Formula
Mean = sum of values ÷ n. Population standard deviation σ = √(Σ(x − mean)² ÷ n). Sample standard deviation s = √(Σ(x − mean)² ÷ (n − 1)).
Math & Statistics
Standard Deviation Calculator
Calculate mean, population standard deviation and sample standard deviation for a small data set, with the squared-difference method visible.
Calculator
Working calculator
Live result4.062 sample SDmean 9 · population SD 3.6332 · squared-difference total 66 across 5 values
Formula used
Mean = sum of values ÷ n. Population standard deviation σ = √(Σ(x − mean)² ÷ n). Sample standard deviation s = √(Σ(x − mean)² ÷ (n − 1)).
This is the method behind the answer, so the result can be checked rather than simply trusted.Visual grid
This number is one point on a larger pattern
Standard Deviation is not just a final answer. It is a step on a line: before and after, input and output, assumption and result.
Micro-timehours, minutes, shiftsHuman scaledays, weeks, projectsMacro-timemonths, years, calendars
InputFormulaResult
4.062 sample SDCalculationTime keeps the path visible: the input, the method and the final number belong together.
CalculationTime
Standard Deviation Calculation Report
4.062 sample SDmean 9 · population SD 3.6332 · squared-difference total 66 across 5 values
Inputs
- Value 1
- 4
- Value 2
- 7
- Value 3
- 9
- Value 4
- 10
- Value 5
- 15
- Value 6
- 0
- Value 7
- 0
- How many values to include
- 5 2 to 7
Method
Mean = sum of values ÷ n. Population standard deviation σ = √(Σ(x − mean)² ÷ n). Sample standard deviation s = √(Σ(x − mean)² ÷ (n − 1)).
- For 4, 7, 9, 10 and 15, the mean is 45 ÷ 5 = 9. Squared differences are 25, 4, 0, 1 and 36, for a total of 66. Population variance is 66 ÷ 5 = 13.2, so population standard deviation is √13.2 ≈ 3.6332. Sample variance is 66 ÷ 4 = 16.5, so sample standard deviation is √16.5 ≈ 4.0620.
Assumptions
- The active count is rounded to a whole number from 2 to 7.
- Only Value 1 through the active count are included; later boxes are ignored.
- The population result divides by n. Use it when the entered values are the whole group being described.
- The sample result divides by n − 1. Use it when the entered values are a sample used to estimate a wider population.
Notes
Use this space on the printed report for client, supplier, classroom, job-location, measurement, quote or approval notes.
Worked example
For 4, 7, 9, 10 and 15, the mean is 45 ÷ 5 = 9. Squared differences are 25, 4, 0, 1 and 36, for a total of 66. Population variance is 66 ÷ 5 = 13.2, so population standard deviation is √13.2 ≈ 3.6332. Sample variance is 66 ÷ 4 = 16.5, so sample standard deviation is √16.5 ≈ 4.0620.
Professional note
Master’s Tip: print the mean, squared-difference total and whether you used population or sample mode. Most standard-deviation mistakes come from using the wrong denominator or silently changing which values were included.
Regional and unit assumptions
Standard or basis: descriptive-statistics arithmetic for population and sample standard deviation. The page follows the common n and n − 1 denominator distinction used in statistics teaching and data summaries; no curriculum, scientific or regulatory standard is claimed.
Assumptions and limitations
- The active count is rounded to a whole number from 2 to 7.
- Only Value 1 through the active count are included; later boxes are ignored.
- The population result divides by n. Use it when the entered values are the whole group being described.
- The sample result divides by n − 1. Use it when the entered values are a sample used to estimate a wider population.
- This calculator uses unweighted numeric values. Frequency tables, grouped classes and weighted standard deviation need a separate method.
Methodology & Accuracy
How this calculator is checked
CalculationTime pages are built around visible arithmetic: the formula, assumptions, worked example and practical limitations are shown so the result can be checked rather than simply trusted.
Formula used
Mean = sum of values ÷ n. Population standard deviation σ = √(Σ(x − mean)² ÷ n). Sample standard deviation s = √(Σ(x − mean)² ÷ (n − 1)).
Standard or basis
Standard or basis: descriptive-statistics arithmetic for population and sample standard deviation. The page follows the common n and n − 1 denominator distinction used in statistics teaching and data summaries; no curriculum, scientific or regulatory standard is claimed.
Where a calculator follows a named legal, trade or industry standard, that standard is cited visibly. Otherwise the page uses transparent general arithmetic and states its limits.Master's Tip
Master’s Tip: print the mean, squared-difference total and whether you used population or sample mode. Most standard-deviation mistakes come from using the wrong denominator or silently changing which values were included.
Related calculators
Questions
How do you calculate standard deviation?
Find the mean, subtract the mean from each value, square those differences, add them, divide by the correct denominator, then take the square root.
What is the difference between population and sample standard deviation?
Population standard deviation divides by n because the entered values are the whole group. Sample standard deviation divides by n − 1 when the entered values are a sample used to estimate a larger population.
Why are differences squared?
Squaring makes negative and positive differences both contribute positively and gives larger deviations more weight before the square root returns the result to the original unit scale.
Can standard deviation be zero?
Yes. If every included value is the same, every difference from the mean is zero, so both population and sample standard deviation are zero.
Should I use standard deviation or range?
Range only compares the smallest and largest values. Standard deviation uses every included value, so it gives a fuller spread check for classroom data, measurements, scores or small reports.
Calculation note
Standard deviation is a measure of spread around the mean. It helps turn a list of numbers into a short record of typical value and variation, which is why it appears in classrooms, measurement checks, quality control and basic data reporting.
Standard deviation measures spread around the mean
The mean tells where the centre of a data set sits. Standard deviation adds a second question: how far do the values typically sit from that centre? Two groups can have the same average but very different spread.
The denominator changes the meaning
Population standard deviation divides by the number of values because the entered values are the full group. Sample standard deviation divides by one fewer value to compensate when a sample is being used to estimate wider variation.
A printable audit trail prevents quiet mistakes
A standard-deviation answer is easier to trust when the report keeps the included values, mean, squared-difference total, denominator choice and final result together. That is especially useful for homework, lab notes, small quality checks and copied spreadsheet results.