CalculationTime

Standard Deviation Calculator

Calculate mean, population standard deviation and sample standard deviation for a small data set, with the squared-difference method visible.

Live math canvas

Your numbers, formula and explanation together

Standard Deviation Calculator: 4.062 sample SD. mean 9 · population SD 3.6332 · squared-difference total 66 across 5 values

Formula applied

The exact method behind this answer

CalculationTime keeps the method visible so the number can be checked instead of blindly trusted.

Mean = sum of values ÷ n. Population standard deviation σ = √(Σ(x − mean)² ÷ n). Sample standard deviation s = √(Σ(x − mean)² ÷ (n − 1)).
  1. Apply the formulaMean = sum of values ÷ n. Population standard deviation σ = √(Σ(x − mean)² ÷ n). Sample standard deviation s = √(Σ(x − mean)² ÷ (n − 1)).4.062 sample SDmean 9 · population SD 3.6332 · squared-difference total 66 across 5 values

Your live breakdown

Current inputs in the calculation

These values come from the controls above and update when the calculator changes.

Value 1
4
Enter the values in the data set. Use active count to choose how many boxes are included.
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
Use 2 through 7. Later boxes are ignored when the active count is lower.

Resulting answer

4.062 sample SD

mean 9 · population SD 3.6332 · squared-difference total 66 across 5 values

Answer
4.062 sample SD
Live support
mean 9 · population SD 3.6332 · squared-difference total 66 across 5 values

Assumptions used

What this answer assumes

Use population standard deviation for the whole group; use sample standard deviation for an estimate from a sample.

  • 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.

Master’s Tip

How to use the result well

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.

Printable record

What belongs in the saved calculation

Save the inputs, result, formula, assumptions, page URL and date together so the calculation can be reviewed later.

Value 1
4
Enter the values in the data set. Use active count to choose how many boxes are included.
Value 2
7
Value 3
9
Value 4
10

Embeddable calculator

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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)).

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.

Spread proof

Sample vs population

Standard deviation starts with the same mean and squared-difference total. Population standard deviation divides by n, while sample standard deviation divides by n minus 1, so the denominator choice is the main interpretation check.

Visible checks

What the page now proves

  • Mean shown
  • Squared-difference total shown
  • Population and sample denominators separated

Assumptions and limitations

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.

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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.

Sources and further readingNIST/SEMATECH e-Handbook of Statistical Methods: Measures of scaleKhan Academy: Population and sample standard deviation review