Mean vs Median vs Mode: Which Average Should You Use?

Short answer: the mean (add up the values and divide by the count) is best for evenly spread data. The median (the middle value when sorted) is better when a few extreme values would pull the mean away from the typical case. The mode (the most frequent value) is the right choice for the most common category or size. When someone says "average", they usually mean the mean, but not always, so ask.

What are the mean, median and mode?

• Mean: the sum of all values divided by how many there are.
• Median: the middle value of the sorted list. With an even count, it is the average of the two middle values.
• Mode: the value that appears most often. A data set can have one mode, several modes, or none.

All three summarise "the centre" of a data set, and they can give very different answers for the same numbers. Mathify's mean, median and mode calculator computes all three together, and the statistics calculator adds the spread.

When does the mean mislead?

When a small number of values are far from the rest. The example below uses made-up salaries in thousands, chosen to show the effect, for eight people at a small company:

32, 34, 35, 36, 38, 40, 41, 250

• Mean: 63.25
• Median: 37

The mean is higher than seven of the eight salaries. One value of 250 pulls it up by more than 26 compared with the median. If you reported 63.25 as the "average salary", most people on that list would not recognise it. The median of 37 describes the typical person far better.

The reverse also happens. Remove the 250 and the mean of the remaining seven is 36.57 with a median of 36. When data is evenly spread, the two are close.

When does the median mislead?

The median ignores how large the extreme values are. If you are budgeting a total cost, the total depends on the sum, and the sum is the mean times the count. The median would hide the expensive item entirely.

Another case: a data set with two clusters. We computed the values 1, 1, 1, 2, 9, 10, 10, 10:

• Mean: 5.5
• Median: 5.5
• Modes: 1 and 10 (each appears three times)

Here the mean and median both sit at 5.5, a number that nobody in the data set is near. The modes show the two real groups. A histogram or a look at the sorted values is the better check.

When is the mode the right answer?

When you want the most common value, especially for categories or sizes. A shop stocking shoes cares about the most frequent size more than the arithmetic centre. For the sizes 6, 7, 7, 8, 8, 8, 9, 10, 13:

• Mode: 8 (appears three times)
• Median: 8
• Mean: 8.44

The mean of 8.44 is not a size anyone wears, and the one large size, 13, pulls it up slightly.

Mode is also the only one of the three that makes sense for labels such as "most common colour", where adding or sorting is meaningless.

Can a data set have no mode?

Yes. If every value appears exactly once, there is no mode. For 4, 8, 15, 16, 23, 42 the mean is 18 and the median is 15.5, and Mathify's calculator reports no mode. A set with two equally frequent values has two modes, as in the bimodal example above.

What about a skewed or capped set?

Test scores show how all three can differ. For 70, 70, 70, 70, 100:

• Mean: 76
• Median: 70
• Mode: 70

Four people scored 70 and one scored 100. The mean of 76 is above what four out of five people got. Whether 76 or 70 is the better description depends on your question: the mean is correct for a class total, and the median or mode describes the typical student.

How should you report an average?

State which one you used, how many values there are, and, where it matters, the spread. The standard deviation calculator measures how far values typically sit from the mean. A mean of 63.25 with a very large spread tells a different story from the same mean with a small spread.

Common mistakes

• Saying "average" without saying mean, median or mode.
• Using the mean for skewed data, such as incomes, prices or house sales, where a few high values dominate.
• Assuming the mean, median and mode are always close. They coincide only for roughly symmetric data with a single peak.
• Averaging averages from groups of different sizes without weighting.
• Reporting a mode when every value appears only once.
• Forgetting that the median needs the values sorted first.

Key takeaways

• Mean: sum divided by count. Sensitive to extreme values.
• Median: middle of the sorted list. Resistant to extremes, but blind to their size.
• Mode: most frequent value. May be one value, several, or none.
• Check the numbers: if the mean and median differ a lot, the data is skewed or has outliers.
• All examples here use invented values to show the arithmetic. Real data sets will differ.

FAQ

Is the average the mean or the median?

In everyday use "average" most often means the mean, but some writers use it loosely for the median or even the mode. If the number matters, check how it was calculated.

Why is the median salary usually quoted rather than the mean?

Because a few very high values can raise the mean well above what most people earn, as in the eight-value example above. The median is not moved by a single extreme value. We are not quoting any real salary data here.

How do I find the median of an even number of values?

Sort the values, take the two in the middle, and average them. For 4, 8, 15, 16, 23, 42 the middle pair is 15 and 16, so the median is 15.5.

Can the mean, median and mode all be the same?

Yes. For a symmetric data set with one peak, such as 2, 4, 4, 4, 6, all three equal 4.

What is the difference between sample and population standard deviation?

The population version divides by the number of values, while the sample version divides by one less. Use the sample version when your data is a sample from a larger group. Mathify's standard deviation page reports both versions.