Math & statistics calculator

Standard Deviation Calculator

Calculate sample and population standard deviation, variance, mean, median, quartiles, outliers, z-scores, standard error, confidence interval, and full descriptive statistics from your dataset. Paste a column from a spreadsheet, read every formula and step, see box-plot and histogram charts, and download a 12-tab Excel workbook.

Transparent assumptions Sample & population · every formula Charts: number line, box plot, histogram z-scores, outliers, CI, empirical rule 12-tab Excel workbook Verify important figures

Enter your dataset SD formula

Exact to the precision you choose — verify figures for coursework, research, or professional reports.

Standard deviation measures how far values typically sit from the mean. Find the mean, square each value’s deviation, average the squares (÷ N for a population, ÷ n−1 for a sample), then take the square root. For 10, 12, 8, 15, 11 the mean is 11.2, the population SD is 2.3152, and the sample SD is 2.5884.

Common datasets & examples

Quick Standard Deviation

Dataset — paste or type values (comma, space, tab, or new line)

5 values detected. Zeros and negatives are kept.

Dataset type

Decimals

Sort

Sample standard deviation (s)

2.59

Divides by n − 1. Use for a sample.

Population standard deviation (σ)

2.32

Divides by N. Use for a full population.

Mean (x̄)

11.20

Σx ÷ n = 56.00 ÷ 5.

Sample variance (s²)

6.70

SD squared (n − 1).

Population variance (σ²)

5.36

SD squared (N).

Median

11.00

Middle value.

Mode

No mode

Every value appears once.

Count (n)

Sum = 56.00.

Range

7.00

Min 8.00 · Max 15.00.

The values show moderate spread around the mean. Compare the standard deviation with the mean (or use the coefficient of variation) to judge whether that spread is large for your context. The sample standard deviation (2.59) is slightly larger than the population standard deviation (2.32) because it divides by n − 1 to correct for sample bias.

The mean and median are close, so the distribution is approximately symmetric — the mean and standard deviation describe it well.

Mean ± standard deviation

Each value plotted against the mean and ±1, ±2, ±3 standard deviation bands (sample SD).

-3σ
3.43-2σ
6.02-1σ
8.61x̄
11.20+1σ
13.79+2σ
16.38+3σ
18.97

Box plot & five-number summary

Minimum, Q1, median, Q3, maximum, with the 1.5×IQR whiskers and any outliers.

Min

8.00

10.00

Median

11.00

12.00

Max

15.00

Add more values to see a meaningful distribution — the histogram appears once there are at least 8 values, where the bin counts start to mean something.

Deviation table

Each value vs the mean, with its z-score (using the sample SD).

Deviation and z-score table
#	Value	xᵢ − mean	(xᵢ − mean)²	\|xᵢ − mean\|	z-score	Outlier?
1	10.00	-1.20	1.44	1.20	-0.46	—
2	12.00	0.80	0.64	0.80	0.31	—
3	8.00	-3.20	10.24	3.20	-1.24	—
4	15.00	3.80	14.44	3.80	1.47	—
5	11.00	-0.20	0.04	0.20	-0.08	—
Σ	56.00	≈ 0	26.80	9.20	n = 5

At a glance

Formula shown: Sample SD s = √(Σ(xᵢ − x̄)² ÷ (n − 1)); population SD σ = √(Σ(xᵢ − μ)² ÷ N).
Scenario support: Quick SD, full descriptive stats, two-dataset comparison, frequency and grouped data, confidence interval, and outlier analysis.
Workbook export: 12-tab Excel (XLSX) workbook
Educational estimate: Planning support from the values you enter — not professional advice.

How to read your result

Choose population (÷ N) when your data IS the entire group you care about, and sample (÷ n−1) when it's a sample meant to represent a larger group; sample SD is always slightly larger, because dividing by n−1 (Bessel's correction) corrects for the fact the mean itself was estimated from the same data. Don't confuse standard deviation (the spread of your data) with standard error (SE = s ÷ √n, the precision of the estimated MEAN) — SE shrinks as your sample grows even though SD does not. Because deviations are squared, a single outlier can inflate the SD sharply — review flagged values rather than deleting them automatically, and lean on the median, IQR, or MAD when data is skewed or outlier-heavy. For roughly bell-shaped data, the empirical rule says about 68% of values fall within 1 SD of the mean, 95% within 2, and 99.7% within 3 — a rough guide, not a guarantee. To compare variability across datasets with different means, use the coefficient of variation (SD ÷ mean × 100) rather than the raw SD.

Standard deviation formula

Population standard deviation

σ = √(Σ(xᵢ − μ)² ÷ N)

Use when the data is the entire population.

Sample standard deviation

s = √(Σ(xᵢ − x̄)² ÷ (n − 1))

Use when the data is a sample of a larger group.

Population / sample variance

σ² = Σ(xᵢ − μ)² ÷ N · s² = Σ(xᵢ − x̄)² ÷ (n − 1)

Variance is SD squared.

Mean

x̄ = Σx ÷ n

The arithmetic average.

What each symbol means

xᵢ — each individual value in the dataset.
μ (mu) — the population mean; x̄ (x-bar) — the sample mean.
N — the number of values in a population; n — the number in a sample.
Σ (sigma) — “sum of”, add up the terms that follow.
(xᵢ − mean)² — the squared deviation of one value from the mean.

Worked example

Take the dataset 10, 12, 8, 15, 11.

Mean: (10 + 12 + 8 + 15 + 11) ÷ 5 = 11.2.

Squared deviations: (10 − 11.2)² = 1.44, (12 − 11.2)² = 0.64, (8 − 11.2)² = 10.24, (15 − 11.2)² = 14.44, (11 − 11.2)² = 0.04.

Sum of squared deviations: 1.44 + 0.64 + 10.24 + 14.44 + 0.04 = 26.80.

Population variance: 26.80 ÷ 5 = 5.36; population SD = √5.36 = 2.3152.

Sample variance: 26.80 ÷ 4 = 6.70; sample SD = √6.70 = 2.5884.

Interpretation: the sample standard deviation is about 2.59, meaning values typically sit about 2.59 units away from the mean of 11.2.

Assumptions & limitations

Assumptions

Every value you paste is treated as a real, exact number on a true numeric scale — zeros and negatives are kept, and non-numeric text is ignored with a count.
Quartiles and the IQR use the type-7 convention (matching Excel QUARTILE.INC); other tools may use a different convention and report slightly different quartiles.
The confidence interval assumes the data is an independent random sample and uses a Student-t critical value for the chosen confidence level.
The empirical rule (68–95–99.7) is shown only as a reference and holds only for roughly bell-shaped (normal) data.

What it does not do

It does not choose sample vs population for you — that choice is yours, and it changes the variance and standard deviation.
It cannot fully validate pasted data: a stray character, a mixed delimiter, or a hidden duplicate can change the result, so confirm the parsed values and count before relying on them.
It does not perform inferential statistics — no hypothesis tests, p-values, regression, or correlation — only descriptive statistics for the data you enter.
It does not judge whether the standard deviation is the right measure of spread; on skewed, heavy-tailed, or outlier-heavy data the median, IQR, or MAD may describe the data more faithfully.

Which calculator should I use? For everyday percentages and reverse percentages, the percentage calculator; for growth between two values, the percentage change calculator; for business margins, the profit margin calculator.

Frequently asked questions

What is standard deviation?

Standard deviation measures how spread out values are around the mean. A small standard deviation means the values are tightly clustered near the average; a large one means they are widely spread.

What is the difference between sample and population standard deviation?

Population SD divides the sum of squared deviations by N and is used for a full population. Sample SD divides by n−1 and is used when the data is a sample of a larger group. Sample SD is always slightly larger.

Why does sample standard deviation divide by n−1?

It applies Bessel’s correction. Because the sample mean is itself estimated from the data, dividing by n would systematically underestimate the population spread; dividing by n−1 corrects most of that bias.

What is standard error?

Standard error is the sample standard deviation divided by the square root of n (SE = s ÷ √n). It measures how precisely the sample mean estimates the true population mean — not the spread of the data itself.

What is the coefficient of variation?

The coefficient of variation is the standard deviation divided by the mean, expressed as a percentage. It compares relative variability across datasets with different averages or units. It is undefined when the mean is 0.

Read the guide

There is no dedicated descriptive-statistics guide yet. This page's formula and assumptions sections above cover sample vs population, standard error, outliers, and the empirical rule in full.

Educational use disclaimer

This standard deviation calculator and its Excel workbook are for education and informational use only. They are not financial, medical, legal, academic, statistical-consulting, or professional advice. Results are mathematically exact to the precision you choose, but only as correct as the data you enter and the method your context requires — quartile conventions, the choice of sample vs population, outlier treatment, confidence level, and rounding all affect the answer. Standard deviation is sensitive to outliers and can mislead on skewed, heavy-tailed, or mixed data; the empirical rule applies only to roughly normal distributions. For graded coursework, research papers, regulatory reporting, or professional analysis, confirm the required formula, the treatment of outliers, the confidence level, and the rounding method with the appropriate source or professional.

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Authorship & verification

Written and maintained by Jay Sudha

Formula and examples verified on 15 June 2026
Educational estimate only

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