# Measures of central tendency (mean, median, mode, and trimmed mean)

## Mean

Use the mean to describe an entire set of observations with a single value representing the center of the data. Many statistical analyses use the mean as a standard reference point. The mean is the sum of all observations divided by the number of observations.

For example, the waiting time (in minutes) of five customers in a bank are: 3, 2, 4, 1, and 2. The mean waiting time is:
On average, a customer waits 2.4 minutes for service at the bank.

## Median

Use the median to describe an entire set of observations with a single value representing the center of the data. Half of the observations are above the median, half are below it. It is determined by ranking the data and finding observation number [N + 1] / 2. If there are an even number of observations, the median is extrapolated as the value midway between that of observation numbers N / 2 and [N / 2] + 1.

## Mode

The mode is the value that occurs most frequently in a set of observations. Minitab also displays how many data points equal the mode. Mode may be used with mean and median to give an overall characterization of your data distribution. While the mean and median require a calculation, the mode is found simply by counting the number of times each value occurs in a data set.

Identifying the mode can help you understand your distribution. A distribution with more than one mode may indicate that you actually have sampled from a mixed population. For example, you may have collected wait time data on customers who are cashing checks and customers who are applying for home equity loans together. To better understand your data, these two cases should be collected separately. If you have more than two modes, the distribution is multimodal.

## Trimmed Mean

The trimmed mean is the mean of the data, without the highest 5% and lowest 5% of the values. Use the trimmed mean to eliminate the impact of very large or very small values on the mean. When the data contain outliers, the trimmed mean may be a better measure of central tendency than the mean.

## Using measures of central tendency to describe skewed distributions

The center of the data is the area where most values in a data set cluster. Central tendency can be described by a number of different statistics, like the mean, trimmed mean, median, or mode. Knowing the central tendency of your data is an important first step in understanding it.

Graphs like histograms, boxplots, and dotplots are useful in visualizing data's central tendency and can assist in deciding which central tendency statistic is most appropriate with a given data set.

Likewise, as distributions stray from normal and become more skewed, the standard deviation becomes more different from the distance between the mean and a typical data value.

The interquartile range is a better measure of spread for highly skewed data than the standard deviation is because the interquartile range is not affected by extreme ranges.

## Comparing the mean and the median

If your data are symmetric, the measures of central tendency (mean and median) will be roughly the same. If the data are asymmetric, the measures may be pulled toward the more extreme observations. Of the measures, the mean is more influenced by extreme values and the median is less influenced.

## How can I display these statistics?

You can use Display Descriptive Statistics to display the mean, median, mode, and trimmed mean. For example, suppose you want to display the mode for the values in C1.
1. Choose Stat > Basic Statistics > Display Descriptive Statistics.
2. In Variables enter C1.
3. Click Statistics. Check Mode (and any other statistic you may want).
4. Click OK in each dialog box.
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