Standard Deviation Calculator
Paste or type your numbers to get the sample and population standard deviation, variance, mean and more, with the working shown step by step.
- Sample (s) & population (σ)
- Step-by-step working
- Paste from Excel
- Frequency tables
Your data
10 values.
Results
Sample standard deviation (s)
9.0363
Population standard deviation (σ)
8.5726
Mean (x̄)
81.1
Sample variance (s²)
81.6556
Center
- Count (n)
- 10
- Sum (Σx)
- 811
- Mean
- 81.1
- Median
- 85
- Mode
- 85 (×3)
Spread
- Sample standard deviation (s)
- 9.0363
- Population standard deviation (σ)
- 8.5726
- Sample variance (s²)
- 81.6556
- Population variance (σ²)
- 73.49
- Sum of squares (SS)
- 734.9
- Mean absolute deviation
- 7.68
- Standard error of the mean
- 2.8575
- Coefficient of variation (s ÷ mean)
- 11.14%
- Minimum
- 66
- Maximum
- 92
- Range
- 26
Quartiles and outliers
- Q1
- 72
- Q3
- 88
- Interquartile range (IQR)
- 16
- Five number summary
- 66, 72, 85, 88, 92
- Outlier fences
- 48 and 112
- Outliers (1.5 × IQR)
- None
Empirical rule (using s)
- Within 1 SD (68% if normal)
- 6 of 10 (60%)
- Within 2 SD (95% if normal)
- 10 of 10 (100%)
- Within 3 SD (99.7% if normal)
- 10 of 10 (100%)
Step-by-step working
Count the values
n = 10
Find the mean
x̄ = Σx ÷ n
Σx = 72 + 85 + 90 + 66 + 85 + 78 + 92 + 88 + 70 + 85 = 811
x̄ = 811 ÷ 10 = 81.1
Subtract the mean from each value and square the result
(x − x̄)²
(72 − 81.1)² = (−9.1)² = 82.81
(85 − 81.1)² = (3.9)² = 15.21
(90 − 81.1)² = (8.9)² = 79.21
…and the same for every value. The table lists them all.
Add up the squared deviations (sum of squares)
SS = Σ(x − x̄)²
SS = 82.81 + 15.21 + 79.21 + 228.01 + 15.21 + 9.61 + 118.81 + 47.61 + 123.21 + 15.21 = 734.9
Divide by n − 1 to get the sample variance
s² = SS ÷ (n − 1)
s² = 734.9 ÷ 9 = 81.6556
Take the square root to get the standard deviation
s = √s²
s = √81.6556 = 9.0363
| # | x | x − x̄ | (x − x̄)² | z |
|---|---|---|---|---|
| 1 | 72 | −9.1 | 82.81 | −1.007 |
| 2 | 85 | 3.9 | 15.21 | 0.4316 |
| 3 | 90 | 8.9 | 79.21 | 0.9849 |
| 4 | 66 | −15.1 | 228.01 | −1.671 |
| 5 | 85 | 3.9 | 15.21 | 0.4316 |
| 6 | 78 | −3.1 | 9.61 | −0.3431 |
| 7 | 92 | 10.9 | 118.81 | 1.2062 |
| 8 | 88 | 6.9 | 47.61 | 0.7636 |
| 9 | 70 | −11.1 | 123.21 | −1.2284 |
| 10 | 85 | 3.9 | 15.21 | 0.4316 |
| Σ | 811 | 0 | 734.9 |
Rows with |z| above 3 are highlighted. The Σ row of x shows Σx.
What is standard deviation?
Standard deviation measures how spread out numbers are around their mean. A small standard deviation means most values sit close to the average; a large one means they are scattered widely. It is the square root of the variance, which is the average squared distance from the mean, so it is in the same units as your data (points, centimetres, dollars). There are two versions: the population standard deviation σ, used when you have every value in a group, and the sample standard deviation s, used when your data is a sample and you want to estimate the spread of the larger population. For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. This calculator works out both versions, shows each step of the formula with your numbers, and adds the other statistics you usually need alongside it.
How to use the standard deviation calculator
- Type or paste your numbers into the data box. Separate them with commas, spaces, tabs or new lines; a column copied from Excel or Google Sheets works as it is.
- Choose Sample or Population. Both results are always shown, and the choice sets which formula the step-by-step working and the z-scores use.
- Read the standard deviation, variance, mean and the full list of statistics. They update as you type.
- Scroll to the working to see the mean, each squared deviation, the sum of squares and the final square root with your numbers. Copy the results or the z-scores with one click.
Standard deviation formulas
- Sample standard deviation
s = √( Σ(x − x̄)² ÷ (n − 1) )
Use when the data is a sample from a larger group. x̄ is the sample mean and n the number of values.
- Population standard deviation
σ = √( Σ(x − μ)² ÷ N )
Use when the data is the whole population. μ is the population mean and N the number of values.
- Frequency table
s = √( Σf(x − x̄)² ÷ (Σf − 1) )
Multiply each squared deviation by its frequency f. For grouped data, x is the class midpoint.
- Shortcut (computational) formula
s² = ( Σx² − (Σx)² ÷ n ) ÷ (n − 1)
Gives the same answer from the sum and the sum of squares, handy on a basic calculator.
Sample or population: which one should you use?
Why use this standard deviation calculator
- Working shown with your numbersEvery answer comes with the formula and each line of arithmetic filled in with your data, plus a table of deviations, so you can check homework or copy the method into an exam answer.
- Sample and population side by sideYou get s and σ (and s² and σ²) at the same time, so there is no guessing which button a teacher or textbook expects. Switch the working between n − 1 and N with one click.
- Paste straight from a spreadsheetCopy a column from Excel, Google Sheets or Numbers and paste it. Commas, spaces, tabs and new lines all work, headers like “Score” are skipped, and multi-column pastes let you pick one column.
- Frequency tables and grouped dataType each value once with how often it appears, or enter class intervals such as 10-19 and the calculator uses their midpoints. No need to repeat a value 30 times.
- See the spread, not just the numberA histogram with the mean and a one standard deviation band, plus a box plot, shows how the data is spread and where any outliers sit.
- Private, free and instantCalculations run in your browser as you type. Nothing is sent to a server, there is no sign-up, no limit on how many numbers you enter and no “show steps” paywall.
What people use a standard deviation calculator for
- Statistics homework. Check answers for AP Statistics, intro stats and lab reports, and copy the step-by-step working.
- Science labs. Summarise repeated measurements, report mean ± standard deviation, and find the standard error for error bars.
- Grades and test scores. See how spread out a class’s scores are and how far one score sits from the average.
- Finance and investing. Measure the volatility of monthly returns or prices; a higher standard deviation means a riskier, bumpier series.
- Quality control. Track how much part sizes, fill weights or process times vary, and spot values outside the usual range.
- Surveys and research. Describe survey responses or experimental groups before running a t-test or building a confidence interval.
The same statistics in Excel, Google Sheets and a TI-84
To check an answer or redo it in a spreadsheet, use these functions (they work the same in Google Sheets). On a TI-83 or TI-84, enter the data in L1, press STAT, CALC, 1-Var Stats; for a frequency table put the counts in L2 and set FreqList to L2.
| Statistic | Excel and Google Sheets | TI-84 (1-Var Stats) |
|---|---|---|
| Sample standard deviation (s) | =STDEV.S(A2:A20) | Sx |
| Population standard deviation (σ) | =STDEV.P(A2:A20) | σx |
| Sample variance (s²) | =VAR.S(A2:A20) | Sx² |
| Population variance (σ²) | =VAR.P(A2:A20) | σx² |
| Mean | =AVERAGE(A2:A20) | x̄ |
| Median | =MEDIAN(A2:A20) | Med |
| Mode | =MODE.MULT(A2:A20) | Not shown |
| Q1 and Q3 | =QUARTILE.INC(A2:A20, 1) and 3, or QUARTILE.EXC | Q1, Q3 (textbook method) |
| Sum of squares (SS) | =DEVSQ(A2:A20) | Not shown (Σx² is) |
| Mean absolute deviation | =AVEDEV(A2:A20) | Not shown |
| Standard error of the mean | =STDEV.S(A2:A20)/SQRT(COUNT(A2:A20)) | Sx ÷ √n |
| Z-score of a value | =STANDARDIZE(x, mean, sd) | (x − x̄) ÷ Sx |
| Percentile from a z-score | =NORM.S.DIST(z, TRUE) | normalcdf(−1E99, z) |
What this calculator does that most others leave out
We compared the top standard deviation, variance, z-score and quartile calculators in October 2026. Most return one number for one mode, show the formula without your numbers, or cap the number of values. This one was built to cover the gaps.
- Both versions at once. Most calculators make you pick sample or population first. Here s, σ, s² and σ² are always on screen.
- Working with your own numbers. Not just the formula: the mean, every squared deviation, the sum of squares and the final square root are written out with your data.
- Frequency and grouped data. Enter value and count pairs or class intervals instead of typing repeated numbers.
- Quartiles that match your class. Choose the textbook / TI-84 method or Excel’s QUARTILE.INC or QUARTILE.EXC.
- Z-scores and outliers. Every value gets a z-score, and values beyond 1.5 × IQR or 3 standard deviations are flagged.
- No limits, fully private. Paste thousands of values or open a CSV file. Everything is processed on your device and never sent anywhere.
More math and statistics tools
Only need the mean, median and mode, or a weighted average for grades? Use the average calculator. For percentages and percent change, try the percentage calculator, and to type formulas like σ = √(Σ(x − μ)² ÷ N) for a report, use the online math editor. Browse all student tools and calculators.
Worked example: standard deviation of 10 quiz scores
Data: 72, 85, 90, 66, 85, 78, 92, 88, 70, 85. 1. Mean: x̄ = 811 ÷ 10 = 81.1. 2. Squared deviations: (72 − 81.1)² = 82.81, (85 − 81.1)² = 15.21, and so on for all ten scores. 3. Sum of squares: SS = 734.9. 4. Sample variance: s² = 734.9 ÷ 9 = 81.6556. Sample standard deviation: s = √81.6556 = 9.0363. 5. If the ten scores are the whole class, divide by 10 instead: σ² = 73.49 and σ = 8.5726. The median and mode are both 85, Q1 is 72 and Q3 is 88.
What to expect from the result
Results are exact to floating-point precision and rounded to the number of decimal places you choose (4 by default); the working uses the same rounding, so tiny last-digit differences from hand calculations are normal. Quartiles depend on the method, so pick the one your course or software uses. The sample standard deviation needs at least two values. Commas are treated as separators, so type 1234 rather than 1,234.
Standard Deviation Calculator questions
Should I use sample or population standard deviation?
Use population (σ, divide by N) when your data includes every member of the group you care about, such as all students in one class. Use sample (s, divide by n − 1) when the data is a subset used to describe a bigger group, such as 50 surveyed customers. Most homework and research data is a sample.
How do you calculate standard deviation by hand?
Find the mean, subtract it from each value and square the result, add the squares, divide by n − 1 for a sample (or N for a population), then take the square root. The calculator shows each of these steps with your own numbers.
How do I calculate standard deviation in Excel or Google Sheets?
Use =STDEV.S(A2:A11) for a sample and =STDEV.P(A2:A11) for a population. Variance is =VAR.S() and =VAR.P(). Google Sheets accepts the same names, plus the older STDEV and STDEVP.
What does a high or low standard deviation mean?
It depends on the scale of the data. Compare it with the mean: the coefficient of variation (SD ÷ mean) shown in the results expresses spread as a percentage, so 5% is very consistent and 50% is very spread out. For test scores, an SD of 5 points means most students scored close together; 20 points means scores varied a lot.
Can I enter a frequency table or grouped data?
Yes. Switch to Frequency table and enter each value with how many times it appears. For grouped data, type a class interval such as 10-19 in the value box and the calculator uses its midpoint (14.5). You can also paste two columns from a spreadsheet.