Standard Deviation Calculator (Find Variance & SD)
Compute population or sample standard deviation, variance, mean, and sum of squares in real time from any numeric series.
Sample vs. Population Standard Deviation Explained
Standard deviation measures how much individual data points vary around the arithmetic mean. Selecting the right formula depends on what your numbers represent:
For deeper analysis and related planning, you can also explore our Area Calculator and Average Calculator.
- Population Standard Deviation (σ): Applied when the data includes every single member of the target group (such as all students enrolled in a specific class, or every production part manufactured in a day). The divisor is total count N.
- Sample Standard Deviation (s): Applied when the numbers represent a smaller subset sampled from a broader population (such as 100 surveyed voters or 20 quality control inspection units). The divisor is n - 1 (Bessel's correction).
How to Calculate Standard Deviation (The Formula)
The population standard deviation formula is:
The sample standard deviation formula incorporates Bessel's correction to eliminate sampling bias:
Step-by-Step Worked Example
Consider the default dataset: 10, 12, 23, 23, 16, 23, 21, 16 (8 values).
- Count and Sum: n = 8, Sum = 144.
- Mean: 144 / 8 = 18.
- Squared Deviations from Mean (18):
(10 - 18)² = 64
(12 - 18)² = 36
(23 - 18)² = 25 (three instances = 75)
(16 - 18)² = 4 (two instances = 8)
(21 - 18)² = 9 - Sum of Squared Differences: 64 + 36 + 75 + 8 + 9 = 192.
- Population Evaluation: Variance = 192 / 8 = 24. Standard Deviation σ = √24 ≈ 4.8990.
- Sample Evaluation: Variance = 192 / (8 - 1) = 192 / 7 ≈ 27.4286. Standard Deviation s = √27.4286 ≈ 5.2372.
The Empirical Rule (68-95-99.7) in Normal Distributions
When data follows a symmetric, bell-shaped normal distribution curve, standard deviation provides an intuitive framework for probabilistic spread:
- 68.27% of all data: Lies within one standard deviation (μ ± 1σ) of the mean.
- 95.45% of all data: Lies within two standard deviations (μ ± 2σ) of the mean.
- 99.73% of all data: Lies within three standard deviations (μ ± 3σ) of the mean.
What is Variance in Statistics?
In statistics, variance measures the average squared difference between each data point and the mean. While standard deviation expresses spread in the original measurement units, variance provides the foundational mathematical metric used in regression analysis, ANOVA tests, and risk assessment models. Standard deviation is simply the square root of variance.
A frequent error is mixing up variance and standard deviation. Variance represents dispersion in squared units (such as squared dollars or squared seconds), which cannot be compared directly to raw data. Another error is neglecting outlier sensitivity: because differences are squared, a single extreme number pulls the standard deviation upward substantially.
Frequently Asked Questions
Why does sample standard deviation divide by (n - 1) instead of n (Bessel's Correction)?
Using a sample mean instead of the true population mean systematically underestimates variability. Dividing by n - 1 (Bessel's correction) corrects this bias, providing an unbiased estimator of population variance.
What is the relationship between variance and standard deviation?
Variance measures dispersion in squared units (average squared difference from the mean). Standard deviation is the square root of variance, restoring the measurement to the original unit of the data.
What does a low versus high standard deviation indicate about a data set?
A low standard deviation shows data points cluster tightly around the mean, reflecting high consistency. A high standard deviation shows data is widely dispersed across a broader numerical range.
How does the empirical rule (68-95-99.7) apply to standard deviation?
In a bell-shaped normal distribution, roughly 68 percent of observations fall within 1 standard deviation of the mean, 95 percent fall within 2 standard deviations, and 99.7 percent fall within 3 standard deviations.
How sensitive is standard deviation to extreme outliers?
Because deviations from the mean are squared, extreme outliers exert a disproportionately large inflationary effect on both variance and standard deviation.
Can data be pasted directly from spreadsheet columns?
Yes. The parser recognizes commas, tabs, spaces, and newline breaks, allowing seamless copy-pasting directly from Microsoft Excel or Google Sheets.
What is the difference between sample and population standard deviation?
Population standard deviation is used when you have data for the entire group you are studying. Sample standard deviation is used when you only have a small sample of the data, and it divides by (N-1) instead of N to correct for statistical bias.
What does a high standard deviation mean?
A high standard deviation means that the numbers in your data set are spread out far away from the mean (average). A low standard deviation means the data points are clustered closely around the mean.