Population Standard Deviation Calculator
Enter every value in the population to get σ, the population variance σ² and the mean μ, with the divide-by-N working shown.
- σ and σ²
- Divide by N
- Step-by-step
- Frequency tables
Your data
6 values.
Results
Population standard deviation (σ)
1.7078
Population variance (σ²)
2.9167
Population mean (μ)
5.5
Count (N)
6
Center
- Count (n)
- 6
- Sum (Σx)
- 33
- Mean
- 5.5
- Median
- 5.5
- Mode
- No mode
Spread
- Sample standard deviation (s)
- 1.8708
- Population standard deviation (σ)
- 1.7078
- Sample variance (s²)
- 3.5
- Population variance (σ²)
- 2.9167
- Sum of squares (SS)
- 17.5
- Mean absolute deviation
- 1.5
- Standard error of the mean
- 0.7638
- Coefficient of variation (σ ÷ mean)
- 31.05%
- Minimum
- 3
- Maximum
- 8
- Range
- 5
Quartiles and outliers
- Q1
- 4
- Q3
- 7
- Interquartile range (IQR)
- 3
- Five number summary
- 3, 4, 5.5, 7, 8
- Outlier fences
- −0.5 and 11.5
- Outliers (1.5 × IQR)
- None
Empirical rule (using σ)
- Within 1 SD (68% if normal)
- 4 of 6 (66.7%)
- Within 2 SD (95% if normal)
- 6 of 6 (100%)
- Within 3 SD (99.7% if normal)
- 6 of 6 (100%)
Step-by-step working
Count the values
N = 6
Find the mean
μ = Σx ÷ N
Σx = 4 + 8 + 6 + 5 + 3 + 7 = 33
μ = 33 ÷ 6 = 5.5
Subtract the mean from each value and square the result
(x − μ)²
(4 − 5.5)² = (−1.5)² = 2.25
(8 − 5.5)² = (2.5)² = 6.25
(6 − 5.5)² = (0.5)² = 0.25
…and the same for every value. The table lists them all.
Add up the squared deviations (sum of squares)
SS = Σ(x − μ)²
SS = 2.25 + 6.25 + 0.25 + 0.25 + 6.25 + 2.25 = 17.5
Divide by N to get the population variance
σ² = SS ÷ (N)
σ² = 17.5 ÷ 6 = 2.9167
Take the square root to get the standard deviation
σ = √σ²
σ = √2.9167 = 1.7078
| # | x | x − μ | (x − μ)² | z |
|---|---|---|---|---|
| 1 | 4 | −1.5 | 2.25 | −0.8783 |
| 2 | 8 | 2.5 | 6.25 | 1.4639 |
| 3 | 6 | 0.5 | 0.25 | 0.2928 |
| 4 | 5 | −0.5 | 0.25 | −0.2928 |
| 5 | 3 | −2.5 | 6.25 | −1.4639 |
| 6 | 7 | 1.5 | 2.25 | 0.8783 |
| Σ | 33 | 0 | 17.5 |
Rows with |z| above 3 are highlighted. The Σ row of x shows Σx.
What is the population standard deviation?
The population standard deviation σ (sigma) measures the spread of a complete group: every student in a class, every product in a production run, every month in a year. It is the square root of the average squared deviation from the population mean μ, so the sum of squares is divided by N, the full count. Because nothing needs to be estimated, there is no n − 1 correction. If your numbers are only part of the group you want to describe, use the sample standard deviation instead. σ is what Excel’s STDEV.P and the TI-84’s σx report, and it is the σ used in z-scores and the normal distribution.
How to find the population standard deviation
- Enter all the values in the population, or use Frequency table if many values repeat.
- Keep the data type on Population so the working divides by N.
- Read σ, σ² and the mean μ, plus the median, range and quartiles.
- Follow the working line by line, then copy the results.
Population standard deviation formulas
- Population standard deviation
σ = √( Σ(x − μ)² ÷ N )
μ is the population mean and N the number of values.
- Population variance
σ² = Σ(x − μ)² ÷ N
The square of σ.
- Population mean
μ = Σx ÷ N
Same arithmetic as the sample mean, different symbol.
- From a frequency table
σ = √( Σf(x − μ)² ÷ Σf )
Weight each squared deviation by its frequency.
Sample or population: which one should you use?
Why use this population standard deviation calculator
- 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.
- 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.
- 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.
- Every summary statistic at onceMean, median, mode, range, quartiles, IQR, outliers, standard error, coefficient of variation, mean absolute deviation and z-scores for each value appear together.
- 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.
When the population standard deviation is the right choice
- Grades and test scores. See how spread out a class’s scores are and how far one score sits from the average.
- Quality control. Track how much part sizes, fill weights or process times vary, and spot values outside the usual range.
- Sports and fitness. Compare how consistent lap times, scores or training numbers are from week to week.
- Finance and investing. Measure the volatility of monthly returns or prices; a higher standard deviation means a riskier, bumpier series.
- Census-style data. Full records such as every sale this month or every employee’s salary.
- Statistics homework. Check answers for AP Statistics, intro stats and lab reports, and copy the step-by-step working.
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: population standard deviation of 6 values
Data: 4, 8, 6, 5, 3, 7 (all six members of a small group). 1. Mean: μ = 33 ÷ 6 = 5.5. 2. Squared deviations: 2.25, 6.25, 0.25, 0.25, 6.25, 2.25. 3. Sum of squares: 17.5. 4. Population variance: σ² = 17.5 ÷ 6 = 2.9167. 5. Population standard deviation: σ = √2.9167 = 1.7078. Treated as a sample, the same data gives s = √(17.5 ÷ 5) = 1.8708.
What to expect from the result
Only use σ when the data really is the whole population; for samples it underestimates the spread. Results are rounded to the decimal places you choose. With a single value σ is 0.
Population Standard Deviation Calculator questions
When should I use population standard deviation?
When your data includes every member of the group you are describing and you are not generalising beyond it: all employees in a company, all games in a season, the full set of exam scores for one class. If the data is a sample used to describe something bigger, use the sample standard deviation.
What is the population standard deviation formula?
σ = √( Σ(x − μ)² ÷ N ), where μ is the mean of all values and N is how many there are. Square each value’s distance from the mean, average the squares, and take the square root.
What is the difference between σ and s?
σ divides the sum of squares by N and describes a whole population; s divides by n − 1 and estimates a population’s spread from a sample. For the same numbers s is slightly larger.
How do I calculate population standard deviation in Excel?
Use =STDEV.P(A2:A50), or =STDEVP in older versions. =VAR.P gives the population variance.
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.