4. Negatively Skewed Distributions have “tails”
to the left (the negative side of the axis)…
Frequency Distributions: Overview
“The Thinker(s)”
Thinking about majoring in geoscience?
Did you know that the average geoscience major who
graduated from U.N.C in 1984ish earns $600,000 annually?
What’s wrong with this picture?
Frequency Distributions: Overview
Positively Skewed Distribution
&
The Measures of Central Tendency
Frequency Distributions: Overview
Soon we will learn about a very important,
positively skewed distribution called
The Chi-Square Distribution (c2).
Frequency Distributions: Overview
Negatively Skewed Distribution
&
The Measures of Central Tendency
Part 2
Z-scores
The Z-Score
1.
If we have a normally distributed (Gaussian distributed)
variable, and if there is not too much skew, we can describe a
particular datum by it’s “z-score”.
2.
A Z-score is a standard deviation from the mean.
3.
Would someone tell us why z-scores are important?
4.
Let’s see how we would compute a z-score by hand…
The Z-Score
1.
Again, a Z-score is a standard deviation from the mean.
2.
Let’s use IQ as an example.
3.
If the standard deviation of IQ’s is 15 points,
one z-score = 15 points above the mean;
two z-scores = (2*15=) 30 points above the mean;
three z-scores = (3*15=) 45 points above the mean;
minus two z-scores = (-2*15=) -30 points below mean.
The Z-Score
1. There is a formula for computing z-scores,
given a score (say, an IQ score), the mean, and
the standard deviation, and vice versa.
2. Z = (raw score - mean) / standard deviation
3. Raw Score = mean + (Z * standard deviation)
4. Let’s do some examples…
The Z-Score
1. If the mean IQ = 100, with a standard
deviation of 15, and Larry’s IQ score is 130,
What’s Larry’s IQ in z-scores?
2. If the z-score associated with Craig’s IQ is -3,
what’s his IQ score?
3. Note: if the standard deviation = 0, then the zscore is “undefined”. ( Can’t divide by zero! )
Part 3
The Relation Between
Z-Scores & Percentile Rank
Z-Scores & Percentile Rank
For ANY Gaussian distribution,
34% of the population falls between 0 and +1 z-score,
and another 34% falls between 0 and -1 z-score.
Z-Scores & Percentile Rank
Question:
What percent of the population falls between
z-scores of zero, and negative infinite?
Z-Scores & Percentile Rank
Question:
So, if your z-score is zero,
what’s your percentile rank?
Z-Scores & Percentile Rank
Question:
If your z-score = +1,
what’s your percentile rank?
Z-Scores & Percentile Rank
Question:
If your z-score = +2,
what’s your percentile rank?
Z-Scores & Percentile Rank
Question:
If your IQ is 85,
what’s your percentile rank?
Z-Scores & Percentile Rank
By convention, if a score is in the top 2.5 percentile,
or the bottom 2.5 percentile,
Psychologists consider the score to be
significantly different from the mean.
Z-Scores & Percentile Rank
1. Suppose we want to know what z-score is associated
with, say, the bottom 2.5 percent of the population.
2. By “eye-balling it”, we know that the z-score should
be approximately -2ish.
3. Let’s learn some commands in EXCEL to convert a
given percentile rank (or a probability) into an exact
z-score…
Z-Scores & Percentile Rank
1.
In excel create one cell called “Proportion” (since a percentile rank is
essentially a proportion).
2.
We’re interested in the 2.5th percentile, which would be a proportion
equaling 0.025 (right?).
3.
Label the neighboring cell “Z-Score”.
4.
Beneath the label use the “=NORMSINV( )” function [or f(x)].
5.
NORMSINV is an abbreviation for Normal Standard Inverse. (Standard
Gaussian ---> Mean = 0, SD = 1)
You can enter the desired proportion directly in the parentheses, or you
can put a cell address there…
6.
Z-Scores & Percentile Rank
1. Remember, this function is just like any other
function…it’s a rule for turning one number into
another number.
2. “You stick in a proportion (corresponding to
percentile rank) and get out the corresponding zscore.”
3. Just to convince yourself, stick in some known values,
like 0.5, and 0.84. These should produce familiar zscores…
Frequency Distributions: Overview
The 50th percentile = the ?? Z-score?
The 84th percentile = the ?? Z-score?
Z-Scores & Percentile Rank
1. Sometimes, psychologists want to know the interquartile range - the raw scores associated with the
25th and 75th percentiles.
2. Can you think of an example of when the interquartile range is used?
3. Here’s the inter-quartile range graphically…
Z-Scores & Percentile Rank
Using the NORMSINV command in Excel,
find the z-scores that correspond to
the 25th & 75th percentile.
Z-Scores & Percentile Rank
1. Finally, let’s do it backwards!!!
2. That is, let’s say that we already have a
z-score, but we want to know the
corresponding percentile rank (i.e., the
corresponding proportion)…
Z-Scores & Percentile Rank
1. We’ll use this command:
“=NORMSDIST( )”
2. NORMSDIST is an abbreviation for Normal
Standard Distribution. (Mean = 0, SD = 1)
3. The function “takes in a z-score” (in the
parentheses), and then “spits out the corresponding
proportion” .
4. Let’s do some examples…
Z-Scores & Percentile Rank
1. What percentile rank (i.e., proportion) is
associated with a z-score of -2?
2. Here’s a tough one…What percentile
rank is associated with an IQ of 145?
Acknowledgments
Images used in this educational presentation
were obtained from Wikimedia Commons, in
accordance with regulations regarding
copyright, use, and dissemination.
http://commons.wikimedia.org/wiki/Main_Page