Measures of central tendency help you find the middle, or the average, of a dataset. The 3 most common measures of central tendency are the mode, median, and mean.

• Mode: the most frequent value.

• Median: the middle number in an ordered dataset.

• Mean: the sum of all values divided by the total number of values.

In addition to central tendency, the variability and distribution of your dataset is important to understand when performing descriptive statistics.

Distributions and central tendency

A dataset is a distribution of n number of scores or values.

Normal distribution

In a normal distribution, data is symmetrically distributed with no skew. Most values cluster around a central region, with values tapering off as they go further away from the center. The mean, mode and median are exactly the same in a normal distribution.

Skewed distributions

In skewed distributions, more values fall on one side of the center than the other, and the mean, median and mode all differ from each other. One side has a more spread out and longer tail with fewer scores at one end than the other. The direction of this tail tells you the side of the skew

In a positively skewed distribution, there’s a cluster of lower scores and a spread out tail on the right. In a negatively skewed distribution, there’s a cluster of higher scores and a spread out tail on the left.

Mode

The mode is the most frequently occurring value in the dataset. It’s possible to have no mode, one mode, or more than one mode.

To find the mode, sort your dataset numerically or categorically and select the response that occurs most frequently.

When to use the mode

The mode is most applicable to data from a nominal level of measurement. Nominal data is classified into mutually exclusive categories, so the mode tells you the most popular category.

For continuous variables or ratio levels of measurement, the mode may not be a helpful measure of central tendency. That’s because there are many more possible values than there are in a nominal or ordinal level of measurement. It’s unlikely for a value to repeat in a ratio level of measurement.

Median

The median of a dataset is the value that’s exactly in the middle when it is ordered from low to high.

Median: Medium

In larger datasets, it’s easier to use simple formulas to figure out the position of the middle value in the distribution. You use different methods to find the median of a dataset depending on whether the total number of values is even or odd.

Median of an odd-numbered dataset

For an odd-numbered dataset, find the value that lies at the position, where n is the number of values in the dataset.

Median of an even-numbered dataset

For an even-numbered dataset, find the two values in the middle of the dataset: the values at the and positions. Then, find their mean.

Mean

The arithmetic mean of a dataset (which is different from the geometric mean) is the sum of all values divided by the total number of values. It’s the most commonly used measure of central tendency because all values are used in the calculation.

Outlier effect on the mean

Outliers can significantly increase or decrease the mean when they are included in the calculation. Since all values are used to calculate the mean, it can be affected by extreme outliers. An outlier is a value that differs significantly from the others in a dataset.

Population versus sample mean

A dataset contains values from a sample or a population. A population is the entire group that you are interested in researching, while a sample is only a subset of that population.

While data from a sample can help you make estimates about a population, only full population data can give you the complete picture.

In statistics, the notation of a sample mean and a population mean and their formulas are different. But the procedures for calculating the population and sample means are the same.

Sample mean formula

The sample mean is written as M or x̄ (pronounced x-bar).

Population mean formula

The population mean is written as μ (Greek term mu).

When should you use the mean, median or mode?

The 3 main measures of central tendency are best used in combination with each other because they have complementary strengths and limitations. But sometimes only 1 or 2 of them are applicable to your dataset, depending on the level of measurement of the variable.

• The mode can be used for any level of measurement, but it’s most meaningful for nominal and ordinal levels.

• The median can only be used on data that can be ordered – that is, from ordinal, interval and ratio levels of measurement.

• The mean can only be used on interval and ratio levels of measurement because it requires equal spacing between adjacent values or scores in the scale.

To decide which measures of central tendency to use, you should also consider the distribution of your dataset.

For normally distributed data, all three measures of central tendency will give you the same answer so they can all be used.

In skewed distributions, the median is the best measure because it is unaffected by extreme outliers or non-symmetric distributions of scores. The mean and mode can vary in skewed distributions.

Frequently asked questions about central tendency

Bhandari, P. (2023, June 21). Central Tendency | Understanding the Mean, Median & Mode. Scribbr. Retrieved July 29, 2024, from https://www.scribbr.com/statistics/central-tendency/

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