1. Measures of Central Tendency
Purpose is to describe a distribution’s typical case – do not say “average” case
Mode
Median
Mean (Average)

2. MEASURES OF DISPERSION
Standard deviation
Uses every score in the distribution
Measures the standard or typical distance from the mean
Deviation score = Xi - X
Example: with Mean= 50 and Xi = 53, the deviation score is = 3

3. The Problem with Summing Deviations From Mean
2 parts to a deviation score: the sign and the number
Deviation scores
add up to zero
Because sum of deviations
is always 0, it can’t be
used as a measure of
dispersion
X Xi - X
8 +5
1 -2
3 0
0 -3
12 0
Mean = 3 

4. Average Deviation (using absolute value)
Works OK, but…
AD =  |Xi – X| N
X |Xi – X|
AD = 10 / 4 = 2.5
X = 3
Absolute Value to get rid of negative values (otherwise it would add to zero)

5. Variance & Standard Deviation
Purpose: Both indicate “spread” of scores in a distribution
Calculated using deviation scores
Difference between the mean & each individual score in distribution
To avoid getting a sum of zero, deviation scores are squared before they are added up.
Variance (s2)=sum of squared deviations / N
Standard deviation
Square root of the variance Xi
(Xi – X)
(Xi - X)2 5 1 2 -2 4 6
 = 20
 = 0
 = 14

6. Terminology
“Sum of Squares” = Sum of Squared Deviations from the Mean =  (Xi - X)2
Variance = sum of squares divided by sample size =  (Xi - X)2 = s2 N
Standard Deviation = the square root of the variance = s

7. Calculating Variance, Then Standard Deviation
Number of credits a sample of 8 students is are taking:
Calculate the mean, variance & standard deviation
Mean = 112/8 = 14
S2 = 72/8 = 9
S = 3 Xi
(Xi – X)
(Xi - X)2 10 -4 16 9 -5 25 13 -1 1 17 3 15 2 4 14 18
 = 112 72

8. Summary Points about the Standard Deviation
Uses all the scores in the distribution
Provides a measure of the typical, or standard, distance from the mean
Increases in value as the distribution becomes more heterogeneous
Useful for making comparisons of variation between distributions
Becomes very important when we discuss the normal curve (Chapter 5, next)

9. Mean & Standard Deviation Together
Tell us a lot about the typical score & how the scores spread around that score
Useful for comparisons of distributions:
Example:
Class A: mean GPA 2.8, s = 0.3
Class B: mean GPA 3.3, s = 0.6
Mean & Standard Deviation Applet

10. Example Using SPSS Output
Hours watching TV for Soc 3155 students:
What is the range & interquartile range?
Is there skew (positive or negative) in this distribution?
What is the most common number of hours reported?
What is the average squared distance that cases deviate from the mean?
Statistics
Hours watch TV in typical week
N Valid 18 Missing 11
Mean
Median
Mode 5.00
Std. Deviation
Variance
Minimum 1.00
Maximum
Percentiles

11. The Normal Curve & Z Scores
3155 Spring ‘07, CLASS #5
[COLLECT HW’s]
[pass out HW’s – DUE A WEEK FROM TODAY]
Stuff today coincides with chapter 5 of Healey.

12. THE NORMAL CURVE Characteristics: Theoretical distribution of scores
Perfectly symmetrical
Bell-shaped
Unimodal
Continuous
There is a value of Y for every value of X, where X is assumed to be continuous variable
Tails extend infinitely in both directions Y
axis
x AXIS

13. THE NORMAL CURVE
Assumption of normality of a given empirical distribution makes it possible to describe this “real-world” distribution based on what we know about the (theoretical) normal curve

14. THE NORMAL CURVE
.68 of area under the curve (.34 on each side of mean) falls within 1 standard deviation (s) of the mean
In other words, 68% of cases fall within +/- 1 s
95% of cases fall within 2 s’s
99% of cases fall within 3 s’s

15. Areas Under the Normal Curve
Because the normal curve is symmetrical, we know that 50% of its area falls on either side of the mean.
FOR EACH SIDE:
34.13% of scores in distribution are b/t the mean and 1 s from the mean
13.59% of scores are between 1 and 2 s’s from the mean
2.28% of scores are > 2 s’s from the mean

16. THE NORMAL CURVE Example:
Male height = normally distributed, mean = 70 inches, s = 4 inches
What is the range of heights that encompasses 99% of the population?
Hint: that’s +/- 3 standard deviations
Answer: 70 +/- (3)(4) = /- 12
Range = 58 to 82

17. THE NORMAL CURVE & Z SCORES
To use the normal curve to answer questions, raw scores of a distribution must be transformed into Z scores
Z scores:
Formula: Zi = Xi – X s
A tool to help determine how a given score measures up to the whole distribution
RAW SCORES:
Z SCORES:

18. NORMAL CURVE & Z SCORES Transforming raw scores to Z scores
a.k.a. “standardizing”
converts all values of variables to a new scale:
mean = 0
standard deviation = 1
Converting raw scores to Z scores makes it easy to compare 2+ variables
Z scores also allow us to find areas under the theoretical normal curve

19. Z SCORE FORMULA Z = Xi – X S10
Xi = 80, S = 10
Xi = 112, S = 10
Xi = 95; X = 86; s=7
Z= 80 – 100 = -2.00 10
Z = 112 – 100 = 1.20
Z= 95 – 86 = 1.29 7

20. USING Z SCORES FOR COMPARISONS
Example 1:
An outdoor magazine does an analysis that assigns separate scores for states’ “quality of hunting” (MN = 81) & “quality of fishing” (MN =74). Based on the following information, which score is higher relative to other states?
Formula: Zi = Xi – X s
Quality of hunting for all states: X = 69, s = 8
Quality of fishing for all states: X = 65, s = 5
Z Score for “hunting”:
81 – 69 = 1.5 8
Z Score for “fishing”:
73 – 65 = 1.6 5
CONCLUSION: Relative to other states, Minnesota’s “fishing” score was higher than its “hunting” score.

21. USING Z SCORES FOR COMPARISONS
Example 2:
You score 80 on a Sociology exam & 68 on a Philosophy exam. On which test did you do better relative to other students in each class?
Formula: Zi = Xi – X s
Sociology: X = 83, s = 10
Philosophy: X = 62, s = 6
Z Score for Sociology:
80 – 83 = - 0.3 10
Z Score for Philosophy:
68 – 62 = 1 6
CONCLUSION: Relative to others in your classes, you did better on the philosophy test

22. Normal curve table
For any standardized normal distribution, Appendix A (p ) of Healey provides precise info on:
the area between the mean and the Z score (column b)
the area beyond Z (column c)
Table reports absolute values of Z scores
Can be used to find:
The total area above or below a Z score
The total area between 2 Z scores

23. THE NORMAL DISTRIBUTION
Area above or below a Z score
If we know how many S.D.s away from the mean a score is, assuming a normal distribution, we know what % of scores falls above or below that score
This info can be used to calculate percentiles

24. AREA BELOW Z
EXAMPLE 1: You get a 58 on a Sociology test. You learn that the mean score was 50 and the S.D. was 10.
What % of scores was below yours?
Zi = Xi – X = 58 – 50 = 0.8 s

25. AREA BELOW Z What % of scores was below yours?
Zi = Xi – X = 58 – 50 = 0.8 s
Appendix A, Column B (28.81%) of area of normal curve falls between mean and a Z score of 0.8
Because your score (58) > the mean (50), remember to add .50 (50%) to the above value
.50 (area below mean) (area b/t mean & Z score) = (78.81% of scores were below yours)
YOUR SCORE WAS IN THE 79TH PERCENTILE
FIND THIS AREA
FROM COLUMN B

26. AREA BELOW Z Example 2: Zi = Xi – X = 44 – 50 = - 0.6 s 10
Your friend gets a 44 (mean = 50 & s=10) on the same test
What % of scores was below his?
Zi = Xi – X = 44 – 50 = - 0.6 s

27. AREA BELOW Z What % of scores was below his?
Z = Xi – X = 44 – 50= -0.6 s
Appendix A, Column C (27.43%) of area of normal curve is under a Z score of -0.6
.2743 (area beyond [below] his Z score) 27.43% of scores were below his
YOUR FRIEND’S SCORE WAS IN THE 27TH PERCENTILE
FIND THIS AREA
FROM COLUMN C

28. Z SCORES: “ABOVE” EXAMPLE
Sometimes, lower is better…
Example: If you shot a 68 in golf (mean=73.5, s = 4), how many scores are above yours?
68 – 73.5 = 4
Appendix A, Column B (41.47%) of area of normal curve falls between mean and a Z score of 1.37
Because your score (68) < the mean (73.5), remember to add .50 (50%) to the above value
.50 (area above mean) (area b/t mean & Z score) = (91.47% of scores were above yours)
FIND THIS AREA FROM COLUMN B

29. Area between 2 Z Scores
What percentage of people have I.Q. scores between Stan’s score of 110 and Shelly’s score of 125? (mean = 100, s = 15)
CALCULATE Z SCORES

30. AREA BETWEEN 2 Z SCORES
What percentage of people have I.Q. scores between Stan’s score of 110 and Shelly’s score of 125? (mean = 100, s = 15)
CALCULATE Z SCORES:
Stan’s z = .67
Shelly’s z = 1.67
Proportion between mean (0) & .67 = = 24.86%
Proportion between mean & 1.67 = = 45.25%
Proportion of scores between 110 and 125 is equal to:
45.25% – 24.86% = 20.39%

31. AREA BETWEEN 2 Z SCORES EXAMPLE 2:
If the mean prison admission rate for U.S. counties was 385 per 100k, with a standard deviation of 151 (approx. normal distribution)
Given this information, what percentage of counties fall between counties A (220 per 100k) & B (450 per 100k)?
Answers:
A: = =
B: = = 0.43
County A: Z of = = 36.21%
County B: Z of 0.43 = = 16.64%
Answer: = 52.85%

32. 4 More Sample Problems
For a sample of 150 U.S. cities, the mean poverty rate (per 100) is 12.5 with a standard deviation of 4.0. The distribution is approximately normal.
Based on the above information:
What percent of cities had a poverty rate of more than 8.5 per 100?
What percent of cities had a rate between 13.0 and 16.5?
What percent of cities had a rate between 10.5 and 14.3?
What percent of cities had a rate between 8.5 and 10.5?
PLEASE TURN THESE IN WITH YOUR NAME ON THEM…