Measures of Central Tendency

Understanding measures of central tendency

While descriptive statistics are used to summarize a data set, measures of central tendency are used to summarize the middle value or where the data tend to cluster or “pile up.” They provide a single, representative value that can be helpful for quickly understanding the general characteristics of a data set. Some key measures of location are:

• Mean: the arithmetic average of a variable.
• Median: the middle value of data when arranged in order.
• Mode: the most frequently occurring value in a data set.
• Geometric mean: a measure that uses products instead of sums.
• $\alpha$-trimmed mean: the arithmetic average of a variable after removing its $\alpha$% low and $\alpha$% high values.

Each measure of central tendency has its strengths and is appropriate in different situations. The choice of which to use depends on the characteristics of your data and the specific goals of your analysis.

These measures summarize the data into one number for location. It is always important to understand variability as well as location. To make a good choice for which measure to use for your data, consider the characteristics of each measure and how much data you have. With smaller sample sizes, it is more likely that one or just a few extreme values will have a big effect on these measures.

What are characteristics of various measures of central tendency?

Characteristics of the mean

• It is the arithmetic average of measurements in the data.
• There is only one mean of the data.
• Its value is influenced by extreme values.
• The mean is applicable only to continuous values.

Characteristics of the median

• It is the central value in set of data: 50% of the data lies above it and 50% lies below it.
• There is only one median.
• The median is not influenced by extreme values.
• The median is applicable only to continuous or ordinal values.

Characteristics of the mode

• It is the most frequent or most probable measurement in the data.
• There can be more than one mode.
• Modes are not influenced by extreme values.
• Modes are applicable for both continuous and categorical (ordinal or nominal) data.

Characteristics of the geometric mean

• It is the n^th root of the product of the measurements in the data.
• It is also the exponential function applied to the arithmetic mean of the logarithms of the data values.
• It is useful when averaging rates.
• If there are any zero values in the data, the geometric mean is zero.
• The geometric mean’s value is influenced by extreme values.
• The geometric mean is applicable only to non-negative continuous values.

Characteristics of the $\alpha$-trimmed mean

• It is the arithmetic average of the data after removing the bottom $\alpha$% and the top $\alpha$% of the data.
• Like the mean, there is only one $\alpha$-trimmed mean of the data once you have determined which $\alpha$ to use.
• Its value is not as influenced by extreme values as the mean is.
• The $\alpha$-trimmed mean is applicable only to continuous values.

When should I use which measure of central tendency?

If your data are continuous and symmetric, the mean is the most common measure of central tendency. For continuous data with a skewed distribution, the median is more commonly used; the mode is often used when the goal is to find the peak. For continuous data with extreme values in either or both tails, consider an $\alpha$-trimmed mean. For continuous data of rates, consider the geometric mean. For numeric ordinal data, the median or the mode are used most often. For categorical nominal data, the mode must be used. See the table below for a summary.