This tutorial uses histograms to illustrate different measures of central
tendency. A histogram
is a type of graph in which the x-axis lists categories or values for a
data set, and the y-axis shows a count of the number of cases falling into
each category. For example, if there are 59 men and 48 women in your class,
you could represent the information with this histogram:
The categories may be non-numeric, as in the histogram above, or may
be numeric, as in the following histogram. The x-axis shows the ages for
respondents to a survey and the y-axis reports the frequency or count for
occurrances of each age.
From the histogram, can you determine what is the "typical" age of the
participants in the survey? This question could be answered
in several different ways, depending on what you really want to know. Do
you want to determine:
The average of the ages?
The age which divides the cases into two equal-sized groups -- the "highs"
vs. the "lows"?
The most common age?
Questions like these are concerned with determining the central
tendency of a group of numbers or data. To answer our question, we
want a single number which can somehow represent all of the ages of the
people who participated in the survey.
Ways to Measure Central Tendency
The three most commonly-used measures of central tendency are the following.
Calculate the frequencies for all of the values in the data.
The mode is the value (or values) with the highest frequency.
Example: For individuals having the following ages -- 18, 18, 19, 20, 20,
20, 21, and 23, the mode is 20.
Check your understanding of these concepts by calculating the mean, median,
and mode of the following three sets of numbers.
Which Measure Should You Use?
This histogram shows the distribution of the number of siblings for survey
respondents. The mode (i.e., most common number of siblings) is easy to
find. Can you also determine the median simply by inspection?
What about the mean?
You should see two copies of the histogram. The upper histogram allows
you to drag the red vertical line to help locate the median. Numbers on
either side of the red line show you how many values exist above and below
The lower histogram allows you to move a triangle within the range of
the distribution which acts like a fulcrum for a see-saw. The mean is located
at the point where the histogram is balanced. Use these tools -- the red
vertical line and the fulcrum -- to find the median and mean of the data.
In a normal distribution, the mean, median, and mode are all the same value.
In various other symmetrical distributions it is possible for the mean
and median to be the same even though there may be several modes, none
of which is at the mean. By contrast, in asymmetrical distributions the
mean and median are not the same. Such distributions are said to
i.e., more than half the cases are either above or below the mean.
Below are some exercises that illustrate the
relationship between mean, median, and mode in skewed distributions. In
each exercise you will be asked to modify a histogram that satisfies certain
conditions. You can change each histogram by dragging the mouse across
it with the button down. You can then check your answer by clicking the
At this point, you should have created a symmetrical distribution, a negatively
skewed distribution, and a positively skewed distribution. If you think
about the three figures, you can deduce a general rule about the relationship
between the symmetry of a distribution of scores and measures of central
tendency. The rule is that, as the symmetry of a distribution increases,
the three measures of central tendency converge on the same value. As the
asymmetry or skewness of a distribution increases, the three measures of
central tendency diverge systematically.
For a positively skewed distribution, the mean will always be the highest
estimate of central tendency and the mode will always be the lowest estimate
of central tendency (assuming that the distribution has only one mode).
For negatively skewed distributions, the mean will always be the lowest
estimate of central tendency and the mode will be the highest estimate
of central tendency.
In any skewed distribution (i.e., positive or negative) the median will
always fall in-between the mean and the mode. As previously discussed in
the section on "choosing an appropriate measure of central tendency", when
dealing with skewed distributions, researchers typically decide between
the mean or median as the best estimate of central tendency. As distributions
go from symmetrical to more skewed, the researcher is more likely to chose
the median over the mean.
Now you should be able to look at real data sets and spot the three Measures of Central
Tendency. Use this activity to examine different variables.