Standard Deviation Calculator

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Free online standard deviation calculator with sample and population modes. Paste your data to compute standard deviation, variance, mean, and see the bell curve plotted with ฯƒ markers. Includes step-by-step formulas, 68โ€“95โ€“99.7 rule, and Python scipy export.

Standard Deviation
Sample divides by nโˆ’1 (Bessel correction); population divides by n
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Paste numbers to compute standard deviation with step-by-step solution.

Bell Curve (Normal Distribution)

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What Is Standard Deviation?

Standard deviation is a measure of how spread out data values are from their mean. A low standard deviation means data points cluster close to the mean, while a high standard deviation means they are spread over a wider range. It is the most commonly used measure of dispersion in statistics.

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Same Units as Data

Unlike variance (squared units), SD is in the same units as your data, making it directly interpretable.

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Bell Curve Foundation

SD defines the shape of the normal distribution โ€” wider curves have larger SD values.

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Sample vs Population

Use s (nโˆ’1) for samples from a larger group; use ฯƒ (n) when you have the complete dataset.

The Standard Deviation Formula

Sample SD:  s = โˆš[ ฮฃ(xi โˆ’ xฬ„)ยฒ / (n โˆ’ 1) ]

Divides by n โˆ’ 1 (Besselโ€™s correction) to give an unbiased estimate of the population variance from a sample.

Population SD:  ฯƒ = โˆš[ ฮฃ(xi โˆ’ ฮผ)ยฒ / n ]

Divides by n because when you have the entire population, there is no need for correction.

Step-by-Step Worked Example

Data: [4, 8, 6, 5, 3, 7, 8, 1]
Step 1: Mean = (4+8+6+5+3+7+8+1)/8 = 42/8 = 5.25
Step 2: Deviations: โˆ’1.25, 2.75, 0.75, โˆ’0.25, โˆ’2.25, 1.75, 2.75, โˆ’3.75
Step 3: Squared: 1.5625, 7.5625, 0.5625, 0.0625, 5.0625, 3.0625, 7.5625, 14.0625
Step 4: Sum of squares = 39.5
Step 5 (sample): sยฒ = 39.5 / 7 = 5.6429
Step 6: s = โˆš5.6429 = 2.3755

The 68โ€“95โ€“99.7 Rule (Empirical Rule)

For normally distributed data, the standard deviation determines how much data falls within specific ranges around the mean:

ฮผ โˆ’1ฯƒ +1ฯƒ โˆ’2ฯƒ +2ฯƒ โˆ’3ฯƒ +3ฯƒ 68.3% 95.4% 99.7%
RangeCoverageMeaning
ฮผ ยฑ 1ฯƒ68.3%About two-thirds of all values
ฮผ ยฑ 2ฯƒ95.4%Nearly all values โ€” outliers are rare
ฮผ ยฑ 3ฯƒ99.7%Virtually all values โ€” beyond is extremely rare

Sample vs Population: When to Use Which

SamplePopulation
Symbolsฯƒ
Divisorn โˆ’ 1n
Use whenAnalyzing a subset of a larger groupYou have every data point in the group
ExampleSurvey of 500 voters from millionsFinal grades of all 30 students in a class
BiasCorrected (unbiased estimate)Exact (no estimation needed)

Besselโ€™s correction (nโˆ’1) exists because the sample mean is calculated from the same data, reducing degrees of freedom by one. This causes the sample variance to underestimate the true variance if you divide by n. Dividing by nโˆ’1 corrects this bias. For large samples (n > 30), the difference becomes negligible.

Interpreting Standard Deviation

Standard deviation alone doesnโ€™t tell you if variability is โ€œhighโ€ or โ€œlowโ€ โ€” it depends on context. Use the Coefficient of Variation (CV) to compare relative spread:

CV:  CV = (s / xฬ„) ร— 100%

Low Variability (CV < 15%)

Data points are tightly clustered around the mean. Common in precise measurements and controlled experiments.

Moderate (15% โ‰ค CV โ‰ค 30%)

Typical spread seen in many natural and social science datasets. Generally acceptable variability.

High Variability (CV > 30%)

Data is widely spread. Common in financial returns, biological variation, and heterogeneous populations.

Frequently Asked Questions

Standard deviation measures how spread out data values are from the mean. A low SD means data points cluster near the mean while a high SD means they are spread over a wider range. It is the square root of variance and is expressed in the same units as the original data.
Sample standard deviation (s) divides by nโˆ’1 using Besselโ€™s correction because a sample underestimates the true variability. Population standard deviation (ฯƒ) divides by n because you have every data point. Use sample SD when analyzing a subset and population SD when you have the complete dataset.
For normally distributed data, about 68.3% falls within one standard deviation of the mean, 95.4% within two, and 99.7% within three. This empirical rule helps you quickly assess how unusual a value is based on how many standard deviations it is from the mean.
Standard deviation is relative to your data. Compare it using the coefficient of variation (CV = SD/mean ร— 100%). A CV below 15% indicates low variability. A CV above 30% indicates high variability. Always consider the context and units of your data when interpreting SD.
Dividing by nโˆ’1 is called Besselโ€™s correction. When computing from a sample, the mean is estimated from the same data, which constrains one degree of freedom. Dividing by nโˆ’1 corrects this bias, giving an unbiased estimate of the population variance. With large samples (n > 30), the difference is negligible.
Variance is the average of squared deviations from the mean. Standard deviation is the square root of variance. SD is preferred for interpretation because it has the same units as the data, while variance is in squared units. Both measure data spread, but SD is more intuitive for most applications.

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