What is the arithmetic mean?
For values x₁ through xₙ, add every observation and divide by n, the number of observations. In x̄ = Σxᵢ / n, x̄ names the mean, Σ means “add,” xᵢ represents each entered value, and n is the count.
In everyday language, average often means the arithmetic mean, but average can also refer to a median, mode, weighted mean, or another measure. This calculator uses the arithmetic mean and keeps the supporting statistics visible so the result can be checked in context.
Calculate the mean in three steps
Use the same seven-number data set shown in the calculator: 10, 2, 38, 23, 38, 23, 21.
- Add the values
10 + 2 + 38 + 23 + 38 + 23 + 21 = 155. Repeated values remain separate observations, so both 23s and both 38s are included.
- Count the values
There are 7 observations, so n = 7. The count—not the number of unique values—becomes the denominator.
- Divide sum by count
155 / 7 = 22.142857… . Report only as many decimal places as the measurements and the decision actually justify.
Mean, median, mode, and range answer different questions
Mean: 22.142857…
The mean uses every value: 155 divided by 7. It is the arithmetic balance point of this data set.
Median: 23
After sorting to 2, 10, 21, 23, 23, 38, 38, the median is the middle value. It describes position rather than using the total.
Modes: 23 and 38
Both values occur twice, more often than any other value. A data set can have more than one mode or no repeated mode.
Range: 36
The range is maximum minus minimum: 38 − 2 = 36. It measures the full span and can change sharply when one extreme value is added.
See how an outlier can move the mean
For 2, 3, 3, and 20, the mean is 7 while the median is 3. The 20 belongs in the arithmetic total, so the mean correctly moves toward it; however, 7 is not close to three of the four observations. That is why the mean can be mathematically correct without describing a typical case well.
Use the descriptive statistics calculator when you need frequencies, quartiles, and several measures of center together. Use the standard deviation calculator when variation around the mean matters as much as the center.
Use the mean with the right data
- Negative numbers, decimals, and genuine zeros can all be averaged. Missing data is not zero: inserting a guessed zero changes both the sum and the count.
- Keep values in compatible units and with the same meaning. A mean of mixed dollars, percentages, and counts has no useful interpretation merely because the arithmetic runs.
- Do not average group averages unless the groups have equal sizes or equal weighting is intentional. Reconstruct the combined sum and count when group sizes differ.
- For a nonterminating quotient, the fraction calculator can preserve an exact ratio before you choose a decimal display. Round the final report according to the source data, not the number of digits on screen.