1. Chapter 3 For Explaining Psychological Statistics, 4th ed. by B. Cohen 1 Chapter 3: Measures of Central Tendency and Variability Imagine that a researcher is interested in measuring motivation on a 1 to 7 scale. The data in the table to the left are the raw motivation scores for 11 participants (i.e., N = 11). Note that there is more than one way to determine the center of a such a distribution of scores.

2. Chapter 3For Explaining Psychological Statistics, 4th ed. by B. Cohen 2 The Simplest Measure of Central Tendency Is the Mode –The mode is defined as the most commonly occurring score in the distribution. –The mode ignores a substantial amount of the data, which is why it is rarely used to summarize quantitative data. –Another drawback is that some distributions have two or more modes. –It is the only measure of central tendency that can be used with categorical level data. –The mode for the motivation data on the previous slide is 4, because it is the most frequently occurring score.

3. Chapter 3For Explaining Psychological Statistics, 4th ed. by B. Cohen 3 The Most Robust (Stable, Reliable) Measure of Central Tendency Is the Median –The median is defined as the score corresponding to the 50 th percentile of the distribution. –Median is not affected by extreme scores. –Use the median as a descriptive measure when the data are skewed or inexact at the extremes. –If a distribution is symmetric, the median will equal the mean. –The median for the motivation data is 4, because half of the scores are larger than 4 and half are smaller. –If a distribution is symmetric and unimodal, the median will equal the mode, as well as the mean.

4. Chapter 3Prepared by Samantha Gaies, M.A.4 Weighted Mean: where = each X score multiplied by its weight and = the total of all the weights The Most Commonly used Measure of Central Tendency for Inferential Statistics Is the Arithmetic Mean Arithmetic Mean:

5. Chapter 3For Explaining Psychological Statistics, 4th ed. by B. Cohen 5 Properties of the mean –If a constant is added to (or subtracted) from every score in a distribution, the mean is increased (or decreased) by that constant. –If every score is multiplied (or divided) by a constant, the mean will be multiplied (or divided) by that constant. –The sum of the deviations from the mean will always equal zero. –The sum of the squared deviations from the mean will be less than the sum of the squared deviations around any other point in the distribution.

6. Chapter 3For Explaining Psychological Statistics, 4th ed. by B. Cohen 6 Measures of Variability Variability refers to how spread out or scattered the scores in a distribution are. Measures of variability are often used in conjunction with measures of central tendency. The spread of the scores is usually measured with respect to a single central location. The more variability associated with a variable, the less precise and reliable is the estimate you get of the population’s location from a sample of a given size.

7. Chapter 3For Explaining Psychological Statistics, 4th ed. by B. Cohen 7 –(Simple) Range: It is found by subtracting the lowest score from the highest score. It is affected by outliers and tends to be very unreliable. Used only when it’s important to know the greatest possible distance among scores in the distribution. –Semi-Interquartile Range (SIQR): It is found by subtracting the 25 th percentile from the 75 th percentile, and then dividing that distance in half. It is not affected by outliers. It is a good descriptive measure, especially when the data are skewed or inexact at the extremes, and is often used in conjunction with the median. –Mean (Absolute) Deviation: Find each score’s deviation from the mean, take its absolute value, and then average all of these values. It is a good descriptive measure, and is fairly robust, but it is rarely used.

8. Chapter 3For Explaining Psychological Statistics, 4th ed. by B. Cohen 8 –Standard Deviation and Variance: Sum of Squares (SS): Find each score’s deviation from the mean, square them all, and then add them all up. Biased Variance (σ 2 ): Divide SS by the total N. Not useful as a descriptive statistic, because it is in squared units. Biased Standard Deviation (σ): Take the square root of the biased variance. Used for describing a population. Unbiased Variance Estimate (s 2 ): Divide SS by the degrees of freedom (i.e., N – 1). Used for estimating the corresponding population parameter from sample data, but not useful as a descriptive statistic. Unbiased Standard Deviation Estimate (s): Take the square root of the unbiased variance. This is the most commonly used measure of variability, especially when drawing inferences from sample data to population parameters.

9. Chapter 3For Explaining Psychological Statistics, 4th ed. by B. Cohen 9 Returning to the motivation data introduced in the first slide, we can demonstrate the calculation of the SS, variance, and standard deviation. The sum in the lower-right corner of the table (30) is the SS. Dividing SS by N, we find that σ 2 = 30/11 = 2.73. Taking the square root, we see that σ = 1.65. Dividing SS by N – 1 yields s 2 = 3.0, and taking the square root yields s = 1.73.

10. Chapter 3For Explaining Psychological Statistics, 4th ed. by B. Cohen 10 The definitional formulas we have been using are summarized in the top half of the table below. The corresponding computational (raw-score) formulas are shown in the bottom half of the table.

11. Chapter 3 For Explaining Psychological Statistics, 4th ed. by B. Cohen 11 If you want to apply the computational formulas to the motivation data, we have included the squared value of each score in the table below to get you started.