Average calculator

How do I calculate the average?

Add the numbers and divide by how many there are. For 2, 4, 4, 4, 5, 5, 7 and 9 the sum is 40 and there are 8 numbers, so the average (the mean) is 40 ÷ 8 = 5.

What is the median?

The middle number once the list is sorted. With an even count, it is the average of the two middle numbers: in 2, 4, 4, 4, 5, 5, 7, 9 those are 4 and 5, so the median is 4.5. The median is less pulled by one very large or small value than the mean.

What is the mode?

The number that appears most often — 4 in the list above, which appears three times. A list can have several modes, or none if every number appears once.

What is the standard deviation?

How spread out the numbers are around the mean. For the list above it is 2 treating the numbers as a whole population, or 2.14 treating them as a sample, which divides by n − 1 instead of n.

How do I average percentages or grades with different weights?

Multiply each value by its weight, add them up and divide by the total weight. For a grade average with credit hours, use the GPA calculator, which does exactly that.

Should I use the mean or the median?

Use the mean when the values are fairly symmetric and every value should count, as with test scores. Use the median when a few extreme values would distort the picture, as with house prices or incomes, because the median depends only on the middle of the sorted list.

How do I find the number I need to reach a target average?

Multiply the target by the total count and subtract the sum so far. With scores of 82, 90, 76, a fourth score needed for an average of 85 is 85 × 4 − 248 = 92.

Is the average of two averages the overall average?

Only when the groups have the same size. Otherwise each group's average has to be weighted by its count: add up all the values in all groups and divide by the total count.

How the calculation works

The mean is the sum divided by the count. The median is the middle value of the sorted list, or the mean of the two middle values when the count is even. The mode is the most frequent value, and there can be more than one or none. The range is the largest value minus the smallest. The standard deviation measures how far the values typically fall from the mean: the population version divides the sum of squared deviations by n, and the sample version divides by n − 1, which is used when the numbers are only a sample from a larger group.

Worked example

Take the list 12, 14, 15, 15, 16, 18, 95. The sum is 185 and there are 7 numbers, so the mean is 185 ÷ 7 = 26.43. Sorted, the middle (fourth) value is 15, so the median is 15. The mode is 15 and the range is 95 − 12 = 83. The sample standard deviation is 30.29. The last value, 95, pulls the mean far above all the other numbers. Without it the mean is 15 and the median is 15, so the median was the better summary of a typical value.

Common mistakes

• Averaging percentages or rates without weights. If one test has 10 questions and another 100, the average of the two percentages is not the percentage of all questions correct.
• Ignoring outliers. A single data entry error or an extreme case can change the mean a lot. Look at the sorted list and compare the mean with the median.
• Using the wrong standard deviation. Use the sample version for a sample drawn from a larger group, and the population version only if the list is everything.

Sources and related calculators

Definitions: OpenStax Introductory Statistics, chapter 2 (Descriptive Statistics: measures of the center and of the spread of the data). Related: standard deviation calculator, weighted average calculator and percentile calculator.