1. Numerical Statistics Measures of Variability
Packet #22

2. Measures in Variability
Why is it important?
Used to better describe data
If a researcher has two samples/populations with the same mean, how are they different?
Can be determined with measures of variability using the range for example
These measures quantify the extent of dispersion in one’s data
Basically, looking at how far apart certain numbers are and why.
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3. Measures in Variability
Range
The difference between the largest and smallest values [+1].
The range measures the spread of the two extreme values.
Interquartile Range
Difference between the 75th and 25th percentile
The range of the middle 50% of scores
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4. Range II Range = Highest – Lowest Range = Highest – Lowest plus 1
Ordered Array 98 80 94 92 78 90 75 86 72 84 70 83 68 82 62 58 36

5. Interquartile Range II
Interquartile range is the score representing the 75th percentile minus the score at the 25th percentile
84 – 70 = 14
Raw Score
Percentile Rank 98
100 80 50 94 95 92 90 78 40 85 75 35 86 72 30 84 70 25 83 68 20 82 65 62 15 58 10 36 5

6. Interquartile Range III
If there is no specific score that falls at the 75th or 25th percentile, one must estimate it through extrapolation.

7. Interquartile Range IV
75th percentile
Determine where in the distribution the 75th percentile would be
Determine how many percentages fall between the known percentages
80 – 60 = 20 percentiles/ticks
Subtract the lower score from the higher score and divide the answer from by the answer of the previous step
(8 – 7) / 20 = 1/20 or 0.05
Percentile
Score
100 10 90 9 80 8 60 7 50 6 40 5 2

8. Interquartile Range V 75th percentile continued
Determine how many “ticks” one will need to either go up, from the lower percentile, or down, from the upper percentile
80 is the closet percentile to 75
Therefore, one must move down 5 ticks
At each tick, one must remove, if moving down, or add, if moving up, the answer from step #3 %
Score 80
8.00 79
8 – 0.05 = 7.95 78
7.95 – 0.05 = 7.90 77
7.85 76
7.80 75
7.75

9. Interquartile Range VI
25th percentile
Utilize the same steps used when solving for the 75th percentile
Percentile
Score
100 10 90 9 80 8 60 7 50 6 40 5 2

10. Interquartile Range VII
25th percentile
Falls between the 10th and 40th percentile
Therefore, the score MUST be between 2 and 5
40 – 10 = 30 percentiles or ticks
Each “tick” is worth
(5 – 2)/30 = 3/30 = 1/10 = 0.1
Percentile
Score 4 30
3.9 29
3.8 28
3.7 27
3.6 26
3.5 25
3.4 24
3.3 23
3.2 22
3.1 21 3 20

11. Interquartile Range VIII
25th percentile
Go up or go down
Does not matter since the distance, to the 25 percentile, from 40 or 10 is 15
This time, the researcher will go UP from the 10th percentile
Percentile
Score 4 30
3.9 29
3.8 28
3.7 27
3.6 26
3.5 25
3.4 24
3.3 23
3.2 22
3.1 21 3 20

12. Interquartile Range IX%
Score 10
2.0 11
2.1 12
2.2 13
2.3 14
2.4 15
2.5 16
2.6 17
2.7 %
Score 18
2.8 19
2.9 20
3.0 21
3.1 22
3.2 23
3.3 24
3.4 25
3.5

13. Measures in Variability II
Deviation
These tell you how far away the “raw score” (the value that you are looking at) is from the mean
x - x
X - mean (sample data)
x - 
X - mu (population data)
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14. Measures in Variability III
Standard Deviation (Variation)
This is the average of all deviations, for all “raw scores,” from your mean
s = standard deviation (sample)
 (pronounced sigma)= standard deviation (population)
Equation for Standard Variation (Deviation)
√ (( (x – mean )2) / (N – 1))
√ (SS / (N – 1))
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15. Measures in Variability IV
Variance
The degree of spread within the distribution
The larger the spread, the larger the variance
This is simply the square of the standard deviation
NOT THE DEVIATION, BUT THE STANDARD DEVIATION
s2 (s squared) denotes the variance of a sample
2 denotes the variance of a population
Equation for Variance
( (x – mean )2) / (N – 1)
SS / (N – 1)
The square root sign is removed since the standard deviation is squared.
Will be written on board to be clearer
**The sum of the squared deviations of n measurements from their mean divided by (n-1)
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16. Variance Chart x (x – x) (x – x)2 2 3.2 -1.2 1.44 3 -0.2 0.04 4 0.8
0.64 5
1.8
3.24 16
6.8

17. Standard Deviation & Variance
√ (( (x – mean )2) / (N – 1))
6.8 / 4 = 1.7
Variance s2
2.89