Mean, Median, Mode Calculator

Mean, median and mode at a glance#

This calculator finds the four most common summary statistics for a data set. The mean is the sum of the values divided by how many there are. The median is the middle value once the numbers are sorted. The mode is the value that appears most often. The range is the largest value minus the smallest.

Take the data set 2, 3, 3, 5, 7. Sorted, it is already in order. The mean is 2 + 3 + 3 + 5 + 7 = 20, divided by 5 values, which is 4. The median is the middle of the five sorted values, which is 3. The mode is 3, because it appears twice and every other value appears once. The range is 7 minus 2, which is 5.

To find the median when the count is even, average the two middle values once sorted. A data set can have more than one mode, or none if every value appears the same number of times.

Enter your numbers in the calculator above for the exact mean, median, mode and range. Results follow standard definitions, so rounding and how you handle ties or repeated values can shift the final figures.

Which measure of center to use#

All three describe the center of a data set, but they answer different questions. The mean uses every value, so it is the right summary for symmetric data without extreme values. The median ignores the size of the outer values and reports only the middle position, so it holds up better when a few very large or very small numbers would drag the mean off. The mode reports the most common value, which is the only one of the three that works on categories rather than numbers.

For incomes or house prices, where a handful of high values pull the average upward, the median is usually the more honest single number. For a fair coin, dice roll, or any roughly even spread, the mean and median sit close together.

Finding the median by hand#

Sort the values from smallest to largest, then find the middle position. With an odd count, the median is the single middle value. With an even count, there is no single middle, so the median is the average of the two middle values.

For the set 4, 8, 15, 16, 23 there are five values, so the median is the third one, 15. For the set 4, 8, 15, 16 there are four values, so the median is the average of the two middle ones, (8 + 15) / 2 = 11.5. The median does not have to be a value that appears in the data.

When the mode is unclear#

A data set can have one mode, several modes, or none. Two values tied for most frequent make it bimodal; three or more make it multimodal. If every value appears the same number of times, there is no mode at all. For 2, 4, 4, 7, 9, 9 both 4 and 9 appear twice, so the set is bimodal with modes 4 and 9.

Reading the three together#

Comparing the three measures tells you about the shape of the data. When the mean is well above the median, the data is skewed by high values; when it is well below, low values pull it down. When the mean, median, and mode all land close together, the data is roughly symmetric. The range, the largest value minus the smallest, then tells you how spread out those values are around the center.

Frequently asked questions#

How do you find the median with an even number of values?#

Sort the values and average the two in the middle. For test scores 55, 60, 65, 70, 75, 80, the two middle values are 65 and 70, so the median is (65 + 70) / 2 = 67.5. The median can be a number that does not appear in the data.

What does it mean when a data set has more than one mode?#

Two values tied for most frequent make the set bimodal; three or more make it multimodal. If every value appears the same number of times, there is no mode. The mean and median are always single numbers, but the mode can be a list or empty.

Why is the median better than the mean for incomes?#

A few very high incomes raise the mean far above what most people earn, so the mean overstates the typical income. The median reports the middle earner and ignores how extreme the top values are, so it represents the typical case more fairly in skewed data.

How do outliers affect the mean, median, and mode?#

Outliers move the mean the most, because the mean adds in every value. The median shifts little, since it depends only on the middle position. The mode does not respond to one extreme value at all, because it depends on frequency, not size.

What is the difference between sample and population data?#

A population is every member of the group you care about; a sample is a subset you actually measure. The mean, median, and mode are calculated the same way for both. The distinction matters more for measures of spread like variance, which use a slightly different divisor for samples.

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

Yes. In a perfectly symmetric data set with a single peak, such as 3, 4, 4, 4, 5, all three equal 4. The further apart the three values are, the more skewed or spread out the data is.