SECTION 2.3 Measures of Central Tendency 73 The Shapes of Distributions A graph reveals several characteristics of a frequency distribution. One such characteristic is the shape of the distribution. A frequency distribution is symmetric when a vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images. A frequency distribution is uniform (or rectangular) when all entries, or classes, in the distribution have equal or approximately equal frequencies. A uniform distribution is also symmetric. A frequency distribution is skewed when the “tail” of the graph elongates more to one side than to the other. A distribution is skewed left (negatively skewed) when its tail extends to the left. A distribution is skewed right (positively skewed) when its tail extends to the right. DEFINITION When a distribution is symmetric and unimodal, the mean, median, and mode are equal. When a distribution is skewed left, the mean is less than the median and the median is usually less than the mode. When a distribution is skewed right, the mean is greater than the median and the median is usually greater than the mode. Examples of these commonly occurring distributions are shown. 5 10 15 20 25 30 35 40 45 1 3 5 7 9111315 Mean Median Mode 5 10 15 20 25 30 35 40 45 1 3 5 7 9111315 Mean Median Symmetric Distribution Uniform Distribution 5 10 15 20 25 30 35 40 45 1 3 5 7 9 11 13 15 Mean Median Mode 5 10 15 20 25 30 35 40 45 1 3 5 7 9 11 13 15 Mean Median Mode Skewed Left Distribution Skewed Right Distribution The mean will always fall in the direction in which the distribution is skewed. For instance, when a distribution is skewed left, the mean is to the left of the median. To explore this topic further, see Activity 2.3 on page 81. 2.3 Study Tip Be aware that there are many different shapes of distributions. In some cases, the shape cannot be classified as symmetric, uniform, or skewed. A distribution can have several gaps caused by outliers or clusters of data. Clusters may occur when several types of data entries are used in a data set. For instance, a data set of gas mileages for trucks (which get low gas mileage) and hybrid cars (which get high gas mileage) would have two clusters. Study Tip The graph of a symmetric distribution is not always bell-shaped (see below). Some of the other possible shapes for the graph of a symmetric distribution are U-, M-, or W-shaped.
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