The Mean Variance Standard Deviation and Z-Scores Chapter 2

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

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 ??

High Variability Almost Flat Low Variability Scores close too µ

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.

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

Definitional Formula

Computational Formula

Variance Sample variance Population variance

8-4 = 4 X 42

Variance Continued

Standard Deviation (SD)

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

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

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

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

Effects of μ and σ

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?

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.

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.