standarddeviationcalculator.net

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Population standard deviation calculator

The population standard deviation, σ, measures the spread of a complete group: every member is in the data, so nothing is being estimated. Enter the values below. The calculator divides by n and shows the sample figure alongside, in case your data turns out to be a sample after all.

Separate numbers with commas, spaces or new lines, or paste a spreadsheet column. Decimals and negatives are fine; write 10:3 for a value that occurs 3 times.

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Calculation type

Not sure which? How to choose sample or population

Standard deviation (population)

2.31391

Your values typically sit about 2.31 above or below their mean of 15.25, in the same units as your data. 8 of 12 values (67%) fall between 12.94 and 17.56, within one standard deviation of the mean; for normally distributed data about 68% would.

Sample SD (s): 2.4168, if these values are a sample from a larger group.

Count (n)
12
Mean (x̄)
15.25
Variance (σ²)
5.35417
Standard error
0.667967
Minimum
11
Q1 (25%)
13.75
Median
15.5
Q3 (75%)
17
Maximum
19
Range
8
More statistics (5)
Relative SD (%RSD)
15.1732%
Coefficient of variation
0.151732
Sum (Σx)
183
Sum of squares, Σ(x − x̄)²
64.25
IQR (Q3 − Q1)
3.25

Data distribution

10 12 14 16 18 20 22 mean 15.25 −1 SD +1 SD 14 — 0.54 SD below the mean17 — 0.756 SD above the mean12 — 1.4 SD below the mean19 — 1.62 SD above the mean15 — 0.108 SD below the mean16 — 0.324 SD above the mean13 — 0.972 SD below the mean18 — 1.19 SD above the mean15 — 0.108 SD below the mean11 — 1.84 SD below the mean17 — 0.756 SD above the mean16 — 0.324 SD above the mean Value

Shaded bands mark ±1, ±2 and ±3 SD from the mean. 8 of 12 values (67%) fall within ±1 SD.

Chart as text

Mean 15.25, population standard deviation σ = 2.31391, from 12 values between 11 and 19.

  • Within ±1 SD (12.94 to 17.56): 8 of 12 values (67%). About 68% for normal data.
  • Within ±2 SD: 12 (100%). About 95% for normal data.
  • Within ±3 SD: 12 (100%). About 99.7% for normal data.
Show the working, step by step

The population standard deviation formula

σ = √( Σ(x − μ)² ÷ N )

where μ is the population mean and N the number of members. Every squared deviation is averaged over all N values. There is no n − 1 correction, because the mean is the true mean of the group rather than an estimate of some larger one.

A worked example: one class's quiz scores

A teacher records the scores of all 12 students in a class on a 20-point quiz: 14, 17, 12, 19, 15, 16, 13, 18, 15, 11, 17, 16. The report is about this class and no other, so the twelve scores are the population.

  1. Sum = 183, so μ = 183 ÷ 12 = 15.25.
  2. Square each score's distance from 15.25. The student who scored 11 contributes (−4.25)² = 18.0625; the two scoring 15 contribute 0.0625 each.
  3. Add them: Σ(x − μ)² = 64.25.
  4. Divide by N = 12: σ² = 5.35417.
  5. Take the square root: σ = 2.31391 points.

Had the teacher treated the class as a sample of all students taking the course, the answer would be s = 2.41680, dividing by 11 instead. Both are loaded above; the result card shows the sample figure beneath the headline.

Real populations, and data that only looks like one

DataPopulation?Why
Heights of the 25 players on a roster, describing the rosterYesEvery member is measured and the claim stops there
All 40 batches produced in a closed lotYesThe lot is finished; nothing further will be added
Monthly sales for every month of 2025Yes, for describing 2025A sample if used to forecast 2026
200 survey respondents out of a cityNoThe conclusion is about the city
Ten repeat measurements of one standardNoThey sample what the instrument would read on repetition

The test is not how many values there are but what you intend to say about them. The sample vs population guide works through more borderline cases.

Population standard deviation in software

  • Excel / Google Sheets: =STDEV.P(A1:A12)
  • TI-84: σx in the 1-Var Stats output
  • Casio: σx or xσn
  • NumPy: np.std(x) — the default, ddof=0
  • pandas: df['x'].std(ddof=0)
  • R: sqrt(mean((x - mean(x))^2)), since sd() always uses n − 1

For the rest of the population summary, the population variance calculator reports σ² as its headline, and the standard deviation calculator switches freely between the two modes.

Common questions

When is my data a whole population?

When the values you have are every member of the group your conclusion is about, and you are not generalising beyond them. The quiz scores of one class, reported for that class; the fill weights of every bottle in a finished lot; census counts for every county in a state. If you would say "students like these" or "the process in general", it is a sample.

Which Excel function gives the population standard deviation?

=STDEV.P(range). The older STDEVP does the same. Google Sheets accepts both spellings. STDEV.S and plain STDEV give the sample version instead.

Is σx on a TI-84 the population standard deviation?

Yes. After 1-Var Stats, σx divides by n and Sx divides by n − 1. Casio calculators label them σx (or xσn) and sx (or xσn−1).

Can I use the population formula for a large sample?

You can, and the error is small: with n = 200 the two differ by 0.25%. But it is still the wrong formula for a sample, and there is no benefit to using it, so reports and exams expect n − 1 whenever the data is a sample, however large.

What is the symbol for population standard deviation?

The Greek letter sigma, σ, and its square σ² for the population variance. The population mean is μ. See the notation guide for how to type them.

Written and reviewed by our editorial team. Last updated . Method and sources: how these numbers are computed.