Measures of Central Tendency:

Mean, Median, & Mode

Mean, Median, & Mode

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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?

- mean
- The sum of the values divided by the number of values--often called the "average."
- Add all of the values together.
- Divide by the number of values to obtain the mean.
- median
- The value which divides the values into two equal halves, with half of the values being lower than the median and half higher than the median.
- Sort the values into ascending order.
- If you have an odd number of values, the median is the middle value.
- If you have an even number of values, the median is the arithmetic mean (see above) of the two middle values.
- mode
- The most frequently-occurring value (or values).
- Calculate the frequencies for all of the values in the data.
- The mode is the value (or values) with the highest frequency.

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 line.

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.

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
``Done'' button.

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.

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Updated March 16, 1998