Standard Deviation and Variance Calculator
Enter up to 12 readings and get the sample and population standard deviation and variance, the mean With the formula, a worked example and typical values.
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Estimate only. Not engineering or safety advice. Check the result against the applicable standard and the equipment data, and have a competent engineer review it before it is relied on. Terms of use.
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About this calculator
This page calculates sample standard deviation, population standard deviation, sample variance and 5 more results from reading 1, reading 2, reading 3, reading 4 and 8 more inputs. Change any value or its unit and the result updates. The steps below show the working for the default values, so you can check each line against your own method.
Limits of this calculator
A small standard deviation means the readings are close together, which is repeatability; it does not mean they are correct, since a consistent offset from the true value gives a small spread and a large error. For the offset use the percentage error calculator. For the mean, median and range see the mean, median and mode calculator.
Cross-check: 26 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review.
Verification status: Engineering reference
Engineering reference. Formula and method documented and tested against separately coded reference calculations. Not yet Verified: the evidence listed on this page is still needed.
What this calculator checks
- The arithmetic of the published formula for the values you entered, in the units shown.
What remains unverified
- That the inputs suit your application. Take ratings, resistances and constants from the manufacturer data sheet or project specification.
Verification register entry
| Calculation author | Rahul Sharma (author) |
|---|---|
| Technical review | No independent technical review has been carried out |
| Individual professional approval | Not recorded |
| Formula reference | not recorded |
| Standard and edition | Not applicable: Pure mathematics or statistics; no electrical standard governs the method |
| Standard review state | Not applicable |
| Scope-screening note | not recorded |
| Cross-check test cases | 26 reference vectors (tests/reference_vectors.json) |
| Review date | not recorded |
| Next review date | not recorded |
| Status | Engineering reference |
Needed before this calculator can be marked Verified: reviewer name, qualification and review date; exact formula reference; formula source (textbook or standard); next review date.
Formula
Worked example
| Input | Value |
|---|---|
| Reading 1 | 230.1 |
| Reading 2 | 229.8 |
| Reading 3 | 230.4 |
| Reading 4 | 229.8 |
| Reading 5 | 231 |
| Reading 6 | 230.2 |
- Mean = 230.216667 from n = 6 readings
- Sum of squared deviations Σ(x − x̄)² = 1.008333
- Population variance = 1.008333 / 6 = 0.168056, so σ = 0.409946
- Sample variance = 1.008333 / 5 = 0.201667, so s = 0.449073
- Standard error = s / √n = 0.449073 / 2.44949 = 0.183333
Assumptions
- Use the sample standard deviation (n − 1) when the readings are a sample of a larger set, which is the usual case for test readings; use the population value only when the readings are the whole set
- The deviations are taken from the mean of the readings entered
- The sample values need at least two readings; with one reading they are shown as a dash
- The coefficient of variation is not defined when the mean is zero
- No distribution is assumed: these are descriptive figures, not a confidence interval
A small standard deviation means the readings are close together, which is repeatability; it does not mean they are correct, since a consistent offset from the true value gives a small spread and a large error. For the offset use the percentage error calculator. For the mean, median and range see the mean, median and mode calculator.
Inputs and results
| Input | Unit | Allowed or shown | Note |
|---|---|---|---|
| Reading 1 | (number) | any number | Enter the readings in any order, in one consistent unit. At least two readings are needed for a sample standard deviation; blank boxes are ignored. |
| Reading 2 | (number) | any number; may be left blank | |
| Reading 3 | (number) | any number; may be left blank | |
| Reading 4 | (number) | any number; may be left blank | |
| Reading 5 | (number) | any number; may be left blank | |
| Reading 6 | (number) | any number; may be left blank | |
| Reading 7 | (number) | any number; may be left blank | |
| Reading 8 | (number) | any number; may be left blank | |
| Reading 9 | (number) | any number; may be left blank | |
| Reading 10 | (number) | any number; may be left blank | |
| Reading 11 | (number) | any number; may be left blank | |
| Reading 12 | (number) | any number; may be left blank |
| Result | Unit | Allowed or shown | Note |
|---|---|---|---|
| Sample standard deviation (s, n − 1) | (number) | Shown to 6 decimal places | |
| Population standard deviation (σ, n) | (number) | Shown to 6 decimal places | |
| Sample variance (s²) | (number) | Shown to 6 decimal places | |
| Population variance (σ²) | (number) | Shown to 6 decimal places | |
| Mean | (number) | Shown to 6 decimal places | |
| Number of readings | (number) | Shown to 0 decimal places | |
| Standard error of the mean (s / √n) | (number) | Shown to 6 decimal places | |
| Coefficient of variation (s / |mean|, %) | (number) | Shown to 4 decimal places |
How it is calculated
Each input is converted to the base unit of its quantity, the formula above is applied in the order shown in the worked example, and each result is shown in the unit you choose. Results are held at full precision and rounded only for display, except where a step selects a standard size, as the worked example shows. A value outside its allowed range is refused with a message instead of giving a result.
How the result changes with reading 1
| Reading 1 | Sample standard deviation | Population standard deviation | Sample variance |
|---|---|---|---|
| 23 | 84.606548 | 77.234858 | 7,158.268 |
| 57.5 | 70.522216 | 64.377681 | 4,973.383 |
| 115 | 47.048641 | 42.949337 | 2,213.574667 |
| 230 | 0.45607 | 0.416333 | 0.208 |
| 460 | 93.800185 | 85.627462 | 8,798.474667 |
| 920 | 281.593693 | 257.058696 | 79,295.008 |
| 2,300 | 844.976099 | 771.354117 | 713,984.608 |
Other inputs are the same as in the worked example above.
Standards and further reading
Complex numbers, rectangular and polar form and angle conversion are standard mathematics, not the subject of a standard. Standards apply when the result feeds an electrical design, and they set the method there.
Frequently asked questions
What do I need to enter?
Enter reading 1, reading 2, reading 3, reading 4, reading 5, reading 6, reading 7, reading 8, reading 9, reading 10, reading 11 and reading 12. Change the unit of a value with its selector and the result stays the same.
What does this calculator assume?
Use the sample standard deviation (n − 1) when the readings are a sample of a larger set, which is the usual case for test readings; use the population value only when the readings are the whole set. The deviations are taken from the mean of the readings entered. The sample values need at least two readings; with one reading they are shown as a dash. The coefficient of variation is not defined when the mean is zero. No distribution is assumed: these are descriptive figures, not a confidence interval.
Which calculators should I use next?
Common next steps are Mean, Median and Mode, Percentage Error, RMS, Peak and Peak-to-Peak and Complex Numbers (Rectangular and Polar). The related links at the bottom of this page open them.
How accurate are the results?
Every figure comes from the same tested engine as the Elecmetric calculators and is meant for estimating. 26 reference cases, calculated from textbook formulas coded separately from the engine by the same author, agree with this calculator. This is not an independent review. Check design values against the standard and the equipment data you work to.
Formula guides and reference
Longer explanations with worked examples and tables, written for the same topic.
- Mean and Standard Deviation Formulas: Statistics formulas for test readings: mean, median, mode, range, standard deviation, variance and standard error
- Measurement Uncertainty Formulas: Measurement uncertainty formulas: the standard uncertainty of a rectangular limit
- Percentage Error Formulas for Measurements: Percentage error formulas for engineers: absolute, relative and percentage error of a meter reading
More Engineering Mathematics calculatorsAll engineering calculatorsStandard Deviation and Variance in Elecmetric