Scatter Plot Maker

Standard Deviation Calculator

Paste a list of numbers to get the standard deviation, variance, mean, median, range, and coefficient of variation, with both sample and population variants and a full step-by-step solution.

Enter your data

12 numeric values parsed
Calculation type

Separate numbers with commas, spaces, or new lines. The calculator skips anything it cannot parse as a number, so you can paste straight from a spreadsheet column.

Sample standard deviation
s = 8.6006
Mean μ = 81.833, n = 12
Variance
73.970
s²
Median
82.000
x̃
Range
28.000
R
CV
10.51%
Bell curve with your data and ±1σ band

Step-by-Step Solution

Using Bessel's correction (n − 1) on the 12 values you entered.

Step 1
Compute the mean
xˉ=1n∑xi\bar{x} = \frac{1}{n}\sum x_i
xˉ=982.0012=81.833333\bar{x} = \frac{982.00}{12} = 81.833333
Step 2
Find the deviations from the mean
ixᵢxᵢ − x̄(xᵢ − x̄)²
172.00-9.833396.6944
285.003.166710.0278
390.008.166766.6944
467.00-14.8333220.0278
574.00-7.833361.3611
688.006.166738.0278
795.0013.1667173.3611
881.00-0.83330.6944
979.00-2.83338.0278
1083.001.16671.3611
1192.0010.1667103.3611
1276.00-5.833334.0278
Σ982.000.0000813.6667
Step 3
Sum the squared deviations
∑(xi−xˉ)2=813.666667\sum (x_i - \bar{x})^2 = 813.666667
Step 4
Divide by the right denominator

For a sample, divide by n−1n - 1 = 11.

s2=813.66666711=73.969697s^2 = \frac{813.666667}{11} = 73.969697
Step 5
Take the square root
s=73.969697=8.600564s = \sqrt{73.969697} = 8.600564
Step 6
Interpret the result

On average, values in this dataset lie about 8.60 units from the mean (81.83). If the data is approximately normal, roughly 68% of values are expected to fall between 73.23 and 90.43 (the shaded band on the chart).

The Standard Deviation Formula

The two formulas differ only in the denominator. The choice depends on whether your data is the entire population or a sample drawn from it.

Sample standard deviation (s)

s=∑(xi−xˉ)2n−1s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}

Use when the data is a sample meant to represent a larger population. The n − 1 denominator (Bessel's correction) removes the small bias that arises because you estimated the mean from the same data.

Population standard deviation (σ)

σ=∑(xi−μ)2n\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{n}}

Use when your data is the complete population. For very large n the two formulas converge - the difference is negligible once n is in the hundreds.

The Empirical Rule (68 - 95 - 99.7)

When a distribution is approximately normal, the standard deviation tells you exactly how the data is partitioned around the mean. Knowing this lets you make quick probability statements without computing anything.

IntervalApprox. share of dataPractical use
μ ± 1σ68.27%A "typical" range. Two-thirds of observations fall here.
μ ± 2σ95.45%Common cut-off for "unusual". 95% confidence intervals use this band.
μ ± 3σ99.73%Used as control limits in manufacturing - anything beyond ±3σ is investigated as a possible process fault.

Heavy-tailed or skewed distributions (income, file sizes, stock returns during crises) violate the empirical rule. Use a histogram or QQ plot to check normality before relying on these percentages.

Worked Example: Six Test Scores by Hand

Take the six scores 4, 8, 6, 5, 3, 7. The arithmetic is small enough to do without a calculator.

1. Mean
xˉ=4+8+6+5+3+76=336=5.5\bar{x} = \frac{4 + 8 + 6 + 5 + 3 + 7}{6} = \frac{33}{6} = 5.5
2. Deviations and their squares
xᵢxᵢ − 5.5(xᵢ − 5.5)²
4-1.52.25
82.56.25
60.50.25
5-0.50.25
3-2.56.25
71.52.25
Σ017.50
3. Sample SD (divide by n − 1 = 5)
s=17.505=3.500=1.871s = \sqrt{\frac{17.50}{5}} = \sqrt{3.500} = 1.871
4. Population SD (divide by n = 6)
σ=17.506=2.917=1.708\sigma = \sqrt{\frac{17.50}{6}} = \sqrt{2.917} = 1.708

Notice how close the two values are even at this small n. The gap shrinks further as n grows. Paste 4, 8, 6, 5, 3, 7 into the calculator above to reproduce both.

Where Standard Deviation Shows Up

Finance - volatility

Annualised SD of daily returns is the standard definition of volatility for stocks, funds, and portfolios. Sharpe ratio divides excess return by this SD.

Quality control - Six Sigma

A process is 'six sigma' when defects fall outside ±6 SD of the target, corresponding to about 3.4 defects per million opportunities.

Education - standardised tests

SAT scores have mean 500 and SD 100 on each section. IQ scales fix mean 100, SD 15. These choices let any individual score be expressed as a z-score.

Sports - performance variability

A pitcher's ERA, a golfer's stroke average, a sprinter's 100m time - SD across games or rounds tells you whose performance is consistent versus volatile.

Climate science - variability vs trend

Temperature trends are debated against year-to-year SD to assess whether a recent year is genuinely unusual or within historical variability.

Data science - feature scaling

Z-score normalisation subtracts the mean and divides by SD, putting every feature on a comparable scale before fitting many ML models.

Common Mistakes to Avoid

  1. Using the population formula on a sample. Dividing by n when you should divide by n − 1 slightly underestimates the SD. The difference is large when n is small.
  2. Forgetting to take the square root. Variance and SD are different quantities - variance is in squared units, SD is in the original units. Most "wrong" SD values are actually variances.
  3. Reporting SD for clearly skewed data. For income, response time, or any heavy-tailed variable, the mean and SD are misleading. Report median and interquartile range instead, or describe the SD alongside a histogram.
  4. Ignoring units when comparing. An SD of 50 means something completely different for daily steps versus daily revenue. Use the coefficient of variation when you want to compare spread across datasets with different scales.
  5. Assuming SD implies normality. The 68 - 95 - 99.7 rule only holds for distributions that are roughly normal. For a uniform distribution, ±1 SD covers roughly 58% of the data, not 68%.

Frequently Asked Questions

What is standard deviation?
Standard deviation measures how spread out a set of numbers is around its mean. A small SD means values cluster tightly around the average; a large SD means they fan out. It is the most widely used measure of dispersion because it is in the same units as the original data.
What is the difference between sample and population standard deviation?
Population SD divides the sum of squared deviations by n (the size of the entire population). Sample SD divides by n − 1 - Bessel's correction - to give an unbiased estimate of the population SD when you only have a sample. Use population SD when your data is the whole population (every student in a class, every product made today). Use sample SD when your data is a subset used to infer about a larger population.
How do you calculate standard deviation by hand?
1. Find the mean of the values. 2. Subtract the mean from each value to get the deviation. 3. Square each deviation. 4. Sum the squared deviations. 5. Divide by n (population) or n − 1 (sample) to get the variance. 6. Take the square root of the variance - that is the standard deviation.
What is variance?
Variance is the average of the squared deviations from the mean. Standard deviation is simply the square root of variance. Variance has squared units (dollars squared, kilograms squared) which is awkward, so standard deviation is reported more often.
When should I use standard deviation vs interquartile range?
Standard deviation is the natural choice when the data is roughly symmetric and free of extreme outliers. When the distribution is skewed or has heavy tails, the interquartile range (IQR) is more robust because it only depends on the middle 50% of the data.
Can standard deviation be negative?
No. Standard deviation is the square root of a non-negative quantity (the average squared deviation), so it is always ≥ 0. An SD of exactly 0 means every value in the dataset is identical to the mean.
What is the empirical rule (68–95–99.7)?
For an approximately normal distribution, about 68% of values fall within ±1 SD of the mean, about 95% within ±2 SD, and about 99.7% within ±3 SD. The calculator above highlights these bands on the bell curve so you can see where your data points sit.
What is the coefficient of variation?
Coefficient of variation (CV) is the standard deviation divided by the mean, expressed as a percentage. It lets you compare relative variability across datasets with very different scales - for example, comparing the spread of daily revenue across two stores with very different average revenues.
How many data points do I need for the standard deviation to be reliable?
Even with as few as 10 values you get a usable estimate, but the sample SD has its own variability that shrinks roughly as 1/√n. For tight estimates you typically want n ≥ 30. For very small samples (n < 5) the sample SD can be far from the true population SD purely by chance.
How is standard deviation used in real life?
It quantifies risk in finance (volatility of returns), grading and assessment (SAT and IQ scores are standardized to fixed means and SDs), quality control (control charts at ±3σ), weather and climate, and is a building block of confidence intervals, hypothesis tests, and most machine-learning normalization.
Why divide by n − 1 instead of n for the sample?
Using the sample mean instead of the true population mean introduces a small downward bias in the sum of squared deviations. Dividing by n − 1 (the degrees of freedom that remain after estimating the mean) corrects this bias and yields an unbiased estimator of the population variance.

Standard Deviation vs MAD vs IQR

Standard deviation is not the only measure of spread, and it is not always the best one. Two robust alternatives are worth knowing - the median absolute deviation (MAD) and the interquartile range (IQR). Each handles outliers and skew differently, and each suits a different question.

MeasureDefinitionSensitive to outliers?Best for
Standard deviation (SD)Square root of the average squared distance from the mean.Very. A single extreme value can dominate the result.Symmetric data, parametric inference, control charts.
MADMedian of the absolute deviations from the median.Resistant. Breakdown point ≈ 50%.Heavy-tailed data, robust outlier detection.
IQRQ3 − Q1: the spread of the middle 50% of the data.Resistant. Ignores the most extreme 25% on each side.Skewed data, boxplots, reporting median ± IQR.

Rule of thumb: if the histogram of your data is roughly bell-shaped, quote mean ± SD. If it has a long tail or visible outliers, quote median ± IQR. MAD is the right choice when you specifically need a robust spread on the same scale as SD - multiply by 1.4826 to get an SD-equivalent estimate.

References and Further Reading

  • Bessel's correction explains why the sample SD divides by n − 1 - Bessel's correction (Wikipedia).
  • The 68–95–99.7 rule, formally the empirical rule, applies to roughly normal distributions - 68–95–99.7 rule.
  • NIST/SEMATECH e-Handbook on dispersion, including SD and related measures - NIST handbook: measures of scale.
  • For robust alternatives when the data is skewed or outlier-heavy - Median absolute deviation.
  • The original case for n − 1 in the sample variance estimator, from R. A. Fisher's 1925 textbook, is the historical reference most undergraduate texts still trace back to.

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