The Mean Variance Standard Deviation and Z-Scores

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

14. Calculate SD2 & SD Variance SD2 = ∑(X-M)2 N SD2 = 66 10 SD2 = 6.60 Step 5

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.