Average Calculator
Paste a column of numbers and get the mean, median, mode, range and both standard deviations at once, each with its arithmetic written out. Includes the population-versus-sample distinction that decides which standard deviation you actually wanted.
Commas, spaces, tabs, semicolons and new lines all separate values, so a column pasted straight out of a spreadsheet works. Currency symbols, percent signs and accounting parentheses are understood; anything left over is listed rather than silently dropped.
(18 + 22 + 31 + … + 83) ÷ 12 = 600 ÷ 12 = 50 The arithmetic mean: the one most people mean by “average”.
Sorted: 18, 22, 31, …, 83, an even count (12), so the median is the average of the 6th and 7th values: (46 + 51) ÷ 2 = 48.5
All 12 values appear exactly once, so nothing is more common than anything else: there is no mode. No mode is a real answer, not a failure. Adding a repeat would create one.
Largest − smallest = 83 − 18 = 65
√(sample variance) = √428.72727272727275 = 20.705730432111608 Divides by n − 1. Use this when your numbers are a sample drawn from something larger.
Σ(x − mean)² ÷ (n − 1) = 4716 ÷ 11 = 428.72727272727275 Divides by n − 1. Use this when your numbers are a sample drawn from something larger.
√(population variance) = √393 = 19.82422760159901 Divides by n. Use this when your numbers ARE the whole group.
Σ(x − mean)² ÷ n = 4716 ÷ 12 = 393 Divides by n. Use this when your numbers ARE the whole group.
(18 × 22 × 31 × … × 83)^(1/12) = 45.487558964974305 The geometric mean: the right average for growth rates, ratios and anything that multiplies.
12 ÷ (1/18 + 1/22 + … + 1/83) = 12 ÷ 0.29584565082861597 = 40.56169143061571 The harmonic mean: the right average for speeds, rates and other per-unit quantities.
18 + 22 + 31 + … + 83 = 600
Common questions
- What is the difference between sample and population standard deviation?
- Sample divides by n − 1, population divides by n. Use sample when your numbers are a subset of some larger group you are trying to describe, which is almost always the case. Use population only when you genuinely have every member. The gap matters most on small sets: for five values the sample figure is about 12% larger. Both are shown, each labelled with its divisor, so nothing is chosen for you.
- Why does the standard deviation of one number say undefined instead of zero?
- Because the sample formula divides by n − 1, and with one value that is division by zero. Returning 0 is the classic bug. It states that a single measurement has no spread, which is not a finding about the data but an artefact of the formula. If that one value really is your entire population, the population standard deviation of 0 is the honest answer, and the tool says so.
- What counts as no mode?
- Only when every value appears exactly once. If several values tie for the most frequent, all of them are reported as modes along with a note that the mode is not distinguishing much. Some calculators return the first, the smallest, or zero, all of which are wrong in a way you cannot see from the answer.
- Which average should I use?
- The arithmetic mean for quantities that add up. The median when outliers would drag the mean somewhere unrepresentative, which is why incomes are reported as medians. The geometric mean for rates of growth and ratios. The harmonic mean for averaging speeds or other rates over a fixed distance. All four are computed, each named, because "average" alone is ambiguous.
- How should I paste my numbers in?
- However you have them. Commas, spaces, tabs, newlines, semicolons and pipes all work, so a copied spreadsheet column goes straight in. Currency symbols, percent signs and accounting parentheses for negatives are understood. Anything that is not a number is skipped and named, rather than silently treated as zero.
- What happens with 1,234: is that one number or two?
- It is read as one number, 1234, because that is the far more common intent. But "3,14" is read as two numbers, since a European decimal comma is equally plausible there. Both readings are genuinely ambiguous, so the tool tells you which one it used and how to write the other. Silently averaging a European column as if the commas were separators is the failure this avoids.
Exact for median, mode, minimum, maximum and range. The mean, variances and standard deviations use compensated summation and are accurate to roughly 15 significant digits, not claimed as exact, because double-precision arithmetic is not.