Central Tendency | Understanding the Mean, Median & Mode

Published on 31 December 2022 by Pritha Bhandari.

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
Mode
Median
Mean
When should you use the mean, median or mode?
Frequently asked questions

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.

Negative Skew Distribution — In this histogram, your distribution is skewed to the left, and the central tendency of your dataset is towards the higher end of possible scores.

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.

Example: Finding the mode

In a survey, you ask 9 participants whether they identify as conservative, moderate, or liberal.

To find the mode, sort your data by category and find which response was chosen most frequently.

To make it easier, you can create a frequency table to count up the values for each category.

Political ideology	Frequency
Conservative	2
Moderate	3
Liberal	4

Mode: Liberal

The mode is easily seen in a bar graph because it is the value with the highest bar.

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.

Example: Ratio data with no mode

You collect data on reaction times in a computer task, and your dataset contains values that are all different from each other.

Participant	1	2	3	4	5	6	7	8	9
Reaction time (milliseconds)	267	345	421	324	401	312	382	298	303

In this dataset, there is no mode, because each value occurs only once.

Median

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

Example: Finding the median

You measure the reaction times of 7 participants on a computer task and categorise them into 3 groups: slow, medium or fast.

Participant	1	2	3	4	5	6	7
Speed	Medium	Slow	Fast	Fast	Medium	Fast	Slow

To find the median, you first order all values from low to high. Then, you find the value in the middle of the ordered dataset – in this case, the value in the 4th position.

Ordered dataset	Slow	Slow	Medium	Medium	Fast	Fast	Fast

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 $\dfrac{(n+1)}{2}$ position, where n is the number of values in the dataset.

\begin{equation*}\dfrac{(5+1)}{2}=3$\end{equation*} — Example

Reaction time (milliseconds)	287	298	345	365	380

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 $\dfrac{n}{2}$ and $(\dfrac{n}{2})+1$ positions. Then, find their mean.

Example

You measure the reaction times of 6 participants and order the dataset.

Reaction time (milliseconds)	287	298	345	357	365	380

The middle positions are calculated using $\dfrac{n}{2}$ and $(\dfrac{n}{2})+1$ , where n = 6.

$\begin{equation*}\dfrac{6}{2}=3\end{equation*}$

$\begin{equation*}(\dfrac{6}{2})+1=4\end{equation*}$

That means the middle values are the 3rd value, which is 345, and the 4th value, which is 357.

To get the median, take the mean of the 2 middle values by adding them together and dividing by 2.

$\begin{equation*}\dfrac{(345+357)}{2}=351\end{equation*}$

Median: 351 milliseconds

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.

\begin{equation*}\sum{x}=287+345+365+298+380=1\,675\end{equation*} — Example: Finding the mean

Participant	1	2	3	4	5
Reaction time (milliseconds)	287	345	365	298	380

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.

\begin{equation*}\sum{x}=832+345+365+298+380=2\,220\end{equation*} — Example: Mean with an outlier

\begin{equation*}\bar{x}=\dfrac{\sum{x}}{n}=\dfrac{2\,220}{5}=444\end{equation*} — Example: Mean with an outlier

Participant	1	2	3	4	5
Reaction time (milliseconds)	832	345	365	298	380

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.

\begin{equation*}\bar{x}=\dfrac{\sum{x}}{n}\end{equation*} — Sample mean formula

\begin{equation*}\mu=\dfrac{\sum{X}}{N}\end{equation*} — Population mean formula

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.

Levels of measurement	Examples	Measure of central tendency
Nominal	Ethnicity Political ideology	Mode
Ordinal	Level of anxiety Income bracket	Mode Median
Interval and ratio	Reaction time Test score Temperature	Mode Median Mean

Levels of measurement

Examples

Measure of central tendency

Nominal

Ethnicity
Political ideology

Mode

Ordinal

Level of anxiety
Income bracket

Mode
Median

Interval and ratio

Reaction time
Test score
Temperature

Mode
Median
Mean

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

What are measures of central tendency?: Measures of central tendency help you find the middle, or the average, of a data set.

The 3 most common measures of central tendency are the mean, median and mode.

The mode is the most frequent value.

The median is the middle number in an ordered data set.

The mean is the sum of all values divided by the total number of values.
Which measures of central tendency can I use?: The measures of central tendency you can use depends on the level of measurement of your data.

For a nominal level, you can only use the mode to find the most frequent value.

For an ordinal level or ranked data, you can also use the median to find the value in the middle of your data set.

For interval or ratio levels, in addition to the mode and median, you can use the mean to find the average value.
What’s the best measure of central tendency to use?: The mean is the most frequently used measure of central tendency because it uses all values in the data set to give you an average.

For data from skewed distributions, the median is better than the mean because it isn’t influenced by extremely large values.

The mode is the only measure you can use for nominal or categorical data that can’t be ordered.

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Pritha Bhandari

Pritha has an academic background in English, psychology and cognitive neuroscience. As an interdisciplinary researcher, she enjoys writing articles explaining tricky research concepts for students and academics.

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