Mean, Median, Mode, Range Calculator

Mean, median and mode are the three common measures of central tendency, each answering the question of what a typical value is in a slightly different way. The range adds a quick sense of how spread out the values are. For the default data, 10, 2, 38, 23, 38, 23 and 21, the mean is about 22.14, the median is 23, there are two modes, 23 and 38, and the range is 36.

Mean

The mean, or arithmetic mean, is what most people call the average. Add all the values and divide by how many there are. For the default list, the sum is 155 and there are 7 values, so the mean is 155 ÷ 7 ≈ 22.142857. The mean of a population is usually written μ, and the mean of a sample x̄.

The mean uses every value, which makes it the most informative measure for symmetric data without extreme values, and it is the basis of standard deviation, regression and most of statistics. That same property makes it sensitive to outliers. If nine employees earn $50,000 and one earns $1,050,000, the mean salary is $150,000, a figure that describes none of them.

A useful way to picture the mean is as a balance point. If the values were equal weights placed along a ruler, the ruler would balance at the mean, because the total distance of values above it equals the total distance of values below it.

Other kinds of mean suit other problems. The weighted mean multiplies each value by a weight before averaging, as in a course grade where the final exam counts more than homework. The geometric mean, the nth root of the product of n values, is the right average for growth rates: an investment that gains 50% one year and loses 50% the next has an arithmetic mean return of 0% but actually loses 25%, which the geometric mean of 1.5 and 0.5, about 0.866, correctly reflects. The harmonic mean, n divided by the sum of reciprocals, averages rates such as speeds over equal distances.

x̄ = (x₁ + x₂ + … + xₙ) ÷ n

x₁ … xₙ

the data values

n

how many values there are

Median

The median is the middle value once the data are sorted from smallest to largest. Half of the values are at or below it and half are at or above it. To find it, sort the list and count in from both ends. With an odd number of values there is a single middle one: the sorted default data are 2, 10, 21, 23, 23, 38, 38, and the fourth value, 23, is the median.

With an even number of values there are two middle values, and the median is their mean. For 4, 7, 9 and 15, the middle pair is 7 and 9, so the median is 8. In general, for n sorted values the median sits at position (n + 1) ÷ 2, and a position ending in .5 means averaging the two neighbors.

The median's great strength is that it ignores how extreme the extreme values are. In the salary example, the median is $50,000 no matter how large the top salary becomes. That is why household income, home prices and similar data with a long tail of large values are usually reported by the median. In a symmetric distribution the mean and median are close; when the mean is well above the median, the data are skewed to the right, and when it is below, they are skewed to the left.

The median is also the second quartile. Splitting the sorted data at the median and taking the median of each half gives the first quartile Q1 and the third quartile Q3. This calculator excludes the overall median from both halves when the count is odd. The distance from Q1 to Q3, the interquartile range, holds the middle half of the data, and values more than 1.5 interquartile ranges beyond the quartiles are flagged as outliers. Different textbooks and software use slightly different quartile conventions, so small differences between tools are normal.

Mode

The mode is the value that appears most often. In the default list, 23 and 38 each appear twice and every other value once, so the data are bimodal with modes 23 and 38. A data set with one mode is unimodal, and one with more than two is multimodal. When every value appears exactly once, there is no mode, because no value is more common than the rest.

Unlike the mean and median, the mode works for categories as well as numbers. You cannot average shirt sizes or favorite colors, but you can say that medium is the most common size sold. Retailers, survey researchers and anyone tallying votes are really looking for the mode.

For continuous measurements, such as weights recorded to the nearest gram, exact repeats may be rare, so the mode is usually found from grouped data: the most common interval in a histogram, called the modal class. Peaks in a histogram are also the visual sign of multiple modes, which can reveal that data come from two different groups, such as the heights of men and women combined.

Range

The range is the simplest measure of spread: the largest value minus the smallest. For the default data it is 38 − 2 = 36. It is quick to compute and easy to explain, and it is useful for things like the spread between the high and low temperatures of a day.

Because it depends only on the two most extreme values, a single unusual measurement can make the range much larger. The interquartile range, which covers the middle 50% of the values, and the standard deviation are more stable measures of spread for most purposes.

Frequently asked questions