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Standard Deviation Calculator
Paste raw data or frequency shorthand, choose sample or population, and see the standard deviation, variance, quartiles, z-scores, empirical-rule coverage and every step of the calculation.
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2SD formulas
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STATISTICS WORKBENCH
Measure spread without losing the why
Frequency shorthand: 10:3, 12:2, 15 means 10 occurs three times, 12 twice, and 15 once. Invalid text is reported instead of silently changing the answer.
SAMPLE STANDARD DEVIATION2.13809
s = √(Σ(x − x̄)² ÷ (n − 1))
variance4.57143
mean5
count8
standard error0.755929
coefficient of variation42.7618%
sum40
DISTRIBUTION
See the spread
DIAGNOSTICS
Go beyond the headline number
Empirical-rule check
68–95–99.7% is a normal-distribution rule of thumb, not a promise for arbitrary data.
Spread summary
Per-value deviations, z-scores & outlier flags
| x | freq | x − mean | (x − mean)² | weighted SS | z | flag |
|---|
Use n − 1
Choose sample when these observations are a subset used to estimate a larger population. Bessel’s correction compensates for estimating the mean from the same sample.
Use N
Choose population when your list is the complete group you want to describe and you are not using it to estimate unseen members.
A useful edge case
Population SD of one value is 0. Sample SD is undefined because n − 1 = 0, so a sample needs at least two observations.
STATISTICS SENSE
Read spread in context
Original units
Standard deviation returns spread to the original unit, making it easier to interpret than variance.
Squared units
Variance is the average squared distance from the mean. It is the quantity under the square root.
Relative spread
Coefficient of variation compares SD with the magnitude of the mean. It is undefined when the mean is zero.
Robust spread
IQR uses the middle 50% and is less sensitive to extreme values. Compare it with SD when outliers are present.
DAILY 12
Build spread intuition
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STANDARD DEVIATION
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FOCUSED PRACTICE
Train one statistics skill at a time
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MIXED
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30sAdding a constant changes the mean, not SD
Every value shifts together, so all distances from the mean stay the same.
Multiplying scales SD by |c|
Multiply every observation by 3 and every deviation triples; variance grows by 9.
Sample SD is larger on the same data
For n > 1, dividing the same squared-deviation total by n − 1 gives a larger variance than dividing by n.
SD cannot be negative
Squared deviations are nonnegative. SD is zero only when every observation is identical.
LOCAL HISTORY
Your recent datasets
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Method, quartiles, privacy & limits
Numerically stable variance
This page uses compensated, mean-centered variance accumulation rather than the fragile shortcut Σx² − n·mean², which can lose precision when values have a large mean but small spread.
Quartile convention
Q1 and Q3 use linear interpolation at positions (n − 1)p, the common type-7 convention used by many statistical tools. Other textbooks may use a different quartile rule.
Outlier flags
Both |z| ≥ 2 and Tukey’s 1.5×IQR fences are shown as screening rules only. A flagged observation is not automatically an error and should not be deleted without context.
Privacy & limits
Calculations run locally. Up to 1,000,000 expanded observations and 20,000 distinct values are accepted; very large magnitudes that overflow safe floating-point variance are rejected instead of returning nonsense.