Central Tendency and Variability

Chapter 2 Central Tendency and Variability

Instructor's Summary of Chapter

Central tendency. The mean is the ordinary average—the sum of the scores divided by the number of scores. Expressed in symbols, M = ΣX/N. Less commonly used indicators of central tendency of a distribution include the mode (the most common single value) and the median (the value of the middle case if you were to line up all the scores from lowest to highest). Variability. The spread of the scores in a distribution can be described by the variance—the average of the squared deviation of each score from the mean: SD2 = Σ(X–M)2/N. Alternatively: SD2 = SS/N, where SS is the sum of squared deviations. The standard deviation is the positive square root of the variance: SD = √SD2. It can be best understood as an approximate measure of the average amount that scores differ from the mean. Controversy: The tyranny of the mean. Some psychologists, especially those associated with behaviorism, humanistic psychology, phenomenology, and qualitative methods, mistrust statistical processes because in the process of creating averages, knowledge about the individual case is lost. Even many who hold these viewpoints, however, acknowledge that statistical analysis does play an important role, but argue that when studying any particular topic, careful study of individuals should come first. How the procedures of this chapter are reported in research articles. Means and standard deviations are commonly reported in research articles, both in the text and in tables. Box 2-1. The sheer joy (yes, joy) of statistical analysis. When several social psychologists were interviewed, many reported that their most enjoyable day-to-day research activity was analyzing data. Box 2-2. Gender, ethnicity, and math performance. Reviews research and thinking on gender and ethnic differences in math and statistics performance, emphasizing lack of differences in underlying abilities.

List of Transparencies

2.1 Stress Ratings Example: (A) calculation of mean, and (B) mean as a balancing point (Figure 2-3) 2.2 Live-in-Country Example: (A) calculation of mean, and (B) calculation of sum of squares, variance, and standard deviation 2.3 Horn Honking Example: (A) calculation of mean, and (B) calculation of mode and median 2.4 Generic Comparisons of Mean and Median 2.5 Horn Honking Examples With and Without an Outlier: (A) calculation of mean and median, and (B) calculation of sum of squares, variance, and standard deviation 2.6 Locations of Mean, Mode, and Median in Different Distributions (Figure 2-9) 2.7 Distributions with (A) same mean but different variance, and (B) different mean but same variance (Figure 2-10) 2.8 Steps for Calculating Variance with Formula 2.9 Review of Central Tendency and Variability

Materials

Lecture outline Transparencies 1.3, 1.7, 1.11, and 2.1 through 2.6

Outline for Blackboard

I. Review/Last Assignment II. Describing Central Tendency III. The Mean: M = ΣX/N IV. The Mode V. The Median VI. Comparing Mean, Mode, Median VII. Review This Class

Instructor's Lecture Outline

I. Review/Last Assignment A. Idea and importance of descriptive statistics. B. Describing a distribution using frequency tables. C. Describing a distribution using histograms and bar graphs.

II. Describing Central Tendency A. Principle: Summarize a distribution of scores as a single number. B. Show TRANSPARENCIES 1.3 and 1.7. C. Previously, we summarized this group of numbers into a table and graph. Now we want to summarize it into a single number that is representative of all the numbers.

III. The Mean: M = ΣX/N A. The mean is the arithmetic average—sum of scores divided by number of scores. B. Usually, it is the best measure for describing the typical value of a distribution. C. Show TRANSPARENCY 2.1: 1. Emphasize value of symbols in statistics and importance of mastering them. 2. Discuss each symbol in formula for mean. 3. Emphasize that the mean is a balancing point. D. Live-in-country example: Show and discuss TRANSPARENCY 2.2.A. E. Horn honking example: Show and discuss TRANSPARENCY 2.3.A.

IV. The Mode A. The mode is the value with the most scores. B. Figuring the mode: Show TRANSPARENCY 2.3.B: 1. Make a frequency table. 2. Find the value/category with the highest frequency. C. The mode is most useful when: 1. There are relatively few values. 2. The values are categories (that is, it is a distribution of a nominal variable). D. Show TRANSPARENCY 1.11. Note that the mode in this example is "romantic partner" because it is the most frequent category.

V. The Median A. The median is the middle score when scores are organized from lowest to highest. B. Figuring the median: Show TRANSPARENCY 2.3.B: 1. Line up all the scores from lowest to highest. 2. Figure how many scores there are to the middle score by adding 1 to the number of scores and dividing by 2. 3. Count up to the middle score or scores. If you have one middle score, this is the median. If you have two middle scores, the median is the average (the mean) of these scores. C. The median is most useful when: 1. The variable is ordinal (rank-ordered). 2. The distribution has a few extreme scores (called outliers). D. Generic example comparisons: Show TRANSPARENCY 2.4:

Copyright © 2013, 2009, 2006, 2003, 1999, 1994 Pearson Education, Inc., Upper Saddle River, NJ 07458. All rights reserved. Page 3 1. Problem with the mean is that it is highly influenced by extreme scores. 2. The median, though, is not affected by extreme scores. E. Horn honking example: Show and discuss TRANSPARENCY 2.5.A.

VI. Comparing Mean, Mode, Median A. Show and discuss TRANSPARENCY 2.6: 1. When a distribution is normal, the mean, mode, and median are in the same location. 2. When a distribution is skewed, the mean is closest to the tail, the mode is at the highest peak, and the median falls between the two.

VII. Review This Class: Use Blackboard outline. Lecture 2.2: Variability

Materials

Lecture outline Transparencies 2.1, 2.2, 2.5, and 2.7 through 2.9

Outline for Blackboard

I. Review/Last Assignment II. Variation III. Deviation Scores and Squared Deviation Scores IV. The Variance: SD2 = Σ(X–M)2/N = SS/N V. The Standard Deviation: SD = √SD2 VI. Miscellaneous Issues VII. Review This Class

Instructor's Lecture Outline

I. Review/Last Assignment A. Idea and importance of descriptive statistics. B. Describing a distribution using frequency tables and graphs. C. Using the mean as a representative value: M = ΣX/N: Show TRANSPARENCY 2.1.

II. Variation A. A distribution can be characterized by how much the scores in it vary from each other—they could be bunched closely together or spread out, for example. B. Knowing the mean and variation in a distribution gives a much better sense of how scores are distributed than the mean alone. C. Examples of possible situations in which the mean SAT is 600 for entering students at a particular college. 1. All have almost exactly 600—there is little variation. 2. About half have SATs of 400 and half of 800—there is great variation. 3. About equal numbers have SATs of 500, 550, 600, 650, 700—there is a moderate amount of variation.

Copyright © 2013, 2009, 2006, 2003, 1999, 1994 Pearson Education, Inc., Upper Saddle River, NJ 07458. All rights reserved. Page 5 D. In general, the amount of variation and mean are independent of each other. E. Show and discuss TRANSPARENCY 2.7.

III. Deviation Scores and Squared Deviation Scores A. One way of determining variation focuses on the extent to which scores differ from the mean. B. A deviation score is the score minus the mean. If a score is 28 and the mean is 25, the deviation score is 3. If a score is 22 and the mean is 25, the deviation score is –3. C. Over an entire distribution, positive and negative deviation scores balance each other out (so they sum to zero). D. For this and other more complicated reasons, we emphasize squared deviations.

2 2 IV. The Variance: SD = Σ(X–M) /N = SS/N A. A widely used measure of variation in a group of scores. B. The average of the squared deviation scores. C. Review steps and formula for calculating the variance. Show TRANSPARENCY 2.8: 1. Subtract the mean from each score. (This gives each score's deviation score.) 2. Square each of these deviation scores. (This gives each score's squared deviation score.) 3. Add up the squared deviation scores. (This gives the sum of squared deviations or SS.) 4. Divide the sum of squared deviations by the number of scores. (This gives the variance, SD2.) D. Live-in-country example: Show and discuss TRANSPARENCY 2.2.B. E. Horn honking example: Show and discuss TRANSPARENCY 2.5.B. F. The problem with the variance is that it is the average amount of squared differences of scores from the mean. So it can be much larger than the average amount that scores differ from the mean. 2 V. The Standard Deviation: SD = √SD A. The positive square root of the variance. B. Approximately the average amount that scores differ from the mean. C. The most widely used descriptive statistic for describing variation. D. Live-in-country example: Show and discuss TRANSPARENCY 2.2.B. E. Horn honking example: Show and discuss TRANSPARENCY 2.5.B.

VI. Miscellaneous Issues A. Definitional formulas, like those you will learn in this class, illustrate the basic logic of what you are calculating or trying to figure out. B. Computational formulas are shortcut methods used in calculations; though mathematically equivalent to definitional formulas, they obscure the basic logic. C. Explaining variability is important in psychological research. Essentially, we are trying to discover those factors that are responsible for differences in the variables of interest. D. Most researchers calculate variance using N–1 rather than N (as you'll learn later on).

VII. Review this Class: Use Blackboard outline and show TRANSPARENCY 2.9.

Stress Ratings Example (from text)

8, 7, 4, 10, 8, 6, 8, 9, 9, 7, 3, 7, 6, 5, 0, 9, 10, 7, 7, 3, 6, 7, 5, 2, 1, 6, 7, 10, 8, 8

Sum of ratings = 193 Number of ratings = 30 Mean = 193 / 30 = 6.43 TRANSPARENCY 2.2

Live-in-Country Example (class questionnaire data)

Question: "Would you prefer to live out in the country with not many people around?"

1 2 3 4 5 6 7 Not at All Moderately Extremely

7, 5, 1, 4, 7, 7, 5, 1, 3, 4, 3, 4, 6, 2, 4, 4, 2, 3, 4, 1, 4, 2, 3, 2, 1, 3, 2, 3, 4, 6, 4, 2, 6, 4, 2, 1, 1, 4, 6, 2, 5, 6, 1, 3, 2, 2, 4, 4, 4, 5, 4, 4, 7, 5, 5, 5, 1, 6, 2, 1, 2, 6, 6, 3, 4, 5, 1, 1, 5, 2, 3, 5, 7, 4, 2, 4, 5, 4, 4, 7, 4, 1, 5, 3, 4, 4, 4, 3, 5, 2, 2, 3, 4, 7, 1, 4, 7, 1, 3, 3, 4, 5, 6, 4, 1, 2, 7, 1, 4, 2

Horn Honking Example (fictional data based on Kenrick & McFarland, 1986)

Interpersonal hostility measured as delay in seconds for 29 cars before honking horn at stalled car after the light has changed to green.

3.5, 2.0, 0.0, 5.0, 0.5, 1.0, 4.0, 3.5, 3.0, 1.5, 1.5, 2.0, 2.5, 3.0, 3.0, 3.5, 4.5, 2.0, 2.5, 4.5, 4.0, 3.5, 3.0, 2.5, 2.5, 3.5, 3.5, 4.0, 3.0 TRANSPARENCY 2.4

Generic Comparison of Mean and Median

1 1 1 1 1 8 9 9 9 9 9  median = 8

7 7 7 7 7 8 50 50 50 50 50  median = 8

Horn Honking Example (fictional data based on Kenrick & McFarland, 1986)

Based on data previously presented: With one additional case of 13 sec: TRANSPARENCY 2.6

Steps for Calculating Variance

 Subtract the mean from each score.

 Square each of these deviation scores.

 Add up the squared deviation scores.

 Divide the sum of squared deviations by the number of scores.

Formula

Review of Central Tendency and Variability

A. Purpose: Reduce a group of numbers—a distribution—to some simple terms— descriptive statistics.

B. Central Tendency: A single number that represents the typical score in a distribution.

1. MEAN is the arithmetic average. (It is the most widely used and generally the best indicator.) a. Add up the scores. b. Divide sum by number of scores. FORMULA AND SYMBOLS: M = ΣX/N

2. MODE is the most frequently occurring score.

3. MEDIAN is the score(s) in the exact middle of the distribution.

C. Variability: A single number that describes how much variation there is in a group of numbers—that is, how spread out or narrow a distribution is.

1. VARIANCE is average of squared deviations from the mean. a. Subtract the mean from each score. b. Square each of these deviation scores. c. Add up the squared deviation scores. d. Divide the sum of squared deviations by the number of scores. FORMULA AND SYMBOLS: SD2 = Σ(X–M)2/N

2. STANDARD DEVIATION is the positive square root of variance. (Intuitively, it is roughly the average amount each score differs from the mean.) a. Compute variance. b. Take the positive square root. FORMULA AND SYMBOLS: SD = √SD2