1. Quantitative Methods in HPELS 440:210
Variability
Quantitative Methods in HPELS
440:210

2. Agenda Introduction Frequency Range Interquartile range
Variance/SD of population
Variance/SD of sample
Selection

3. Introduction Statistics of variability:
Describe how values are spread out
Describe how values cluster around the middle
Several statistics  Appropriate measurement depends on:
Scale of measurement
Distribution

4. Basic Concepts Measures of variability:
Frequency
Range
Interquartile range
Variance and standard deviation
Each statistic has its advantages and disadvantages

5. Agenda Introduction Frequency Range Interquartile range
Variance/SD of population
Variance/SD of sample
Selection

6. Frequency Definition: The number/count of any variable
Scale of measurement:
Appropriate for all scales
Only statistic appropriate for nominal data
Statistical notation: f

7. Frequency Advantages: Disadvantages: Ease of determination
Only statistic appropriate for nominal data
Disadvantages:
Terminal statistic

8. Calculation of the Frequency  Instat
Statistics tab
Summary tab
Group tab
Select group
Select column(s) of interest OK

9. Agenda Introduction Frequency Range Interquartile range
Variance/SD of population
Variance/SD of sample
Selection

10. Range
Definition: The difference between the highest and lowest values in a distribution
Scale of measurement:
Ordinal, interval or ratio

11. Range Advantages: Disadvantages: Ease of determination
Terminal statistic
Disregards all data except extreme scores

12. Calculation of the Range  Instat
Statistics tab
Summary tab
Describe tab
Calculates range automatically OK

13. Agenda Introduction Frequency Range Interquartile range
Variance/SD of population
Variance/SD of sample
Selection

14. Interquartile Range
Definition: The difference between the 1st quartile and the 3rd quartile
Scale of measurement:
Ordinal, interval or ratio
Example: Figure 4.3, p 107

16. Interquartile Range Advantages: Disadvantages: Ease of determination
More stable than range
Disadvantages:
Disregards all values except 1st and 3rd quartiles

17. Calculation of the Interquartile Range  Instat
Statistics tab
Summary tab
Describe tab
Choose additional statistics
Choose interquartile range OK

18. Agenda Introduction Frequency Range Interquartile range
Variance/SD of population
Variance/SD of sample
Selection

19. Variance/SD  Population
The average squared distance/deviation of all raw scores from the mean
The standard deviation squared
Statistical notation: σ2
Scale of measurement:
Interval or ratio
Advantages:
Considers all data
Not a terminal statistic
Disadvantages:
Not appropriate for nominal or ordinal data
Sensitive to extreme outliers

20. Variance/SD  Population
Standard deviation:
The average distance/deviation of all raw scores from the mean
The square root of the variance
Statistical notation: σ
Scale of measurement:
Interval or ratio
Advantages and disadvantages:
Similar to variance

21. Calculation of the Variance  Population
Why square all values?
If all deviations from the mean are summed, the answer always = 0

22. Calculation of the Variance  Population
Example: 1, 2, 3, 4, 5
Mean = 3
Variations:
1 – 3 = -2
2 – 3 = -1
3 – 3 = 0
4 – 3 = 1
5 – 3 = 2
Sum of all deviations = 0
Sum of all squared deviations
Variations:
1 – 3 = (-2)2 = 4
2 – 3 = (-1)2 = 1
3 – 3 = (0)2 = 0
4 – 3 = (1)2 = 1
5 – 3 = (2)2 = 4
Sum of all squared deviations = 10
Variance = Average squared deviation of all points  10/5 = 2

23. Calculation of the Variance  Population
Step 1: Calculate deviation of each point from mean
Step 2: Square each deviation
Step 3: Sum all squared deviations
Step 4: Divide sum of squared deviations by N

24. Calculation of the Variance  Population
σ2 = SS/number of scores, where SS =
Σ(X - )2
Definitional formula (Example 4.3, p 112) or
ΣX2 – [(ΣX)2]
Computational formula (Example 4.4, p 112)

25. Computational formula
Step 4: Divide by N

26. Computation of the Standard Deviation  Population
Take the square root of the variance

27. Agenda Introduction Frequency Range Interquartile range
Variance/SD of population
Variance/SD of sample
Selection

28. Variance/SD  Sample Process is similar with two distinctions:
Statistical notation
Formula

29. Statistical Notation Distinctions Population vs. Sample
N = n

30. Formula Distinctions Population vs. Sample
s2 = SS / n – 1, where SS =
Σ(X - M)2
Definitional formula
ΣX2 - [(ΣX)2]
Computational formula
Why n - 1?

31. N vs. (n – 1)  First Reason
General underestimation of population variance
Sample variance (s2) tend to underestimate a population variance (σ2)
(n – 1) will inflate s2
Example 4.8, p 121

32. Actual population σ2 = 14
Average biased s2 = 63/9 = 7
Average unbiased s2 = 126/9 = 14

33. N vs. (n – 1)  Second Reason
Degrees of freedom (df)
df = number of scores “free” to vary
Example:
Assume n = 3, with M = 5
The sum of values = 15 (n*M)
Assume two of the values = 8, 3
The third value has to be 4
Two values are “free” to vary
df = (n – 1) = (3 – 1) = 2

34. Computation of the Standard Deviation of Sample  Instat
Statistics tab
Summary tab
Describe tab
Calculates standard deviation automatically OK

35. Agenda Introduction Frequency Range Interquartile range
Variance/SD of population
Variance/SD of sample
Selection

36. Selection When to use the frequency
Nominal data
With the mode
When to use the range or interquartile range
Ordinal data
With the median
When to sue the variance/SD
Interval or ratio data
With the mean

37. Textbook Problem Assignment
Problems: 4, 6, 8, 14.