Arithmetic Mean Calculator

What is the arithmetic mean?

The arithmetic mean, what most people simply call the average, is the single value you would get if the total were shared out equally among all the items. Add up the numbers, then divide by how many there are.

It is one of three common measures of the centre of a data set. The others are the median, the middle value when the list is sorted, and the mode, the value that occurs most often. The mean, median and mode page compares all three side by side.

Arithmetic mean formula

mean = (x1 + x2 + … + xn) ÷ nn is the number of values in the list

Example: the values 12, 15, 21, 18 and 24 add up to 90. There are 5 of them, so the mean is 90 ÷ 5 = 18.

A second example: five test scores of 72, 85, 90, 68 and 85 total 400, giving a mean of 400 ÷ 5 = 80. Notice that the mean does not have to equal any of the scores in the list.

How to use the calculator

1. Type or paste your numbers into the box. Commas, spaces, tabs and new lines all separate values. Use a dot for decimals; negatives are fine.
2. Press Find the mean. The first line is the mean and the second is the count of numbers.

The mean is printed with up to twelve decimals, so a repeating result such as 16.666666666667 shows its rounding rather than hiding it. Whole-number means appear without a decimal part, and results with a few decimals are padded (3.50 rather than 3.5). Separators are only commas and white space: a semicolon is not a separator, so 3;5;10 is read as the single number 3510. Anything inside an entry that is not part of a number is dropped, and an entry with no digits at all counts as 0.

Useful properties of the mean

• Balance point. The differences between each value and the mean always add up to zero. For 12, 15, 21, 18, 24 and the mean 18 the differences are −6, −3, +3, 0, +6, which total 0.
• Shifting. Adding a constant to every value adds the same constant to the mean: add 5 to each number above and the mean becomes 23.
• Scaling. Multiplying every value by a constant multiplies the mean by it: doubling the list doubles the mean to 36.
• Updating. You can fold a new value into an existing mean without re-adding everything: new mean = (old mean × n + new value) ÷ (n + 1). Adding 30 to the list above gives (18 × 5 + 30) ÷ 6 = 20.
• Best single guess. The mean is the number that minimises the sum of the squared distances to the data, which is why it underpins variance and regression.

When the mean can mislead

Because every value counts equally, a few extreme numbers can drag the mean far from where most of the data sits. Take five annual salaries in thousands: 40, 42, 45, 47 and 250. The mean is 424 ÷ 5 = 84.8, yet four of the five people earn less than 50. The median, 45, describes the group far better.

• Skewed data. Incomes, house prices and waiting times have a long tail on one side, which pulls the mean above the median.
• Mixed groups. If a list blends two different populations, the mean may fall in a gap where no real observation lives.
• Averages of averages. Averaging group means ignores group sizes. Use the weighted mean calculator when the groups differ in size.
• Rates and growth. For speeds over equal distances use the harmonic mean, and for compounding growth factors use the geometric mean.

A good habit is to look at the whole list, or at a histogram, before trusting any one summary number.

Averaging angles and times of day

Quantities that wrap around, such as compass bearings or clock times, break the ordinary mean. The arithmetic mean of 350° and 10° is 180°, pointing the opposite way from both directions. The sensible answer is 0° (north), which you get by converting each angle to a point on a circle, averaging the points, and converting back.

Times behave the same way. Two people going to bed at 23:00 and 01:00 have a naive average of 12:00 noon, while the circular average is midnight. For data like this, use a circular mean instead of this calculator.