1. Basic Statistics
Measures of Variability

2. Measures of Variability
The Range Deviation Score The Standard Deviation The Variance

3. STRUCTURE OF STATISTICS
TABULAR
Continuing with numerical approaches.
DESCRIPTIVE
GRAPHICAL
NUMERICAL
STATISTICS
NUMERICAL
CONFIDENCE INTERVALS
INFERENTIAL
TESTS OF HYPOTHESIS

4. STRUCTURE OF STATISTICS NUMERICAL DESCRIPTIVE MEASURES
TABULAR
CENTRAL TENDENCY
DESCRIPTIVE
GRAPHICAL
NUMERICAL
VARIABILITY
SYMMETRY

5. STRUCTURE OF STATISTICS NUMERICAL DESCRIPTIVE MEASURES
CENTRAL TENDENCY
RANGE
VARIABILITY
VARIANCE
NUMERICAL
SYMMETRY
STANDARD DEVIATION

6. Do you think your project will succeed ?
We need the variability of IQs in the class!
You are an elementary school teacher who has been assigned a class of fifth graders whose mean IQ is 115. Because children with IQ of 115 can handle more complex, abstract material, you plan many sophisticated projects for the year.
Do you think your project will succeed ?
General population
85% 55 70 85
100
115
130
145

7. Can you conclude that the pay for two occupations is equal?
Is the average salary enough? We need the variability!
Having graduated from college, you are considering two offers of employment. One in sales and the other in management. The pay is about same for both. After checking out the statistics for salespersons and managers at the library, you find that those who have been working for 5 years in each type of job also have similar averages.
Can you conclude that the pay for two occupations is equal?
Sales
management
Much more
Much less
$20,000

8. Single score Group of scores Central Tendency measures Mean IQ=118
IQ of 100 students

9. ? ? ?
More homogeneous
Central Tendency Measures
Measures of Central Tendency do not tell you the differences that exist among the scores

10. Same Mean---Different Variability So What?4
How many are out here? 2 1 3 60
Central Tendency

11. 1. The Range = The difference between the largest (Xmax) and the smallest (Xmin).
25, 21, 22, 23, 28, 26, 24, 29 21 22 23 24 25 26 28 29
Range = 29 –21 = 8
A large range means there is a lot of variability in data.

12. ? Range = 29 –10 = 19 Drawbacks of The Range Range = 29 –26=3
10, 28, 26, 27, 29 ?
Range = 29 –26=3 10 26 27 28 29
Range = 29 –10 = 19
The Range depends on only the two extreme scores

13. Range and Extreme Observations
Because the range is determined by just two scores in the group, it ignores the spread of all scores except the largest and smallest.
One aberrant score or outlier can be greatly increase the range

14. Before you determine the Range, all scores must be arranged in order
Range and Measurement Scales
Before you determine the Range, all scores must be arranged in order
Country
Code
SES F
Age
1, 2, 3,
1, 2, 3,
1, 2, 3,
1, 2, 3,
3-1=2
3-1=2
3-1=2
3-1=2
1=American 2=Asian 3=Mexican
1=Upper 2=Middle 3=Lower

15. Differences among Scores
3. The Variance
Differences among Scores 78 49 49 41 88 35 66 53 44 66 49 27 83 53 95 27
Differences among Scores 81 35 42 62 72 57 41 81 49 49 63 35 41 53 ? 77 35 78 88 66

16. How can you determine the variability of each individual in group?
Total Variability = Sum of Individual Variability
How can you determine the variability of each individual in group? 72 22 22 55 70 3 12 67
The amount of Individual difference entirely depends on comparison criteria.

17. Can you figure out total amount of differences among scores ?
Can you figure out how much each score is different from other scores ?
Can you figure out total amount of differences among scores ?

18. You need a Common Criteria for computing Total Variability46 48 47
Mean
Score 53 ? 49
Reference score 51 50 45 52

19. You need a Common Criteria for computing Total Variability-3 46 -1 48 -2 47
Deviation Scores 49 +4 53 ? 49
Reference score +2 51 50 +1 45 -4 +3 52
A Deviation score tells you that a particular score deviate, or differs from the mean

20. DEVIATION SCORE = (Xi - Mean)
A score a great distance from the mean will have large deviation score. 2 1 3
Mean A B C D E F

21. Sum of Deviation Scores
Total amount of variability?!
Sum of all distance values!
mathematically
No way!
conceptually

22. The idea makes sense…but
If you compute the sum of the deviation scores, the sum of the deviation scores equals zero!
Sum of Deviation scores = (-4) + (-3) + (-2) (1) + (2) + (3) + (4) = 0

23. The Sum of Absolute Deviation Scores
The sum of absolute deviations is rarely used as a measure of variability because the process of taking absolute values does not provide meaningful information for inferential statistics.

24. Sum of Squares of deviation scores “SS”
Conceptually
And
Mathematically

25. SS= Sum of Squares of Deviation Scores, SS
Instead of working with the absolute values of deviation scores, it is preferable to
(1) square each deviation score and
(2) sum them to obtain a quantity know as the Sum of Squares.
SS=(-4)+(-3)+(-2)+(-1)+0+(1)+(2)+(3)+(4) =
=60 2 2 2 2 2 2 2 2 2
SS= i

26. So !
Group of scores
“B”
Group of scores
“A”
SS(A)=30
SS(B)=40
Can you say that the variability of the data in Group B is greater than the data in Group A?

27. 3, 4 3, 4 =.50 What happens to SS when we look at some data? Group A
Group B
Mean = 3.5
Mean = 3.5 2 2 2 2
SS = ( ) + ( )
=.50
SS = ( ) + ( ) ( ) + ( )
=1.00 2 2

28. SS tends to increase as number of data(N) increase.
Mean i
i=1
SS tends to increase as number of data(N) increase.
SS is not appropriate for comparing variability among groups having unequal sample size.
How can you overcome the limitation of SS

29. The resulting value will be Mean of the Deviation Scores (Mean Square)
If SS is divided by N
The resulting value will be Mean of the Deviation Scores (Mean Square)
VARIANCE

30. 3, 4 3, 4 Group A Group B Mean = 3.5 Mean = 3.52 2 2 2
V = ( ) + ( )
=.50/2 = .25
V = ( ) + ( ) +
( ) + ( )
= 1.00/4 = 2.5 2 2

31. Variance
Population Variance
Sample Variance

32. POPULATION VARIANCE Individual value Population mean Sigma Square
Population size

33. SAMPLE VARIANCE Sample variance Sample size-1 Degree of freedom
Sample Mean
Individual value
Sample variance
Sample size-1
Degree of freedom
The sample variance (S2) is used to estimate the
population variance (2)

34. Why n-1 instead n ?

35. population
100 =
Sampling n
Sampling error
sample
?<100<?

36. Ideally, a sample variance would be based on (x - m)2
Ideally, a sample variance would be based on (x - m)2. This is impossible since m is not known if one has only a sample of n cases. m is substituted by .
The value of the squared deviations is less from X than from any other score .
Hence, in a sample, the value of (X-X) n would be less than n.
> n n

37. > Ideal sample variance
One could correct for this bias by dividing by a factor somewhat less than n

38. 7 sample n=5 We know that ? must equal 25.
If we know that the mean is equal to 5, and the first 4 scores add to 18, then the last score MUST equal 7.
We know that ? must equal 25.
n-1 are free to change
Degree of freedom

39. 4. Standard Deviation SD
Positive square root of the variance
Population
Sample

40. The Standard Deviation and the Mean with Normal Distribution

41. Relationship between m and s
Normal Distribution m
m-3s
m-2s
m-1s
m+1s
m+2s
m+3s
Relationship between m and s

42. S 68% 95% 99.9% Relationship between and Normal Distribution -3S -2S

43. EMPIRICAL RULE
For any symmetrical, bell-shaped distribution, approximately 68% of the observations will lie within  1 standard deviation of the mean; approximately 98% within  2 standard deviations of the mean; and approximately 99.9% within  3 standard deviations of the mean.

44. You can approximately reproduce your data!
If a set of data has a Mean=50 and SD=10, then…
68%
95%
99% 20 30 40 50 60 70 80