Measures of Central Tendency (College Board AP® Psychology): Study Guide

What are measures of central tendency?

• Measures of central tendency are statistical tools used to describe the central or typical value of a data set

• They summarize large amounts of data into a single representative score, making it easier to identify patterns and draw conclusions

• There are three measures of central tendency:

  • The mean — the arithmetic average

  • The median — the middle value

  • The mode — the most frequently occurring value

• Choosing the most appropriate measure depends on the nature of the data set and whether extreme scores (outliers) are present

Mean

• The mean is calculated by adding up all the values in a data set and dividing the total by the number of values

• The mean represents the arithmetic average of the data set

How to calculate the mean

• Add up all values in the data set

• Divide the total by the number of values

• Example:

  • Data set: 4, 6, 7, 9

  • 4 + 6 + 7 + 9 = 26

  • 26 ÷ 4 = 6.5

  • Mean = 6.5

How to interpret the mean

• The mean tells us the average score across all participants

  • It represents what a typical participant scored

• Example:

  • If the mean score on an anxiety scale for a group receiving CBT is 12 and the mean for a control group is 24, this suggests that participants receiving CBT reported lower anxiety on average than those who did not

• When comparing means across conditions, the larger or smaller the difference between them, the more meaningful the finding may be — though statistical significance must also be considered

When to use the mean

• Use the mean when the data set does not contain extreme scores (outliers) and when all scores are reasonably close together

• The mean is the most appropriate measure when data is normally distributed

• Avoid the mean when the data set contains outliers — extreme scores will distort the mean and make it unrepresentative of the data set as a whole

Evaluation of the mean

Strengths

• The mean is the most sensitive measure of central tendency as it takes every score in the data set into account

  • This makes it the most representative and reliable measure of central tendency when data is normally distributed

Limitations

• The mean is sensitive to extreme scores (outliers) so it can only be used when the scores are reasonably close 

  • This means that it would not be a suitable measure for some data sets

• The mean score may not appear in the data set itself

  • E.g. a mean of 6.5 from the data set above does not correspond to any actual score in the set

Median

• The median is the middle value of a data set when all values are arranged in numerical order

• The median represents the positional average — the point that divides the data set exactly in half

How to calculate the median

For an odd number of values:

• Arrange the values in ascending order

• Identify the middle value

• Example:

  • Data set: 20, 43, 56, 78, 92, 67, 48

  • Ordered: 20, 43, 48, 56, 67, 78, 92

  • Median = 56 (the 4th value in a set of 7)

For an even number of values:

• Arrange the values in ascending order

• Identify the two middle values

• Add them together and divide by 2

• Example:

  • Data set: 15, 16, 18, 19, 22, 24

  • Two middle values: 18 and 19

  • 18 + 19 = 37 ÷ 2 = 18.5

  • Median = 18.5

How to interpret the median

• The median tells us the midpoint of the data set

  • Half of all scores fall above it and half fall below it

• Example:

  • If the median score on a stress scale is 15 for one group and 28 for another, this suggests that the typical participant in the second group reported considerably higher stress levels than the typical participant in the first group

• The median is particularly useful for interpretation when the data set contains outliers, as it gives a more accurate picture of the typical score than the mean would

When to use the median

• Use the median when the data set contains outliers or extreme scores that would distort the mean

• The median is the most appropriate measure when data is skewed — when scores are not evenly distributed around the center

Evaluation of the median

Strengths

• The median is not affected by outliers

  • This means that it gives a more accurate picture of the typical score when extreme values are present in the data set

• The median is more appropriate than the mean for skewed data sets

  • This is because the median remains unaffected by skew and therefore gives a more accurate picture of the center of the data

Limitations

• The mean does not take all scores into account

  • Because it only identifies the middle value, it ignores the actual values of all other scores

  • This makes it less sensitive than the mean

• It is time-consuming to calculate with large data sets

  • This is because all values must be arranged in order before the median can be identified

Mode

• The mode is the most frequently occurring value in a data set

• The mode identifies the most common score rather than the average or middle value

How to calculate the mode

• Count how many times each value appears in the data set

• The value that appears most frequently is the mode

• Example:

  • Data set: 3, 3, 3, 4, 4, 5, 6, 6, 6, 6, 7, 8

  • 6 appears four times — more than any other value

  • Mode = 6

A data set may have

• No mode — if all values occur equally often

• One mode — the most common scenario

• Two modes (bimodal) — if two values occur equally often and more frequently than all others

• More than two modes (multimodal) — if three or more values occur equally often

How to interpret the mode

• The mode tells us the most common score in the data set

  • This is the value that occurred most frequently among participants

• Example:

  • If the modal response on a survey about sleep duration is 6 hours, this tells us that more participants reported sleeping 6 hours per night than any other amount

• A bimodal distribution suggests that the data set contains two distinct clusters of scores

  • This may indicate that two different subgroups within the sample responded differently, which is worth investigating further

When to use the mode

• Use the mode when the researcher is interested in the most common or most popular value rather than the average

• The mode is the only measure of central tendency that can be used with categorical data — data that falls into named categories rather than numerical values

  • E.g. most common eye color, most frequently chosen answer on a multiple choice question

Evaluation of the mode

Strengths

• The mode is not affected by extreme values

• The mode is the only measure of central tendency that can be applied to categorical data

Limitations

• A data set may include two or more modes, which makes it difficult to identify a single typical value

  • This reduces the usefulness of the measure

• The mode may be unrepresentative on small data sets

  • A value that happens to occur twice may be identified as the mode even if it does not reflect the typical score in the data set

Choosing the right measure