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.
Your results will appear here: the standard deviation first, then the rest of the summary and a chart.
Standard deviation (population)
2.31391
2.31
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
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
More statistics
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.
- Sum = 183, so μ = 183 ÷ 12 = 15.25.
- 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.
- Add them: Σ(x − μ)² = 64.25.
- Divide by N = 12: σ² = 5.35417.
- 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
| Data | Population? | Why |
|---|---|---|
| Heights of the 25 players on a roster, describing the roster | Yes | Every member is measured and the claim stops there |
| All 40 batches produced in a closed lot | Yes | The lot is finished; nothing further will be added |
| Monthly sales for every month of 2025 | Yes, for describing 2025 | A sample if used to forecast 2026 |
| 200 survey respondents out of a city | No | The conclusion is about the city |
| Ten repeat measurements of one standard | No | They 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:
σxin the 1-Var Stats output - Casio:
σxorxσn - NumPy:
np.std(x): the default,ddof=0 - pandas:
df['x'].std(ddof=0) - R:
sqrt(mean((x - mean(x))^2)), sincesd()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.
Related calculators
-
Sample standard deviation
Dividing by n − 1, for data drawn from a larger group.
-
Population variance
σ², the population spread in squared units.
-
Sample vs population
Which denominator to use, and how much it changes the answer.
-
Standard deviation calculator
The general calculator, with both modes and a dot plot.
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.
What is the difference between a standard deviation and a population standard deviation?
"Standard deviation" on its own usually means the sample version, s, which divides by n − 1. The population standard deviation σ divides by N and is used when the data is the entire group. On the same data σ is slightly smaller.
When does it matter whether my data are a sample or the whole population?
It matters whenever your numbers might be all of the thing you are describing. The two formulas give different answers because the sample version (n − 1 divisor) corrects for the fact that a sample's own mean sits closer to its values than the true population mean does. If the list is genuinely every member of the group, the population formula is the right one and the n − 1 correction is not needed; if the list is a sample from a larger group, the sample formula is the standard choice, which is also what most software defaults to.
Population standard deviation calculator FAQ
Common questions
Can I calculate population standard deviation without Excel?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
How do I calculate population standard deviation when values repeat?
When values repeat, you can enter each occurrence individually or use a frequency table. Both give the same result. The frequency table standard deviation calculator is more efficient for datasets with many repeated values.
Is there a free population standard deviation calculator?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
Can I calculate population standard deviation from a list of numbers?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
How can I calculate population standard deviation step by step?
To calculate standard deviation: (1) Find the mean of the dataset. (2) Subtract the mean from each value. (3) Square each deviation. (4) Sum the squared deviations. (5) Divide by n − 1 (sample) or n (population). (6) Take the square root. See how to calculate standard deviation for a worked example.
What is the easiest way to calculate population standard deviation?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
Can a population standard deviation calculator show the working?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
Can I calculate population standard deviation on my phone?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
Which inputs do I need for population standard deviation?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
Can I paste comma-separated values into a population standard deviation calculator?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
How do I calculate population standard deviation for decimal values?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
Can I calculate population standard deviation for negative numbers?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
How do I calculate population standard deviation for a small dataset?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
How do I calculate population standard deviation for a large dataset?
For a large population or dataset, the difference between sample SD (n − 1) and population SD (n) becomes negligible: with n = 1000 the gap is only about 0.1%. For large datasets, use the standard deviation calculator or the large dataset standard deviation calculator which uses Welford's numerically stable method.
Can I download the results from a population standard deviation calculator?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
How accurate is an online population standard deviation calculator?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
What should I do if a population standard deviation calculator gives an unexpected result?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
What is the a Casio scientific calculator formula for population standard deviation?
The population standard deviation formula is σ = √[ Σ(x − μ)² / N ]. The symbols are: σ = population SD, μ = population mean, x = each value, N = population size. The divisor is N because every member is included. See standard deviation formula for more detail.
Can I calculate population standard deviation from a frequency table?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
How do I calculate population standard deviation online?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
How do I check my population standard deviation calculation?
Standard deviation is a measure of how spread out a set of numbers is around their mean. It is the square root of the average squared distance from the mean, expressed in the same units as the original data. A small standard deviation means values cluster close to the mean; a large one means they are widely scattered. See what is standard deviation for a full explanation with examples.
What is the a TI-84 calculator formula for population standard deviation?
The population standard deviation formula is σ = √[ Σ(x − μ)² / N ]. The symbols are: σ = population SD, μ = population mean, x = each value, N = population size. The divisor is N because every member is included. See standard deviation formula for more detail.