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# Measures of Central Tendency for Numerical Data Study Guide (page 2)

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Updated on Oct 5, 2011

#### Median Note

If N, the number of values in the population, is odd, the median is the st value in the ordered list of population values. If N is even, the median is any value between the th and the st values in the ordered list of population values; usually, the average of the two values is taken as the median.

#### Example

Find the median height of the 62 high school orchestra members. Compare the mean and median.

#### Solution

Because an even number of orchestra members exists, the median height is the average of the 31st and 32nd values in the ordered list of orchestra members' heights. The stem-and-leaf plot makes these values easy to find. Simply start at the top of the plot and count the number of leaves, always working from the stem out. Continue until the 31st and 32nd values have been identified. The 31st value is 66.6 inches, and the 32nd value is 67.1 inches. Any value between 66.6 and 67.1 is a median value. However, we will follow tradition and average the two: = 66.85.That is, the median orchestra member height is 66.85 inches.

Notice that in this case, the mean and median are close, but not identical. For the median, exactly half of the population values are less than 66.85 inches, and half are greater than 66.85 inches. For the mean, 29 of the values are less than the mean, and 33 are greater than the mean, which is still close to half of the values. Sometimes, the mean and median are much further apart. We will consider what differences in the mean and median indicate about the distribution in the next lesson.

## Mode

The mode is the most frequently occurring value in the population and is another measure of central tendency. This measure tends to be most useful for discrete random variables. For the orchestra members' heights, three members have heights of 59.7 inches. Similarly, we have three other groups, each with three members, with heights of 68.7, 68.9, and 70.5 inches. Thus, there are four modes.

Which measure of central tendency is the best? Each provides a little different information. The mean is the most common measure, but it is influenced by extreme values. One extreme value can have a big impact on the mean, especially if the population does not have many members. An unusually small population value may cause the mean to be quite a bit smaller than it would have been if that value was not in the population. Similarly, an unusually large population value may tend to inflate the mean. In contrast, the median and mode are not affected by these unusual values. The mode is often not useful because it may not be unique.

## Sample Measures of Central Tendency

The population mean, median, and mode are parameters. To find their values, we must know all of the population values. Unless the population is small, as was the case when our population of interest was the orchestra members at one specific high school, this rarely happens. In practice, we cannot usually find the population mean, median, or mode. We estimate these parameters by finding the sample mean, sample median, and sample mode, respectively. The mean, median, and mode of the sample values are the sample mean, sample median, and sample mode. These statistics are each an estimator of their population counterpart. That is, the sample mean is an estimate of the population mean; the sample median is an estimate of the population median; and the sample mode is an estimate of the population mode. We will learn more about the characteristics of these estimates and their uses later. For now, let's concentrate on how to find them.

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