What this standard deviation calculator does
This tool takes a list of numbers and computes the standard deviation — a measure of how spread out the values are around the mean. It supports both the population form (σ, dividing the sum of squared deviations by n) and the sample form (s, dividing by n−1 to correct for sampling bias). The result panel includes the mean, the variance, the standard deviation, the count, and a complete step-by-step variance table showing (x − x̄) and (x − x̄)² for every value. It also visualizes the 68-95-99.7 rule (empirical rule) with the specific numeric intervals for your data.
The formulas
Variable definitions
- xi — the i-th value.
- x̄ — the mean of the list.
- n — the number of values.
- σ2, s2 — population variance, sample variance.
- σ, s — population standard deviation, sample standard deviation.
Population vs sample — which to use
The choice between population and sample standard deviation depends on whether your list is the entire group of interest or only a part of it.
| Case | Use | Denominator |
|---|---|---|
| You have all the values | Population σ | n |
| You have a subset of a larger group | Sample s | n − 1 |
| n is very large (say > 100) | Either — results almost identical | close enough |
Bessel's correction: when you use a sample, the sample mean is drawn from the same data, so it sits closer to the sample values than the true population mean would. That makes Σ(x − x̄)² systematically underestimate the true spread. Dividing by n−1 instead of n compensates. For n = 10, the sample SD is about 5% larger than the population SD; for n = 100, less than 0.5% larger.
In BISE class 9 and 10, most textbook problems present the whole class or the whole exam set, and use the population formula. But some problems explicitly frame the data as a "sample from a larger group" and use n−1. Read the problem statement to know which one to apply.
Worked examples
Example 1: Small symmetric list (population)
- Mean x̄ = 40 / 5 = 8.
- Deviations: −4, −2, 0, +2, +4.
- Squared deviations: 16, 4, 0, 4, 16. Sum = 40.
- Population variance σ² = 40 / 5 = 8.
- Population SD σ = √8 ≈ 2.828.
- Sample SD s = √(40 / 4) = √10 ≈ 3.162.
Example 2: BISE marks (population)
- Sum = 677; n = 10; mean = 67.7.
- Σ(x − x̄)² = 1418.1 (compute each squared deviation, add them up).
- Population variance σ² = 1418.1 / 10 = 141.81.
- Population SD σ = √141.81 ≈ 11.909.
- So the typical marks are about 67.7 ± 11.9. Most students fall between roughly 56 and 80.
Example 3: With an outlier
- Mean = 1080 / 6 = 180.
- Deviations include −170, −168, −166, −164, −162, +820.
- Σ(x − x̄)² = 28900 + 28224 + 27556 + 26896 + 26244 + 672400 = 810220.
- Population SD = √(810220 / 6) ≈ 367.5.
- Standard deviation is almost as large as the mean itself — the single outlier dominates the spread.
Example 4: All values the same
- Mean = 5.
- Every deviation is 0.
- Variance = 0; standard deviation = 0.
- No spread at all — every value is identical.
Example 5: Comparing two classes
Class B: 50, 65, 75, 85, 95 (mean 74)
- Both classes have the same mean of 74.
- Class A: Σ(x − x̄)² = 40. Population variance = 8. SD ≈ 2.83.
- Class B: Σ(x − x̄)² = 1450. Population variance = 290. SD ≈ 17.03.
- Class B is much more spread out — the same mean but very different consistency.
Example 6: Sample SD for a subset
- This is a sample, not the whole class → use s (divide by n−1).
- Mean = 363 / 5 = 72.6.
- Σ(x − x̄)² ≈ 4.26 + ... = 380.8.
- Sample variance s² = 380.8 / 4 = 95.2.
- Sample SD s = √95.2 ≈ 9.757.
The 68-95-99.7 rule (empirical rule)
When a dataset follows a roughly bell-shaped (normal) distribution, standard deviation gives a quick sense of how the values are distributed around the mean:
- About 68% of values lie within 1 standard deviation of the mean (x̄ ± σ).
- About 95% lie within 2 standard deviations (x̄ ± 2σ).
- About 99.7% lie within 3 standard deviations (x̄ ± 3σ).
This is why a value more than 2σ from the mean is often treated as unusual. In BISE problems, this rule helps you decide whether a single very high or very low mark is consistent with the rest of the class or genuinely exceptional.
Common mistakes to avoid
- Forgetting to square the deviations. Σ(x − x̄) without squaring is always 0 — that is a property of the mean, not a bug. Standard deviation is built from squared deviations.
- Dividing by n when the data is a sample. If your data is a subset of a larger group, use n−1 (Bessel's correction). Using n when you should use n−1 gives an SD that is too small.
- Dividing by n−1 when the data is the whole population. The opposite error — using n−1 when n was correct understates the population SD.
- Forgetting the square root. Variance is in squared units; standard deviation is back in the original units. Without the square root, you have variance, not standard deviation.
- Rounding the mean before squaring. Keep full precision for x̄ while computing (x − x̄)² for each value, then round only at the final SD. Otherwise small rounding errors compound.
- Confusing standard deviation with standard error. Standard deviation measures the spread of individual values. Standard error measures the uncertainty in the sample mean (SE = s/√n). They are different quantities.
- Assuming standard deviation is always meaningful. For heavily skewed data (like income), the SD can be misleading — median and IQR are better measures of spread in that case.
- Treating a SD of 0 as an error. If every value is identical, the SD is exactly 0. That is correct, not a bug.
FAQ
What is standard deviation?
Standard deviation measures how spread out the values in a dataset are around the mean. A small standard deviation means the values cluster close to the mean; a large standard deviation means they are spread over a wide range. It is the square root of the variance and has the same units as the original data.
What is the difference between population and sample standard deviation?
Population standard deviation (σ) uses the whole population and divides the sum of squared deviations by n. Sample standard deviation (s) uses a subset and divides by n−1 (Bessel's correction) to correct for the fact that a sample tends to underestimate the spread. For large n the two are almost equal; for small n the sample SD is noticeably larger.
Why divide by n−1 for a sample?
When you sample a subset of data, you compute the mean from that same sample. The sample mean is closer to the sample values than the true mean is, so the sum of squared deviations underestimates the true variance. Dividing by n−1 instead of n corrects for this bias. This is called Bessel's correction.
What does the 68-95-99.7 rule mean?
For data that follows a normal (bell-shaped) distribution, about 68% of values lie within one standard deviation of the mean, about 95% within two standard deviations, and about 99.7% within three. This is a useful quick check for whether a value is unusual — anything more than two standard deviations from the mean is often considered an outlier.
Can standard deviation be zero?
Yes. Standard deviation is zero when every value in the list is identical — there is no spread. For example, the list 5, 5, 5, 5 has mean 5 and standard deviation 0. This is a valid result, not an error.
What is variance?
Variance is the average of the squared deviations from the mean. It is the square of the standard deviation. Variance is measured in squared units (e.g. metres-squared if the data is in metres), which is why standard deviation — in the original units — is often more intuitive to interpret.
Related tools
Related Hira Academy resources
Practice statistics with our Class 9 notes and Class 10 notes, and track term progress with the free Student Portal.