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.

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.

How do I calculate average annual growth rate or average percentage increase?

Choose Growth rates and enter the observed change for each complete year as a percentage. For 20% and -10%, the arithmetic mean annual change, or AAGR, is 5%; the compound annual equivalent is about 3.923048%, because 100 becomes 120 then 108. Choose Months for monthly observations: the mean is labelled monthly and the compound annual equivalent uses 12 months per year. A part-year annual equivalent describes the entered factors and does not predict repetition. If you only have starting and ending amounts plus elapsed time, use the CAGR page in ROI Calculator; two endpoints do not reconstruct individual annual observations. In Excel, =AVERAGE(D2:D6) averages annual percentages stored as decimals. Format its result as a percentage. This browser field reads 20% as the numeric percentage 20, so its mean of 5 represents 5%.

How do I average percentages with different group sizes?

Use All statistics for equally weighted percentages: 50% and 100% have a mean of 75%. For an overall percentage when one group has 10 observations and the other has 90, choose Weighted mean and enter 50 10 on one line and 100 90 on the next. The result is 95%. The denominator or group size is the weight; averaging the percentages equally would answer a different question. For course categories and grade targets, use Grades Calculator with the known category weights.

Can I average decimal amounts, ages, salaries or temperatures?

Yes, when you supply the numeric observations in one consistent unit. Decimals and negative temperatures can use the ordinary arithmetic mean. Currency symbols are removed from numbers, but currencies are not converted and income statistics are not looked up. Enter each observation, choose the statistic you need and read any notice about a skipped or ambiguous entry.

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. It includes the population-versus-sample distinction that decides which standard deviation you actually wanted. Switch to Weighted mean for values with different weights, or Growth rates to compare an arithmetic mean of observed annual or monthly percentages with their compound annual equivalent.

Arithmetic mean · 12 values50median 48.5 · range 18 to 83 · sample SD ≈ 20.7057

What to work out

Values: 18, 22, 31, 39, 44, 46, 51, 58, 62, 70, 76, 83

Featured result

Sample or population

Count: 12

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.

Meansum ÷ count
50
(18 + 22 + 31 + … + 83) ÷ 12 = 600 ÷ 12 = 50 The arithmetic mean: the one most people mean by “average”.
Medianthe middle of the sorted list
48.5
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
Modeno value repeats
None
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.
Rangelargest − smallest
65
Largest − smallest = 83 − 18 = 65
Standard deviation (sample)divides by n − 1 · your choice
20.705730432111608
√(sample variance) = √428.72727272727275 = 20.705730432111608 Divides by n − 1. Use this when your numbers are a sample drawn from something larger.
Variance (sample)divides by n − 1 · your choice
428.72727272727275
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.
Standard deviation (population)divides by n
19.82422760159901
√(population variance) = √393 = 19.82422760159901 Divides by n. Use this when your numbers ARE the whole group.
Variance (population)divides by n
393
Σ(x − mean)² ÷ n = 4716 ÷ 12 = 393 Divides by n. Use this when your numbers ARE the whole group.
Geometric meanfor growth rates and ratios
45.487558964974305
(18 × 22 × 31 × … × 83)^(1/12) = 45.487558964974305 The geometric mean: the right average for growth rates, ratios and anything that multiplies.
Harmonic meanfor speeds and other per-unit rates
40.56169143061571
12 ÷ (1/18 + 1/22 + … + 1/83) = 12 ÷ 0.29584565082861597 = 40.56169143061571 The harmonic mean fits positive rates across equal quantities, such as speeds over equal distances; equal-time speeds instead use the arithmetic mean.
Sumevery value added up
600
18 + 22 + 31 + … + 83 = 600

Sample or population is not a formatting choice. Sample divides by n − 1, population by n, and on eight numbers the two answers differ by about 7%. Use sample when your numbers are a subset of something larger, which is almost always. A tool that shows one of them and calls it “standard deviation” has picked for you without saying so. Both are above, each labelled with the divisor it used.

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.
How do I calculate average annual growth rate or average percentage increase?: Choose Growth rates and enter the observed change for each complete year as a percentage. For 20% and -10%, the arithmetic mean annual change, or AAGR, is 5%; the compound annual equivalent is about 3.923048%, because 100 becomes 120 then 108. Choose Months for monthly observations: the mean is labelled monthly and the compound annual equivalent uses 12 months per year. A part-year annual equivalent describes the entered factors and does not predict repetition. If you only have starting and ending amounts plus elapsed time, use the CAGR page in ROI Calculator; two endpoints do not reconstruct individual annual observations. In Excel, =AVERAGE(D2:D6) averages annual percentages stored as decimals. Format its result as a percentage. This browser field reads 20% as the numeric percentage 20, so its mean of 5 represents 5%.
How do I average percentages with different group sizes?: Use All statistics for equally weighted percentages: 50% and 100% have a mean of 75%. For an overall percentage when one group has 10 observations and the other has 90, choose Weighted mean and enter 50 10 on one line and 100 90 on the next. The result is 95%. The denominator or group size is the weight; averaging the percentages equally would answer a different question. For course categories and grade targets, use Grades Calculator with the known category weights.
Can I average decimal amounts, ages, salaries or temperatures?: Yes, when you supply the numeric observations in one consistent unit. Decimals and negative temperatures can use the ordinary arithmetic mean. Currency symbols are removed from numbers, but currencies are not converted and income statistics are not looked up. Enter each observation, choose the statistic you need and read any notice about a skipped or ambiguous entry.

Uses browser double precision with compensated sums and stable variance calculations. Median positions and mode frequencies are exact for the parsed numbers; arithmetic results can round. Unrepresentable inputs and intermediate results are explained.