Central tendency: Mean, median and mode

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 mode, median, and mean.

  • Mode: the most frequent value.
  • Median: the middle number in an ordered data set.
  • Mean: the sum of all values divided by the total number of values.

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

Distributions and central tendency

A data set 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.

Example: Normal distribution
You survey a sample in your local community on the number of books they read in the last year.

A histogram of your data shows the frequency of responses for each possible number of books. From looking at the chart, you see that there is a normal distribution.

Normal_Distribution

The mean, median and mode are all equal; the central tendency of this data set is 8.

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.

In this histogram, your distribution is skewed to the right, and the central tendency of your data set is on the lower end of possible scores.

In a positively skewed distribution, mode < median < mean.

Positive Skew Distribution

In this histogram, your distribution is skewed to the left, and the central tendency of your data set is towards the higher end of possible scores.

In a negatively skewed distribution, mean < median < mode.

Negative Skew Distribution

Mode

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

To find the mode, sort your data set 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.

Showing the mode in a bar chart

 

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 data set 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 data set, there is no mode, because each value occurs only once.

Median

The median of a data set 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 categorize 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 data set – in this case, the value in the 4th position.

Ordered data set Slow Slow Medium Medium Fast Fast Fast

Median: Medium

In larger data sets, 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 data set depending on whether the total number of values is even or odd.

Median of an odd-numbered data set

For an odd-numbered data set, find the value that lies at the (n+1)/2 position, where n is the number of values in the data set.

Example
You measure the reaction times in milliseconds of 5 participants and order the data set.

Reaction time (milliseconds) 287 298 345 365 380

The middle position is calculated using (n+1)/2, where n = 5.

(5+1)/2 = 3

That means the median is the 3rd value in your ordered data set.

Median: 345 milliseconds

Median of an even-numbered data set

For an even-numbered data set, find the two values in the middle of the data set: the values at the n/2 and (n/2) + 1 positions. Then, find their mean.

Example
You measure the reaction times of 6 participants and order the data set.

Reaction time (milliseconds) 287 298 345 357 365 380

The middle positions are calculated using n/2 and (n/2) + 1, where n = 6.

6/2 = 3

(6/2) + 1 = 4

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 two.

(345 + 357)/2 = 351

Median: 351 milliseconds

Mean

The arithmetic mean of a data set 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.

Example: Finding the mean
Participant 1 2 3 4 5
Reaction time (milliseconds) 287 345 365 298 380

First you add up the sum of all values:

x = 287 + 345 + 365 + 298 + 380 = 1675

Then you calculate the mean using the formula ⅀x/n. There are 5 values in the dataset, so n = 5.

Mean (x̄) = 1675/5 = 335

Mean: 335 milliseconds

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 data set.

Example: Mean with an outlier
In this data set, we swap out one value with an extreme outlier.

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

x = 832 + 345 + 365 + 298 + 380 = 2220

Mean (x̄) = ⅀x/n = 2220/5 = 444

Due to the outlier, the mean becomes much higher, even though all the other numbers in the data set stay the same.

Mean: 444 milliseconds

Population versus sample mean

A data set 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.

The sample mean is written as M or x̄ (pronounced x-bar). For calculating the mean of a sample, use this formula:

x̄ = ⅀x/n

  • x̄:  sample mean
  • x: sum of all values in the sample data set
  • n: number of values in the sample data set

The population mean is written as μ (Greek term mu). For calculating the mean of a population, use this formula:

μ = ⅀ X/N

  • μ: population mean
  • X: sum of all values in the population data set
  • N: number of values in the population data set

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 data set, 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

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

    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

    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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