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Standard Deviation Calculator

Population & sample SD, variance, and mean โ€” with every step shown.

8 values detected

Population SD (ฯƒ)
2
Sample SD (s)
2.1381
Mean (xฬ„)
5
Count (n)
8

Step 1: Mean = 40 รท 8 = 5

Step 2: Squared deviations from the mean sum to 32

Step 3: Population variance = 32 รท 8 = 4

Step 4: Population SD = โˆš4 = 2

Step 5: Sample SD divides by n-1 = 7: โˆš(32 รท 7) = 2.1381

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What Standard Deviation Tells You

Standard deviation measures spread: how far numbers typically stray from the average. Two classes can both average 75%, but if one has an SD of 3 and the other 15, the first class is tightly bunched while the second ranges from failing to perfect scores.

Population vs. Sample: Which One?

  • Population (ฯƒ): you have every value โ€” divide by N.
  • Sample (s): you have a subset and want to estimate the whole โ€” divide by nโˆ’1.
  • When in doubt for homework, most statistics classes expect the sample version.

The 68-95-99.7 Rule

For bell-shaped data, about 68% of values fall within one SD of the mean, 95% within two, and 99.7% within three. That is why SD is the backbone of grading curves, quality control, and polling margins.

Need the average first? Try our average calculator.

Frequently Asked Questions

What is the difference between population and sample standard deviation?

Population SD (ฯƒ) divides the sum of squared deviations by N and describes a complete dataset. Sample SD (s) divides by nโˆ’1 to correct for bias when estimating from a sample.

How is standard deviation calculated step by step?

Find the mean, subtract it from each value, square each difference, average those squares (this is the variance), then take the square root.

Why do we divide by nโˆ’1 for sample standard deviation?

Dividing by nโˆ’1 (Bessel's correction) compensates for the fact that a sample tends to underestimate the true spread of the full population.

Is this standard deviation calculator free?

Yes โ€” every tool on Math Solver Free is free with no sign-up, and all calculations run in your browser.