1. The Mean Variance Standard Deviation and Z-Scores
Chapter 2

2. Chapter Outline Representative Values Variability
Mean, Variance, Standard Deviation, and Z Scores in Research Articles

3. High or Low Variability
Data Sets:
Data 1: 10, 10, 10, 10, 11 ??
Data 2: 10, 11, 13, 15, 17 ??
Data 3: 10, 20, 30, 40, 50 ??
Data 4: 10, 50, 90, 100, 150 ??

4. High
Variability
Almost Flat
Low
Variability
Scores close
too µ

5. Standard Deviation (SD)
Variance (SD2)
Measure of how spread out a set of scores are
average of the squared deviations from the mean
Standard Deviation (SD)
Most widely used way of describing the spread of a group of scores
the positive square root of the variance
the average amount the scores differ from the mean
To calculate SD:
Take the square root of SD2.

6. Formulas for SD2 and SD Standard Deviation (SD): Variance (SD2):
average of the squared deviations from the mean
SD2 = ∑(X-M)2 N
Standard Deviation (SD):
√SD2

7. Definitional Formula

8. Computational Formula

9. Variance
Sample
variance
Population
variance

10. 8-4 = 4 X 42

11. Variance Continued

12. Standard Deviation (SD)

13. Computing SD2 & SD Step 1: Data Step 2 Deviation scores: X - M X
SD2 = ∑(X-M)2 N
Step 1:
Data
Step 2
Deviation scores:
X - M
7 – 6 = 1
8 – 6 = 2
3 – 6 = -3
1 – 6 = -5
6 – 6 = 0
9 – 6 = 3 X 7 8 3 1 6 9

14. Calculate SD2 & SD Variance SD2 = ∑(X-M)2 SD2 = 66 SD2 = 6.60 Step 5 N10
SD2 = 6.60

15. Amount of Variation and Mean are Independent
Can have a distribution with same means BUT DIFFERENT SDs
Can have a Distribution with same
SDs BUT DIFFERENT MEANS

16. Variability How spread out the scores are in a distribution
amount of spread of the scores around the mean
Distributions with the same mean can have very different amounts of spread around the mean.
Mean1 = 50, SD = 3
Mean2 = 50, SD = 20
Distributions with different means can have the same amount of spread around the mean.
Mean1 = 25, SD = 3
Mean2 = 50, SD = 3

17. Effects of μ and σ

19. How Are You Doing?
What do the SD2 and SD tell you about a distribution of scores?
What are the formulas for finding the variance and standard deviation of a group of scores?

20. Key Points
The mean (M = (∑X) / N) is the most commonly used way of describing the representative value of a group of scores.
The mode (most common value) and the median (middle value) are other types of representative values.
Variability refers to the spread of scores on a distribution.
Variance and standard deviation are used to describe variability.
The variance is the average of the squared deviations of each score from the mean ([∑ (X-M)2] / N).
The standard deviation is the square root of the variance(√SD2).
A Z score is the number of standard deviations that a raw score is above or below the mean (Z = (X-M) / SD).
Means and standard deviations are often reported in research articles.

21. Key Points
The mean (M = (∑X) / N) is the most commonly used way of describing the representative value of a group of scores.
The mode (most common value) and the median (middle value) are other types of representative values.
Variability refers to the spread of scores on a distribution.
Variance and standard deviation are used to describe variability.
The variance is the average of the squared deviations of each score from the mean ([∑ (X-M)2] / N).
The standard deviation is the square root of the variance(√SD2).
A Z score is the number of standard deviations that a raw score is above or below the mean (Z = (X-M) / SD).
Means and standard deviations are often reported in research articles.