1. VARIABILITY Distributions Measuring dispersion
Variance and standard deviation

2. Review: Distribution
Case no.
Age
Height
M/F 1 23 68 M 2 22 64 F 3 69 4 25 71 5 27 6 72 7 24 65 8 66 9 10 11 21 12 62 13 14 15 16 56 17 18 70 19 20 26 60 52 31 61 28 29 30 67
Summary
statistics
mean = 24
mean = 67
%M 39 %F 61
An arrangement of cases according to their score or value on one or more variables
Categorical variable
Continuous variable

3. Dispersion How do cases “disperse” (arrange themselves) around the mean?
officers

4. Three statistics that measure dispersion Measure how cases “disperse” (arrange themselves) around the mean
Average deviation  (x - ) n
Average distance between the mean and the values (scores) for each case
Uses absolute distances (no + or -)
Affected by extreme scores
We’ll never use it in class
Variance (s2): A sample’s cumulative dispersion
 (x - )
n  we always use n-1 (our sample sizes are always small)
Standard deviation (s): A standardized form of variance, comparable between samples
 (x - )
n  we always use n-1 (our sample sizes are always small)
Square root of the variance
Expresses dispersion in units of equal size for that particular distribution
Less affected by extreme scores
Mean 2.3
officers

5. This is not an acceptable graph – it’s only to illustrate dispersion
Variability exercise
Sample 1 (n=10)
Officer Score Mean Diff. Sq.
____________________________________________________
Sum
Variance (sum of squares / n-1) s2 .99
Standard deviation (sq. root of variance) s .99
Random sample of patrol officers, each scored 1-5 on a cynicism scale
This is not an acceptable graph – it’s only to illustrate dispersion

6. Sample 2 (n=10)
Officer Score Mean Diff. Sq ___ ___ ___
2 1 ___ ___ ___
3 1 ___ ___ ___
4 2 ___ ___ ___
5 3 ___ ___ ___
6 3 ___ ___ ___
7 3 ___ ___ ___
8 3 ___ ___ ___
9 4 ___ ___ ___
10 2 ___ ___ ___
Sum ____
Variance s2 ____
Standard deviation s ____
Another random sample of patrol officers, each scored 1-5 on a cynicism scale
Compute ...

7. Two random samples of patrol officers, each scored 1-5 on a cynicism scale
Sample 1 (n=10)
Officer Score Mean Diff. Sq.
Sum
Variance (sum of squares / n-1) s2 .99
Standard deviation (sq. root of variance) s .99
Sample 2 (n=10)
Officer Score Mean Diff. Sq.
Sum
Variance (sum of squares / n-1) s
Standard deviation (sq. root of variance) s .97
These are not acceptable graphs – they’re only used here to illustrate how the scores disperse around the mean

8. VARIABILITY
Shape of distributions
Flat, peaked, normal

9. “Flat” distributions Mean A poor 3.65 descriptor
Dispersion (aka, “variability”): How scores or values arrange themselves around the mean
When scores are more dispersed (i.e., “variability” is greater) a distribution’s shape gets flatter
Greater distance between most scores and the mean
Many scores are at a considerable distance from the mean
The mean loses value as a “summary statistic”
Arrests
Mean A poor  descriptor

10. “Peaked” and “normal” distributions
Dispersion (aka, “variability”): How scores or values arrange themselves around the mean
Peaked: If most scores cluster about a certain value the shape of the distribution is called “peaked”
Normal: If the clustering of scores is around the mean the distribution is called “normal”
In social science research it turns out that scores or values for many variables are normally or near-normally distributed
This allows use of the mean to describe the underlying datasets
That’s why means are called a “summary statistic” - they can “summarize” the values of samples or populations
Arrests
Mean Not a good  descriptor
Peaked distribution (but not “normal”)
Arrests
Mean A good  descriptor
Peaked and “normal” distribution

11. Characteristics of normal distributions
Unimodal and symmetrical: shapes on both sides of the mean are identical
68.26 percent of the area “under” the curve – meaning percent of the cases – falls within one “standard deviation” (+/ ) from the mean
The fact that a distribution is “normal” or “near-normal” does NOT imply that the mean is of any particular value. All it implies is that scores distribute themselves around the mean “normally”.
Means depend on the data. In this distribution the mean could be any value.
By definition, the standard deviation score that corresponds with the mean of a normal distribution - whatever the mean might be - is zero. ( = 0)
Mean (whatever it is)
Standard deviation (always 0 at the mean)

12. How well do means represent (summarize) a sample?
If variable “no. of tickets” was “normally” distributed most cases would fall inside a bell-shaped curve. Here they don’t.
Number of tickets
Frequency
B D F H K
A C E G I J L M
-1 SD mean SD
13 officers scored on numbers of tickets written in one week
In a normal distribution about 66% of cases would fall within 1 SD of the mean.
13 X .66 = 9 cases
But here only 7 cases (Officers D-J) do, while nearly as many (6) don’t. Scores are very dispersed, making the distribution mostly flat. So here the mean is NOT a good shortcut for describing how officers performed.
Officer A: 1 ticket
Officers B & C: 2 tickets each
Officers D & E: 3 tickets each
Officers F & G: 4 tickets each
Officers H & I: 5 tickets each
Officer J: 6 tickets
Officers K & L: 7 tickets each
Officer M: 9 tickets
Mean = SD = 2.33

13. 13 officers scored on numbers of tickets written in one week
Here, 9 of 13 cases (officers C-K) do fall within 1 SD of the mean.
The distribution is near-normal because most officers wrote close to the same number of tickets. The cases “cluster” around the mean.
So, for this sample the mean is a decent summary statistic - a good shortcut for describing officer performance
D G
E H J
A B C F I K L M
-1 SD mean SD
Number of tickets
Frequency
Here most cases do fall inside the bell-shaped curve. Variable “no. of tickets” seems near-normally distributed
Officer A: 1 ticket
Officer B: 2 tickets
Officer C: 3 tickets
Officers D, E, F: 4 tickets each
Officers G, H, I: 5 tickets each
Officers J & K: 6 tickets each
Officer L: 7 tickets
Officer M: 9 tickets
Mean = SD = 2.1

14. Going beyond description…
When variables are normally or near-normally distributed, the mean, variance and standard deviation can help describe datasets
But they are also useful in explaining why things change; that is, in testing hypotheses
You want to test the hypothesis that college-educated cops are more effective: college  greater effectiveness
Independent variable: college (Y/N)
Dependent variable: effectiveness (scale 1-5)
You go to the XYZ police dept., draw two samples of patrol officers - one of college grads, the other of non-college grads - and test each officer for effectiveness.
On a scale of 1 (ineffective) to 5 (highly effective) this is how they scored:
10 college grads (mean 3.7)
10 non-college (mean 2.8)
The difference between means is in the hypothesized direction. But does that “prove” that college grads are more effective? To determine whether the difference in means is “statistically significant,” meaning large enough to prove the value of education, we need to know each sample’s variance. Don’t worry - we’ll cover this later!
Are college-educated cops more effective?
College grads
Non-college grads

15. Exam information
You must bring a regular, non-scientific calculator with no functions beyond a square root key.
You will be asked to apply concepts including research question, hypothesis and variables to the “college education and police job performance" article.
You will be given data and asked to create graph(s) depicting the distribution of a single variable.
You will compute basic statistics, including mean, median, mode and standard deviation. All computations must be shown on the answer sheet.
You will be given the formula for variance (s2). You must use and display the procedure described in the slides and practiced in class for manually calculating variance (s2) and its square root, known as standard deviation (s).
This is a relatively brief exam. You will have one hour to complete it. We will then take a break and move on to the next topic.