Standard Deviation Calculator
The standard deviation calculator finds the standard deviation, variance, and mean of any dataset — showing every step of the working, including the deviation from the mean for each value. Enter your numbers below, separated by commas, and get a complete step-by-step breakdown instantly.
This free standard deviation calculator supports both sample and population data. It automatically applies Bessel's correction (dividing by n−1) for sample data, and shows the result alongside the variance, the mean, and a full deviation table. No signup required.
Step-by-Step Solution
What Is Standard Deviation?
Standard deviation measures how spread out the values in a dataset are around the mean. A low standard deviation means the values are clustered close to the mean. A high standard deviation means the values are spread out over a wider range. It is one of the most widely used measures of variability in statistics, written with the symbol σ (sigma) for a population, or s for a sample. Statistics How To's breakdown of standard deviation is a good next stop for more worked examples and edge cases.
Standard Deviation vs Variance
| Term | Meaning |
|---|---|
| Variance | The average of the squared deviations from the mean. Measured in squared units (e.g. squared dollars, squared points). |
| Standard deviation | The square root of the variance. Measured in the same units as the original data — which is why it's easier to interpret than variance. |
Variance and standard deviation always measure the same underlying spread — standard deviation is simply the more interpretable version, since taking the square root returns the value to the original units of the data.
Standard Deviation Formula Explained
Population Standard Deviation
Use the population formula when your dataset includes every member of the group you care about — for example, test scores for an entire class.
Calculate the mean first with the Average Calculator if you don't already have it.
Sample Standard Deviation (Bessel's Correction)
Use the sample formula when your dataset is only a subset of a larger group — for example, 30 patients out of an entire hospital's records. Dividing by n−1 instead of n is called Bessel's correction, and it corrects for the tendency of a sample to underestimate the true variability of the full population.
How to Use This Standard Deviation Calculator
- Type or paste your numbers into the input field, separated by commas — for example: 2, 4, 4, 4, 5, 5, 7, 9.
- Select Population or Sample, depending on whether your data represents the entire group or only part of it.
- Click Calculate. The calculator shows the mean, the deviation of each value, the variance, and the standard deviation — with every step of the working.
- Use the Coefficient of Variation tab to see the relative standard deviation (RSD) as a percentage of the mean.
- Click Reset to clear all inputs and start a new calculation.
The calculator accepts positive numbers, negative numbers, and decimals. There is no limit on dataset size.
Worked Examples — Step by Step
Example 1: Population Standard Deviation
Find the population standard deviation of the dataset: 2, 4, 4, 4, 5, 5, 7, 9
- Data: 2, 4, 4, 4, 5, 5, 7, 9 (N = 8)
- Mean = (2+4+4+4+5+5+7+9) / 8 = 40 / 8 = 5
- Calculate deviations from mean and square them:
| Value | Mean | Deviation (x−μ) | Deviation² |
|---|---|---|---|
| 2 | 5 | −3 | 9 |
| 4 | 5 | −1 | 1 |
| 4 | 5 | −1 | 1 |
| 4 | 5 | −1 | 1 |
| 5 | 5 | 0 | 0 |
| 5 | 5 | 0 | 0 |
| 7 | 5 | +2 | 4 |
| 9 | 5 | +4 | 16 |
| Sum of Deviation²: | 32 | ||
- Sum of squared deviations = 32
- Divide by N = 8 to get the variance: 32 ÷ 8 = 4
- Take the square root to get the standard deviation: √4 = 2
Example 2: Sample Standard Deviation (with Bessel's Correction)
A researcher measures 5 tree heights (in feet) from a larger forest: 6, 8, 10, 12, 14. Since this is a sample, not the entire forest, use n−1.
- Write out the sample: 6, 8, 10, 12, 14 (n = 5 values)
- Find the sample mean: (6+8+10+12+14) ÷ 5 = 50 ÷ 5 = 10
- Find each deviation from the mean (x − x̄): −4, −2, 0, 2, 4
- Square each deviation: 16, 4, 0, 4, 16
- Add the squared deviations: 16+4+0+4+16 = 40
- Divide by (n − 1), not n: 40 ÷ (5 − 1) = 40 ÷ 4 = 10
- Take the square root: √10 ≈ 3.162
Why n−1 matters: if this same data were treated as a full population (dividing by N=5 instead of n−1=4), the variance would be 40÷5 = 8 and the standard deviation would be √8 ≈ 2.828 — a smaller, slightly less accurate estimate of the true spread in the forest.
Example 3: Converting Variance to Standard Deviation
A dataset has a known variance of 64. Find the standard deviation.
- Start with the variance: 64
- Standard deviation is the square root of variance: σ = √64
- Calculate: √64 = 8
Example 4: Relative Standard Deviation (RSD)
A dataset has a mean of 50 and a standard deviation of 5. Find the relative standard deviation as a percentage.
- Write the known values: mean = 50, standard deviation = 5
- Divide the standard deviation by the mean: 5 ÷ 50 = 0.1
- Multiply by 100 to express as a percentage: 0.1 × 100 = 10%
Example 5: The Empirical Rule (68-95-99.7)
An IQ test is designed with a mean of 100 and a standard deviation of 15. Use the empirical rule to find the score ranges.
- Write the known values: mean = 100, standard deviation = 15
- 68% of scores fall within 1 standard deviation: 100 − 15 = 85 to 100 + 15 = 115
- 95% of scores fall within 2 standard deviations: 100 − 30 = 70 to 100 + 30 = 130
- 99.7% of scores fall within 3 standard deviations: 100 − 45 = 55 to 100 + 45 = 145
When to Use Standard Deviation
Standard deviation is used anywhere you need to understand how spread out a dataset is, not just its average. Common situations include:
- School and test scores: teachers use standard deviation to see how consistent a class's scores are, and to compare one class against another with the same average.
- Scientific research: standard deviation reports the precision of experimental measurements — a small standard deviation means repeated measurements are consistent.
- Investing and finance: standard deviation of returns measures portfolio risk and volatility — a stock with a higher standard deviation is considered riskier, even if its average return is the same.
- Quality control: manufacturers track the standard deviation of product measurements to catch inconsistent production before it reaches customers.
- Grading on a curve: standard deviation and the empirical rule help teachers understand what percentage of a class scored within a certain range.
Population vs Sample: Which Should You Use?
Choosing the right formula matters — using the wrong one gives a slightly wrong answer, especially with small datasets.
| Rule | When it applies |
|---|---|
| Use Population (÷N) when: | Your dataset includes every member of the group you're studying — for example, the ages of every employee at a small company. |
| Use Sample (÷n−1) when: | Your dataset is only a subset of a larger group you want to draw conclusions about — for example, a survey of 200 people meant to represent an entire country. |
As a dataset gets larger, the difference between dividing by n and n−1 shrinks. For very large samples, the two formulas produce nearly identical results. For small samples, the choice matters more, which is why Bessel's correction exists in the first place.
The Empirical Rule, Standard Error, and Relative Standard Deviation
The Empirical Rule (68-95-99.7 Rule)
For data that follows a normal distribution (a symmetric bell curve), the empirical rule describes how much of the data falls within each range of standard deviations from the mean:
| Range | Percentage of data |
|---|---|
| Within 1 standard deviation | About 68% of all values |
| Within 2 standard deviations | About 95% of all values |
| Within 3 standard deviations | About 99.7% of all values |
Standard Deviation vs Standard Error
These two terms are often confused. Standard deviation describes the spread of the raw data itself. Standard error describes how much a sample mean is likely to vary from the true population mean, and it shrinks as the sample size grows — standard deviation does not shrink with a larger sample.
Relative Standard Deviation (RSD)
Relative standard deviation — also called the coefficient of variation — expresses the standard deviation as a percentage of the mean, using the same conversion the Percentage Calculator performs. This makes it possible to compare the spread of two datasets that use different units or very different averages.
Common Mistakes When Calculating Standard Deviation
- Using the wrong denominator: dividing by N when the data is actually a sample (or by n−1 when it's a full population) is the most common error — always confirm which formula applies before calculating.
- Forgetting to square the deviations: raw deviations from the mean always sum to zero, which is why they must be squared before adding — skipping this step gives a meaningless result.
- Reporting variance instead of standard deviation: variance and standard deviation are related but not interchangeable — forgetting the final square root step reports a number in the wrong units.
- Confusing standard deviation with standard error: these measure different things and don't shrink the same way as sample size changes — see the comparison above before choosing which one to report.
Frequently Asked Questions
- What is standard deviation?
- Standard deviation is a number that shows how spread out the values in a dataset are around the mean. A small standard deviation means the values are close together. A large standard deviation means the values are spread out over a wide range. It is written as σ for a population and s for a sample.
- How do you calculate standard deviation?
- Find the mean of the dataset, subtract the mean from each value to get the deviations, square each deviation, add them up, divide by N (population) or n−1 (sample), then take the square root of the result. For example, the dataset 2, 4, 4, 4, 5, 5, 7, 9 has a mean of 5 and a population standard deviation of 2.
- What is the difference between standard deviation and variance?
- Variance is the average of the squared deviations from the mean, measured in squared units. Standard deviation is the square root of the variance, which brings the result back into the same units as the original data.
- What is the difference between sample and population standard deviation?
- Population standard deviation divides by N and is used when your data covers an entire group. Sample standard deviation divides by n−1 (Bessel's correction) and is used when your data is only a subset meant to represent a larger group.
- What does the empirical rule (68-95-99.7 rule) mean?
- For data that follows a normal distribution, about 68% of values fall within 1 standard deviation of the mean, 95% fall within 2 standard deviations, and 99.7% fall within 3 standard deviations. For example, with a mean of 100 and a standard deviation of 15, 68% of values fall between 85 and 115.
- What is the difference between standard deviation and standard error?
- Standard deviation measures the spread of the raw data itself and does not change based on sample size. Standard error measures how much a sample mean is likely to differ from the true population mean, and it gets smaller as the sample size increases.
- What is relative standard deviation (RSD)?
- Relative standard deviation, also called the coefficient of variation, expresses the standard deviation as a percentage of the mean: RSD = (standard deviation ÷ mean) × 100. It is useful for comparing the spread of two datasets that use different units.
- Can standard deviation be negative?
- No. Standard deviation is always zero or a positive number, because it comes from a square root of squared values, which can never be negative. A standard deviation of zero means every value in the dataset is exactly the same.
Related Calculators
Average CalculatorFind the mean, median, mode, and weighted average of any dataset — the same mean used in every standard deviation calculation. Square Root Calculator
Simplify square roots to exact radical form — useful when a variance doesn't reduce to a whole number. Percentage Calculator
Convert a relative standard deviation into a percentage, or calculate percent increase, decrease, and percent error.