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Measures of Central Tendency

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Researchers often summarize their data using a particular measure of central tendency—one score that represents the entire set of data points.

The most straightforward measure is the mode—the most frequently occurring score—which is useful when computing totals for categorical data, such as professional occupation. Here, the mode is noted as writer, since they constitute the majority of what careers were reported.

Another measure is the median—the true middle point in a set of numerical data that are arranged in order of magnitude. Here, the median of salaries is $65,000…with half of employees paid at or above…and the other half at or below…this midpoint. In cases with an even number of values, the average of the two middle numbers determines the median.

Finally, the most common measure for numerical data is the mean—the arithmetic average—which is equal to the total sum of all numerical scores, divided by the number of data points.

For example, the total sum of all incomes would be divided by the number of people employed—in this instance, yielding a mean of $140,000. Because this measure takes into account all values in the data set for calculation, researchers usually prefer to report the mean.

In a normal distribution, the mode, mean, and median have approximately equivalent values and are positioned exactly in the center of the curve. However, in skewed distributions—when either very high or very low scores are more prevalent and weighted more heavily—these measurements are not equal.

If the majority of the participants reported low incomes, the mean would rise due to a few extremely high-income scores. Thus, the mean’s sensitivity to extreme values could result in an artificially high or low representation of the data. In this case, the median or mode would serve as a more accurate measurement.

In the end, the best measure of central tendency is one that considers the pattern of distribution in the data at hand.

Overview

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The "center" of a data set is also a way of describing location. The two most widely used measures of the "center" of the data are the mean (average) and the median. The words "mean" and "average" are often used interchangeably. The substitution of one word for the other is common practice. The technical term is "arithmetic mean" and "average" is technically a center location. However, in practice among non-statisticians, "average" is commonly accepted for "arithmetic mean."

Another measure of the center is the mode. The mode is the most frequent value. If a data set has two values that occur the same number of times, then the set is bimodal.

Calculating the Mean and Median

To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts (previously discussed under box plots in this chapter). The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.

The mean can also be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the sample mean is an x with a bar over it (pronounced "x bar").

The Greek letter μ (pronounced "mew") represents the population mean. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random.

You can quickly find the location of the median by using the expression (n+1)/2. The letter n is the total number of data values in the sample. If n is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If n is an even number, the median is equal to the two middle values added together and divided by 2 after the data has been ordered. For example, if the total number of data values is 97, then (n+1)/2 = (97+1)/2 = 49. The median is the 49th value in the ordered data. If the total number of data values is 100, then (n+1)/2 = (100+1)/2 = 50.5. The median occurs midway between the 50th and 51st values. The location of the median and the value of the median are not the same.

This text is adapted from Barbara Illowsky, Ph.D., Susan Dean, Collaborative Statistics. OpenStax CNX.

Transcript

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Researchers often summarize their data using a particular measure of central tendency—one score that represents the entire set of data points.

The most straightforward measure is the mode—the most frequently occurring score—which is useful when computing totals for categorical data, such as professional occupation. Here, the mode is noted as writer, since they constitute the majority of what careers were reported.

Another measure is the median—the true middle point in a set of numerical data that are arranged in order of magnitude. Here, the median of salaries is $65,000…with half of employees paid at or above…and the other half at or below…this midpoint. In cases with an even number of values, the average of the two middle numbers determines the median.

Finally, the most common measure for numerical data is the mean—the arithmetic average—which is equal to the total sum of all numerical scores, divided by the number of data points.

For example, the total sum of all incomes would be divided by the number of people employed—in this instance, yielding a mean of $140,000. Because this measure takes into account all values in the data set for calculation, researchers usually prefer to report the mean.

In a normal distribution, the mode, mean, and median have approximately equivalent values and are positioned exactly in the center of the curve. However, in skewed distributions—when either very high or very low scores are more prevalent and weighted more heavily—these measurements are not equal.

If the majority of the participants reported low incomes, the mean would rise due to a few extremely high-income scores. Thus, the mean’s sensitivity to extreme values could result in an artificially high or low representation of the data. In this case, the median or mode would serve as a more accurate measurement.

In the end, the best measure of central tendency is one that considers the pattern of distribution in the data at hand.

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