Free tool Standard deviation calculator and formula, step by step
Paste a list of numbers in any format: one per line, separated by spaces or commas, or copied from a spreadsheet. You get both kinds of standard deviation, the variance, mean, median, mode, quartiles and more, plus the formula filled in with your own numbers, step by step.
- Sample (s) and population (σ)
- 17 descriptive statistics
- Runs locally
Paste your numbers
Results update as you type. Anything that is not a number is skipped and listed, so a stray header or unit will not break the calculation.
Separate values with spaces, line breaks, tabs, commas or semicolons. Negative numbers and decimals are fine. Do not use thousands separators: “1,200” is read as 1 and 200.
Software disagrees on quartiles. Pick the one that matches the tool you compare against.
Results
Sample standard deviation (s)
5.1001
Divides by n − 1. Use for a sample. Excel STDEV.S.
Population standard deviation (σ)
4.8384
Divides by n. Use when you have every value. Excel STDEV.P.
Mean (x̄)
30.3
Count (n)
10
Sample variance (s²)
26.011
Population variance (σ²)
23.41
Median
29.5
Mode
None
Minimum
23
Maximum
41
Range
18
Max − min.
Sum
303
Q1 (25th percentile)
27.25
By the selected method.
Q3 (75th percentile)
32.5
By the selected method.
Interquartile range
5.25
Q3 − Q1.
Standard error of the mean
1.6128
s ÷ √n.
Coefficient of variation
16.83%
s ÷ |mean|.
Standard deviation, step by step
The formulas below with your numbers substituted.
| Step | Calculation |
|---|---|
| Count | n = 10 |
| Sum | Σx = 23 + 31 + 27 + 35 + 29 + 26 + 41 + 30 + 28 + 33 = 303 |
| Mean | x̄ = Σx ÷ n = 303 ÷ 10 = 30.3 |
| Sum of squared deviations | Σ(x − x̄)² = (23 − 30.3)² + (31 − 30.3)² + (27 − 30.3)² + (35 − 30.3)² + (29 − 30.3)² + (26 − 30.3)² + (41 − 30.3)² + (30 − 30.3)² + (28 − 30.3)² + (33 − 30.3)² = 234.1 |
| Population variance | σ² = Σ(x − x̄)² ÷ n = 234.1 ÷ 10 = 23.41 |
| Population standard deviation | σ = √σ² = √23.41 = 4.83839 |
| Sample variance | s² = Σ(x − x̄)² ÷ (n − 1) = 234.1 ÷ 9 = 26.0111 |
| Sample standard deviation | s = √s² = √26.0111 = 5.10011 |
Deviation table
Each value’s distance from the mean and its square: the raw material of the variance.
| # | x | x − x̄ | (x − x̄)² |
|---|---|---|---|
| 1 | 23 | −7.3 | 53.29 |
| 2 | 31 | 0.7 | 0.49 |
| 3 | 27 | −3.3 | 10.89 |
| 4 | 35 | 4.7 | 22.09 |
| 5 | 29 | −1.3 | 1.69 |
| 6 | 26 | −4.3 | 18.49 |
| 7 | 41 | 10.7 | 114.49 |
| 8 | 30 | −0.3 | 0.09 |
| 9 | 28 | −2.3 | 5.29 |
| 10 | 33 | 2.7 | 7.29 |
23, 26, 27, 28, 29, 30, 31, 33, 35, 41Share these inputs: the values go after the # in the link, so they never reach a server.
https://tapcub.com/tools/standard-deviation-calculator
Version v1.0Updated Update history
What the spread is telling you
The mean says where the data sit; the standard deviation says how far a typical value is from there. The default example is ten days of sign-ups.
Sample vs population SD
Ten days are a sample of all days, so the sample SD, 5.10 sign-ups, is the one to report. The population SD (4.84) is smaller because it divides by n instead of n − 1; with large n the two converge.
Mean ± SD
For roughly bell-shaped data, about 68% of values fall within one SD of the mean and about 95% within two: here roughly 25 to 35 on a typical day, and 20 to 40 almost always. A day outside that range is worth a look.
Median and quartiles
The median (29.5) and the middle 50% (Q1 27.25 to Q3 32.5) are not pulled by extremes. If the mean is far from the median, the data are skewed and the SD will overstate the typical spread on one side.
Coefficient of variation and standard error
CV compares spread across metrics with different scales: 16.8% here. The standard error (1.61) is how precisely the mean is known, not how much the days vary. Confusing the two is a classic reporting error.
Standard deviation is sensitive to outliers. One bot-driven spike can double it; check the sorted values and the deviation table before reading too much into a large SD.
Paste, choose, read
There is no calculate button; everything refreshes as you type.
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1
Paste the numbers
Copy a column from Excel or Google Sheets, or type values separated by spaces. Full-width digits and Chinese commas also work.
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2
Pick the quartile method if you need quartiles
Match the software you compare with: Excel QUARTILE.INC and R use the inclusive method, SPSS and QUARTILE.EXC the exclusive one, TI calculators the median of halves. SD, mean and median do not depend on this choice.
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3
Report the right SD
Use the sample SD unless the list is the entire population (every employee, every day of the year). Copy the steps table if you need to show your working.
Square the deviations, average them, take the root
Both standard deviations come from the same sum of squared deviations from the mean. The population version divides by n; the sample version divides by n − 1 (Bessel’s correction), because a sample mean sits closer to its own data than the true mean does, and dividing by n would underestimate the spread.
- Steps: find the mean, square each deviation from it, add them up, divide by n or n − 1, take the square root
- Variance is the SD squared, in squared units (seconds², dollars²)
- Excel / Google Sheets: STDEV.S and VAR.S for samples, STDEV.P and VAR.P for populations; the old STDEV is the sample version
- Quartiles: inclusive uses position (n − 1)p, exclusive (n + 1)p, both interpolating between neighbours
- This calculator uses a two-pass algorithm, so huge numbers with small differences (timestamps, IDs) do not lose precision
Population standard deviation
σ = √( Σ(x − μ)² ÷ N )
2, 4, 4, 4, 5, 5, 7, 9: μ = 5, Σ(x − μ)² = 32, σ = √(32 ÷ 8) = 2
Sample standard deviation
s = √( Σ(x − x̄)² ÷ (n − 1) )
Same data as a sample: s = √(32 ÷ 7) = 2.138
Standard error and CV
SE = s ÷ √n CV = s ÷ |x̄|
SE = 2.138 ÷ √8 = 0.756; CV = 2.138 ÷ 5 = 42.8%
Standard deviation of ten days of sign-ups
The default data: daily sign-ups over ten days. Mean = 303 ÷ 10 = 30.3. Each value’s deviation is squared and summed.
| Day | Sign-ups x | x − 30.3 | (x − 30.3)² |
|---|---|---|---|
| 1 | 23 | −7.3 | 53.29 |
| 2 | 31 | 0.7 | 0.49 |
| 3 | 27 | −3.3 | 10.89 |
| 4 | 35 | 4.7 | 22.09 |
| 5 | 29 | −1.3 | 1.69 |
| 6 | 26 | −4.3 | 18.49 |
| 7 | 41 | 10.7 | 114.49 |
| 8 | 30 | −0.3 | 0.09 |
| 9 | 28 | −2.3 | 5.29 |
| 10 | 33 | 2.7 | 7.29 |
| Total | 303 | 0 | 234.1 |
Population: σ² = 234.1 ÷ 10 = 23.41, σ = 4.838. Sample: s² = 234.1 ÷ 9 = 26.01, s = 5.100. Deviations always sum to 0, which is why they are squared. Example data.
How standard deviations get misreported
The formula is simple; picking and reading it is where errors creep in.
Using the population formula on a sample
Survey answers, a week of traffic, a test group: these are samples. Dividing by n understates the spread, most noticeably with fewer than about 30 values.
Reporting the standard error as the spread
“Mean 30 ± 1.6” sounds tight, but 1.6 is the SE. Day-to-day spread is the SD, 5.1. Say which one you report.
Ignoring outliers
One crawler burst or one bulk order can dominate the sum of squares. Look at the sorted values; report the median and IQR alongside the SD for skewed data.
Comparing quartiles from different tools
Excel, SPSS and a TI-84 can give three different Q1s for the same data. None is wrong; they use different definitions. State the method.
Spread and percentiles without exporting
Most metrics in a dashboard are averages. TapCub also shows how values are distributed, so you can see the spread behind the average.
Distributions and percentiles
Items per order, sessions before purchase, any numeric property: shown as a distribution with percentiles.
Learn moreSegments
Split a distribution by channel, device or any rule-based segment to see where the spread comes from.
Learn moreBots counted separately
Crawler spikes inflate variance. They are kept out of your reports by default.
Learn moreStandard deviation questions
Still have a question?
Chat with the team behind TapCub — we usually reply within a few hours on business days.
What is the standard deviation formula, and how do you calculate it?
Sample SD: s = √( Σ(x − x̄)² ÷ (n − 1) ); population SD: σ = √( Σ(x − μ)² ÷ N ). Find the mean, subtract it from each value, square each difference, add the squares, divide by n − 1 (or N), and take the square root. For 2, 4, 4, 4, 5, 5, 7, 9: mean 5, squares sum to 32, sample SD = √(32 ÷ 7) = 2.138, population SD = √(32 ÷ 8) = 2.
Should I use sample or population standard deviation?
Sample, in almost every practical case: survey respondents, a test group, a few weeks of data. Use the population formula only when the list contains every member of the group you are describing, such as all 40 employees of a team.
Why divide by n − 1?
The sample mean is calculated from the same data, so the data are on average closer to it than to the true mean. Dividing by n would therefore underestimate the variance; n − 1 corrects the bias exactly for the variance (Bessel’s correction).
What is the difference between variance and standard deviation?
Variance is the average squared deviation; standard deviation is its square root. The SD is in the same units as the data (seconds, dollars), which makes it easier to read. Variance is what you add up when combining independent sources of variation.
Why does my quartile differ from Excel?
There are several definitions. Excel’s QUARTILE and QUARTILE.INC use the inclusive method, QUARTILE.EXC the exclusive one, SPSS also exclusive, TI-83/84 the median of each half. Pick the matching method in the calculator; the SD, mean and median are identical under every method.
What is a good or normal standard deviation?
There is no universal value: it depends on the units and the mean. To compare spread across metrics, use the coefficient of variation (SD ÷ mean). To judge a single value, ask how many SDs it is from the mean; more than 2–3 is unusual for roughly bell-shaped data.
How is standard error different from standard deviation?
The SD describes how spread out individual values are. The standard error (SD ÷ √n) describes how precisely the mean is estimated, and it shrinks as n grows. Use SE for confidence intervals and tests, SD to describe the data.
Update history
New features, fixes and data refreshes for this tool, newest first.
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· v1.0
LaunchedFirst release.
Missing something or found an error? Tell us — requests that come up often go into the next update. All tool updates
From one list to a comparison
Once you know the mean and spread of a metric, the next questions are usually how precise the mean is and whether two groups differ.