# © 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 4. Measuring Averages.

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© 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 4. Measuring Averages

© 2008 McGraw-Hill Higher Education Central Tendency Statistic A statistic that provides an estimate of the typical, usual, or normal score found in a distribution of raw scores Provides a sense of average

© 2008 McGraw-Hill Higher Education The Three Measures of Central Tendency The three central tendency statistics are the mean, the median, and the mode Each has strengths and weaknesses that make it useful with certain score distributions The relative values of the three central tendency statistics inform us of the shape of a score distribution

© 2008 McGraw-Hill Higher Education Central Tendency Statistics and Levels of Measurement The mean and median are appropriate with interval/ratio variables The mode is appropriate with variables of all levels of measurement With nominal variables, means and medians are meaningless

© 2008 McGraw-Hill Higher Education The Mean The mean is the sum of all scores in a distribution divided by the number of scores observed; the arithmetic average Most useful central tendency statistic because it allows for many mathematical operations Applies to interval/ratio variables

© 2008 McGraw-Hill Higher Education Calculating the Mean Sum the individual scores of the interval/ratio variable X, then divide this sum by the number of observations (i.e., the sample size, n) Weaknesses of the mean: Its calculation is affected by outliers and skews

© 2008 McGraw-Hill Higher Education Combined Mean of Two Groups To get the combined mean of two groups of different sizes (n), multiply n times the group mean for each group and sum to obtain the total sum of X; divide this sum by the total n Do NOT simply average the two group means

© 2008 McGraw-Hill Higher Education The Median The median is the middle score in a ranked distribution; the score for which half of the cases fall above and half fall below It is equal to the 50 th percentile It is a location score Useful with interval/ratio variables Best central tendency statistic to report when a distribution of scores is skewed

© 2008 McGraw-Hill Higher Education Calculating the Median Rank the scores from smallest to largest Divide the sample size by 2 to get near the middle score in the ranked distribution If n is odd, the median will be an actual case in the sample; however, if n is even, the median is located between two middle scores and is calculated by taking the mean of those two scores

© 2008 McGraw-Hill Higher Education Weaknesses of the Median The median is insensitive to the values of the scores in a distribution. Thus, two distinctly different score distributions may have the same median The median is sensitive to a change in sample size. If new cases are added, the median may drastically change

© 2008 McGraw-Hill Higher Education The Mode The mode is the most frequently occurring or “most popular” score in a distribution It is useful with variables of all levels of measurement The mode is a score, X, not a frequency, f Do not confuse the mode with “the majority of scores” The mode is easy to spot in charts

© 2008 McGraw-Hill Higher Education Calculating the Mode Compile scores into a frequency distribution Identify the value of X with the most cases On a histogram, it is the score of X for the highest column; on a polygon, the highest peak; on a pie chart, the largest slice

© 2008 McGraw-Hill Higher Education Weaknesses of the Mode The mode is the least useful measure of the three because of its narrow informational scope The mode is insensitive to the values of scores in a distribution The mode is also insensitive to sample size

© 2008 McGraw-Hill Higher Education Frequency Distribution Curves A substitute for a frequency histogram or polygon in which we replace these graphs with a smooth curve The area under the curve represents the total number of subjects in a population, and is equal to a proportion of 1.00 or a percentage of 100 percent The relative locations of the mean, median, and mode on the X-axis are predictable for certain shapes of distribution curves

© 2008 McGraw-Hill Higher Education Three Common Shapes of Frequency Distributions 1. Normal distribution or “normal curve” 2. Negatively skewed distribution 3. Positively skewed distribution

© 2008 McGraw-Hill Higher Education Features of the Normal Curve The mean, median, and mode of the variable are equal and centered in the curve The curve is symmetrical and bell-shaped When a distribution is not skewed or otherwise oddly shaped, the mean is the central tendency statistic of choice

© 2008 McGraw-Hill Higher Education Features of a Negatively Skewed Distribution Has extreme scores in the low or negative end The mean will have the lowest value of X, the mode the highest, and the median will fall between

© 2008 McGraw-Hill Higher Education Features of a Positively Skewed Distribution Has extreme scores in the high or positive end The mean will have the highest value of X, the mode the lowest, and the median will fall between When a distribution is skewed, the median is the statistic of choice. It minimizes error in describing a skewed distribution because it falls between the mean and the mode

© 2008 McGraw-Hill Higher Education Statistical Follies Mixing subgroups of subjects can result in a distorted mean For example, the average age of first- graders (around 6 years old) and their mothers (averaging around 31 years old) is 18.5 years, an age no one in either group approximates