Measures of the Center: Means, Medians, and the Mode

Another way to summarize a data set would be to calculate a statistic that summarized the contents into a single value that we would think of as the typical or most representative value.

The table of percentiles suggests one such value: the 50th percentile, or median.
Because the median is located at the 50th percentile, it represents the middle of the distribution:

Half of the values are less than the median, and half are greater than the median. Based on the results from Figure 4-16, the median house price in the Albuquerque sample from 1993 was ,000 for no northeast sector homes, ,000 for northeast sector homes, and ,000 overall.

The exact calculation of the median depends on the number of observations in the data set. If there is an odd number of values, the median is the middle value, but if there is an even number of values, the median is equal to the sum of the two central values divided by 2.

Another commonly used summary measure is the average. The average, or mean, is equal to the sum of the values divided by the number of observations.

This value is usually represented by the symbol x (pronounced “x-bar”), a convention we’ll repeat throughout the course of this book.

Describing Your Data 155 The total number of observations in the sample is represented by the symbol n, and each individual value is represented by x followed by a subscript.

The fi rst value is x1, the second value is x2, and so forth, up to the last value (the nth value), which is represented by xn.

The formula calls for us to sum all of these values, an operation represented by the Greek symbol S (pronounced “sigma”), a summation symbol. In this case, we’re instructed to sum the values of xi , where i changes in value from 1 up to n; in other words, the formula tells us to calculate the value of x1 1 x2 1 … 1 xn. The average, or mean, is equal to this expression divided by the total number of observations.

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