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Learn statistics
Standard deviation is the most common measure of spread in statistics. These pages explain what it means, how to calculate it by hand and in software, and the ideas built on it: variance, standard error, z-scores and the normal curve.
A suggested order
If you are starting from scratch, this path takes about an hour:
- Read what standard deviation is to get the idea before the arithmetic.
- Work through how to calculate it by hand once, with a pencil.
- Learn sample vs population, the choice that trips people up most.
- Connect it to variance and z-scores.
- Check yourself with the quiz, then learn the keystrokes for your own tool in the software guides below.
Start here
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What is standard deviation?
What the number means, its units, and how to read a high, low or zero SD.
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How to calculate it
The six steps on real numbers, both formulas worked in full, and practice problems.
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The standard deviation formula
Each term taken apart, why the deviations are squared, and the shortcut formula’s trap.
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Sample vs population
The question that decides n or n − 1, and how much it changes the answer.
Core concepts
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SD vs variance
The square root that separates them, and when each is the right one to report.
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Symbols and notation
σ, s, μ, x̄ and Σ: what each means and how to type it.
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The empirical rule
The 68–95–99.7 bands for normal data, with the exact percentages.
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Z-scores
Measuring any value in standard deviations from the mean, and turning it into a percentile.
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Seeing the SD on a graph
Your data as a histogram with the ±1, ±2 and ±3 SD bands shaded.
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Standard error
Why the precision of a mean is s ÷ √n, and how it differs from the SD.
Reference and practice
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Formula reference
Every formula the site’s calculators use, grouped, each linked to its tool.
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FAQ
Short answers to the questions people ask most about standard deviation.
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Quiz
Check your understanding with questions that explain each answer.
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Use cases
How the SD is used in grading, quality control, lab work, research, finance and sports.
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Methodology
The algorithms, precision, quartile method and privacy behind the calculators.
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References
The textbooks, standards and papers the site’s methods and figures come from.
Software and calculators
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Excel and Google Sheets
STDEV.S, STDEV.P, conditional SD, and the errors that catch people.
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TI-84
Keystrokes, Sx versus σx, and frequency lists.
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Casio calculators
fx-991CW, fx-991EX, ES PLUS and MS keystrokes, and which result is sx or σx.
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Python
NumPy, pandas and the statistics module, and the ddof default.
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R
sd(), var(), the population SD, NA handling and by-group summaries.
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Standard deviation in psychology
IQ and T-score scales, APA-style reporting, and Cohen’s d.
All the software guides are also collected on the guides page. To run a calculation, go to all calculators.