1. Variability
Pick up little assignments from Wed. class.

2. Variability Variability Example
How tightly clustered or how widely dispersed the values are in a data set.
Example
Data set 1: [0,25,50,75,100]
Data set 2: [48,49,50,51,52]
Both have a mean of 50, but data set 1 clearly has greater Variability than data set 2.

3. Variability: The Range
The Range is one measure of variability
The range is the difference between the maximum and minimum values in a set
Example
Data set 1: [1,25,50,75,100]; R: = 100
Data set 2: [48,49,50,51,52]; R: = 5
The range ignores how data are distributed and only takes the extreme scores into account
RANGE = (Xlargest – Xsmallest) + 1

4. Quartiles Split Ordered Data into 4 Quarters = first quartile
= second quartile= Median
= third quartile
25%
25%
25%
25%

5. Quartiles Md Q1 Q3
75%
25%

6. Variability: Interquartile Range
Difference between third & first quartiles
Interquartile Range = Q3 - Q1
Spread in middle 50%
Not affected by extreme values

7. Standard Deviation and Variance
How much do scores deviate from the mean?
deviation =
Why not just add these all up and take the mean? X
X- 1 6
 = 

8. Standard Deviation and Variance
Solve the problem by squaring the deviations! X
X-
(X-)2 1 -1 -2 4 6 +4 16
5.5
 = 2
Variance =

9. Standard Deviation and Variance
Higher value means greater variability around 
Critical for inferential statistics!
But, not as useful as a purely descriptive statistic
hard to interpret “squared” scores!
Solution  un-square the variance!
Standard Deviation =

10. Variability: Standard Deviation
The Standard Deviation tells us approximately how far the scores vary from the mean on average
estimate of average deviation/distance from 
small value means scores clustered close to 
large value means scores spread farther from 
Overall, most common and important measure
extremely useful as a descriptive statistic
extremely useful in inferential statistics
The typical deviation in a given distribution

11. Variability: Standard Deviation
Standard Deviation can be calculated with the sum of squares (SS) divided by n

12. Sample variance and standard deviation
Sample will tend to have less variability than popl’n
if we use the population fomula, our sample statistic will be biased
will tend to underestimate popl’n variance

13. Sample variance and standard deviation
Correct for problem by adjusting formula
Different symbol: s2 vs. 2
Different denominator: n-1 vs. N
n-1 = “degrees of freedom”
Everything else is the same
Interpretation is the same

14. Definitional Formula:
Variance:
deviation
squared-deviation
‘Sum of Squares’ = SS
degrees of freedom
Standard
Deviation:

15. Variability: Standard Deviation
let X = [3, 4, 5 ,6, 7]
M = 5
(X - M) = [-2, -1, 0, 1, 2]
subtract M from each number in X
(X - M)2 = [4, 1, 0, 1, 4]
squared deviations from the mean
S (X - M)2 = 10
sum of squared deviations from the mean (SS)
S (X - M)2 /n-1 = 10/5 = 2.5
average squared deviation from the mean
S (X - M)2 /n-1 = = 1.58
square root of averaged squared deviation

16. Variability: Standard Deviation
let X = [1, 3, 5, 7, 9]
M = 5
(X - M) = [-4, -2, 0, 2, 4 ]
subtract M from each number in X
(X - M)2 = [16, 4, 0, 4, 16]
squared deviations from the mean
S (X - M)2 = 40
sum of squared deviations from the mean (SS)
S (X - M)2 /n-1 = 40/4 = 10
average squared deviation from the mean
S (X - M)2 /n-1 = = 3.16
square root of averaged squared deviation

17. In class example
Work on handout

18. Standard Deviation & Standard Scores
Z scores are expressed in the following way
Z scores express how far a particular score is from the mean in units of standard deviation

19. Standard Deviation & Standard Scores
Z scores provide a common scale to express deviations from a group mean

20. Standard Deviation and Standard Scores
Let’s say someone has an IQ of 145 and is 52 inches tall
IQ in a population has a mean of 100 and a standard deviation of 15
Height in a population has a mean of 64” with a standard deviation of 4
How many standard deviations is this person away from the average IQ?
How many standard deviations is this person away from the average height?
3 each, positive and negative

21. Homework
Chapter 4
8, 9, 11, 12, 16, 17