Measuring the Center: Mean, Median, and Mode, Explained

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Measuring the Center: Mean, Median, and Mode

Research Methods in Psychology · Chapter 14: Descriptive StatisticsSection 14.2 · lesson 85 of 949 min · free with a free account

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In this lesson

Three answers to the same question, and when each is the right one.

  • Compute the mean, median and mode for a dataset
  • Match each measure to a level of measurement
  • See how the three behave on skewed distributions
  • Detect when the mean is misleading

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Transcript

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Descriptive Statistics

Measuring the Center: Mean, Median, and Mode

You have a distribution. Now you need a number that stands for it. There are three candidates, and choosing between them is a real decision. By the end of this section you will compute the mean, median, and mode. You will identify the level of measurement appropriate for each one. You will recognize how the three behave on symmetric and on skewed distributions. You will choose the most informative measure for a given research situation. And you will detect the situations where the mean actively misleads. That last skill turns up far outside statistics classes.

The Question

Where Is the Center

Once a frequency distribution makes the shape of the data visible, one question follows. Where is the center? Three measures of central tendency give three different answers. The mean balances all values equally. The median identifies the middle position, once the values are sorted. And the mode picks out the value, or values, that occur most often. Notice that none of these is wrong. Each captures a real and useful sense of center. And each has situations where it is the right summary, and others where it misleads. Choosing among them is one of the most fundamental skills in descriptive statistics.

Measure One

The Mean

Start with the arithmetic mean. It is the sum of all values, divided by the number of values. For a sample of n observations, add up all the scores and divide by n. That is the most familiar measure of central tendency. It is also the most widely used in inferential statistics. One notational point is worth fixing now, because it recurs throughout the course. The letter M is conventional for sample means. The Greek letter mu is reserved for the corresponding population value. When such a population value exists, that is. Sample statistics get Roman letters. Population parameters get Greek ones.

Two Properties

Why the Mean Is Powerful

The mean has two properties that make it powerful. The first is that it uses every value in the dataset. So it responds to any change, anywhere in the data. Move one score, and the mean moves. The second property is more subtle, and more important. The deviations of the individual values from the mean always sum to zero. Always. That is a mathematical guarantee, not a tendency. Which makes the mean the balancing point of the distribution. Together, those properties make the mean the foundation of much of what follows. Variance, standard deviation, regression, and most parametric inferential tests.

The Image

The Balancing Point of a Ruler

Here is an image that makes the balancing property intuitive. Imagine the values of a distribution placed as weights along a ruler. Each observation is a weight, sitting at its own score. The mean is the point at which the ruler balances. Now pull one weight far out toward an extreme. The balance point shifts toward it. The mean moves to compensate. One value has changed, and the summary of the whole dataset has changed with it. That sensitivity is the mean's strength when the data are well behaved. And it is precisely its weakness when extreme values distort the picture.

Measure Two

The Median

The median takes a different approach entirely. It is the middle value, once the observations are arranged from smallest to largest. With an odd number of observations, that is simply the middle one. With an even number, it is the average of the two middle values. By construction, the median divides the distribution into two equal halves. Half the observations lie below it, and half above. Now here is the defining property, and it is worth dwelling on. The median depends only on the position of values in the ordered sequence. Not on the specific values themselves. Double the largest observation in your dataset, and the median does not move at all. That robustness makes it the preferred measure for skewed data and for outliers.

Measure Three

The Mode

The mode is the value, or values, that occur most frequently. A distribution may have one mode, two, or several. Unimodal, bimodal, or multimodal. And the mode has one property the other two lack completely. It is the only measure of central tendency that applies to categorical data. Ask people their preferred political party, or their favorite color. The mean and median of those answers have no meaning whatsoever. But the modal category, the one chosen most often, is a genuinely useful summary. For continuous data the mode is less commonly reported. Individual values rarely repeat, so the raw mode is usually meaningless. Researchers refer instead to the modal interval, the histogram bin with the largest count.

On Skewed Data

How the Three Separate

On a perfectly symmetric distribution with a single peak, something convenient happens. The mean, median, and mode all coincide. One number, three names. The interesting differences appear when the distribution is asymmetric. And they follow a reliable pattern. The mean is pulled toward the long tail, because it uses every value. The median lies between the mean and the mode. And the mode sits at the location of highest frequency, the peak. On a positively skewed distribution, the order runs mode, median, mean. On a negatively skewed distribution, the order reverses. Which is why reporting all three tells a reader about the asymmetry, not just the center.

Side by Side

Seven Properties Compared

Seven properties separate the three measures cleanly. Does it use all the values? The mean does. The median uses position only, and the mode uses frequency only. Is it sensitive to outliers? The mean is, highly. The median is robust, and the mode is essentially insensitive. For nominal data, only the mode is suitable. For ordinal data, the median and the mode work, and the mean needs caution. For interval and ratio data, the mean and median both work well. As a foundation for further statistics, the mean stands alone. And one last row that catches people out. The mean and median are always uniquely defined. The mode is not. A dataset can have several modes, or in principle none that is informative.

Consideration One

Level of Measurement

Three considerations guide the choice. Start with level of measurement. Nominal data are categories without inherent order. They admit only the mode, because the other two require arithmetic on the values. Ordinal data permit the median and the mode. But not, strictly speaking, the mean. The reason is that differences between adjacent ordinal categories are not equal. The gap from strongly disagree to disagree need not match disagree to neutral. Interval and ratio data have meaningful differences, so all three measures apply. Now a practical note about what researchers actually do. Reporting a mean for Likert-scale responses is common in practice. It should be done with awareness of the limitation, not in ignorance of it.

Considerations Two and Three

Shape, and the Question Being Asked

The second consideration is the shape of the distribution. On approximately symmetric distributions, the mean is usually most informative. It incorporates all the data, and it supports further analysis. On strongly skewed distributions, the mean is pulled toward the long tail. It may not represent a typical observation at all, so the median is more representative. On bimodal or multimodal distributions, no single number captures the center well. Consider reporting the locations of the separate peaks instead. The third consideration is the theoretical question you are asking. Take average waiting times in an emergency department. If the question concerns total demand on resources, you want the mean. If the question concerns the typical patient's experience, you want the median.

The Salary Example

When the Arithmetic Is Right and the Impression Is False

One example makes the whole issue concrete, and it is a famous one. Take the mean salary at a small company. Replace one ordinary employee with a single extraordinarily paid one. The mean salary rises by hundreds of dollars per year. The arithmetic is entirely correct. The impression created is false. Most employees still earn exactly what they earned before. The median pay stays the same, and that is the point. The median answers the question, what does a typical employee earn? Finally, a rule that applies whichever measure you choose. Report central tendency together with a measure of variability. A mean alone leaves readers unable to judge how well it summarizes the data.

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FAQ

Questions students ask us

About Measuring the Center: Mean, Median, and Mode

What is the difference between the mean, median, and mode?
The mean is the sum of the values divided by how many there are, the median is the middle value once the data are sorted, and the mode is the most frequent value. For the scores 3, 5, 5, 7, and 10, the mean is 30 / 5 = 6, the median is 5, and the mode is 5. With an even number of scores, the median is the average of the two middle values.
Which measure of central tendency should be used for nominal or ordinal data?
Nominal data allow only the mode, while ordinal data allow the median and the mode. Categories such as political party or favorite color cannot be averaged or ranked, so the most frequent category is the meaningful summary. With ordinal data, such as Likert-scale responses, the gaps between adjacent categories are not necessarily equal, so a mean should be used with caution. Interval and ratio data support all three measures.
How does a skewed distribution affect the mean and median?
In a skewed distribution the mean is pulled toward the long tail, so it can sit far from a typical value, while the median resists extreme scores. In positive skew the usual order from lowest to highest is mode, median, mean, and in negative skew it reverses. If one employee at a small company earns a huge salary, the mean salary jumps while the median barely moves, so the median better describes typical pay.
Why is the mean called the balancing point of a distribution?
The mean is the balancing point because the deviations of all values from it sum to zero. For the scores 2, 4, and 9, the mean is 5, and the deviations of -3, -1, and 4 add to zero. If the values were weights on a ruler, the ruler would balance at the mean. Moving one score far out shifts the mean toward it, which is why the mean is sensitive to outliers.

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Research Methods in Psychology (Psychology 111) carries a NCCRS credit recommendation for 3 semester credits. ACE and NCCRS have over 2,100 cooperating colleges that will consider accepting credits; the final decision is with the universities.
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