Standard Deviation Calculator — Calculate SD, Variance & Mean
Enter a list of numbers to instantly calculate population or sample standard deviation, variance, mean, sum, min, max, and range. Supports any delimiter — commas, spaces, semicolons, or newlines. A step-by-step breakdown table shows every deviation and squared deviation so you can follow the math.
Enter Your Data
Enter numbers separated by commas, spaces, semicolons, or newlines.
Use Population when your data represents the entire group (divides by N).
Frequently Asked Questions
What is standard deviation?
Standard deviation is a measure of how spread out numbers are around their mean (average). A low standard deviation means data points cluster close to the mean; a high standard deviation means they are spread out widely. It is represented by σ (sigma) for a population and s for a sample.
What is the formula for standard deviation?
Population standard deviation: σ = √(Σ(xᵢ − x̄)² / N). Sample standard deviation: s = √(Σ(xᵢ − x̄)² / (N−1)). Where xᵢ is each value, x̄ is the mean, and N is the count. The only difference is the denominator: N for population, N−1 (Bessel's correction) for sample.
What is the difference between population and sample standard deviation?
Population standard deviation (σ) is used when your data represents the entire group you are studying. Sample standard deviation (s) is used when your data is a subset drawn from a larger population. Sample SD divides by N−1 instead of N to correct for the bias introduced by estimating the population mean from sample data.
How do I calculate standard deviation step by step?
1. Find the mean by adding all values and dividing by the count. 2. Subtract the mean from each value to get the deviations. 3. Square each deviation. 4. Sum the squared deviations. 5. Divide by N (population) or N−1 (sample) to get the variance. 6. Take the square root to get the standard deviation.
What does a standard deviation of 0 mean?
A standard deviation of 0 means all values in the dataset are identical — there is no spread at all. For example, the dataset [5, 5, 5, 5] has a standard deviation of 0 because every value equals the mean.
What is variance and how is it related to standard deviation?
Variance is the average of the squared deviations from the mean (σ² for population, s² for sample). Standard deviation is simply the square root of variance. Variance is useful mathematically but is in squared units, while standard deviation is in the same unit as the original data, making it more interpretable.
What is the 68-95-99.7 rule?
For normally distributed data, approximately 68% of values fall within ±1 standard deviation of the mean, 95% within ±2 standard deviations, and 99.7% within ±3 standard deviations. This is also called the empirical rule and is very useful for quickly interpreting spread.
How many data points do I need to calculate standard deviation?
You need at least 1 data point to calculate population standard deviation (with 1 point, it equals 0). For sample standard deviation you need at least 2 data points, because the formula divides by N−1, which would be 0 for a single value. More data points generally give a more reliable estimate.
Standard Deviation Formula
Standard deviation measures how spread out values are around the mean. A low standard deviation means the values cluster tightly around the average; a high standard deviation means they are spread widely.
Population Standard Deviation (σ)
σ = √( Σ(xᵢ − x̄)² / N )
Used when your data set is the entire population. Divides by N.
Sample Standard Deviation (s)
s = √( Σ(xᵢ − x̄)² / (N−1) )
Used when your data is a sampledrawn from a larger population. Divides by N−1 (Bessel's correction) to reduce bias.
Where:
- xᵢ — each individual data value
- x̄ — arithmetic mean of all values
- N — total number of values
- Σ — sum over all values
Step-by-Step Example
Calculate the population standard deviation of: 2, 4, 4, 4, 5, 5, 7, 9
- Find the mean:
x̄ = (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) / 8 = 40 / 8 = 5
- Compute squared deviations:
(2−5)² = 9, (4−5)² = 1, (4−5)² = 1, (4−5)² = 1, (5−5)² = 0, (5−5)² = 0, (7−5)² = 4, (9−5)² = 16
- Sum the squared deviations:
9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32
- Divide by N (population):
Variance = 32 / 8 = 4
- Take the square root:
σ = √4 = 2
When to Use Population vs Sample
| Scenario | Use |
|---|---|
| Test scores for your entire class | Population (σ) |
| Survey of 500 voters representing all voters | Sample (s) |
| Heights of every player on a team | Population (σ) |
| Quality control sample from a production run | Sample (s) |
| Daily temperatures for an entire month | Population (σ) |
Rule of thumb: If you collected data from every member of the group, use Population. If you only measured a subset, use Sample.
Variance vs Standard Deviation
Variance and standard deviation both measure spread, but they differ in units:
- Variance (σ² or s²) — the average of the squared deviations from the mean. Its unit is the square of the original unit (e.g., cm²). Squaring emphasizes large deviations and makes the math tractable.
- Standard Deviation (σ or s) — the square root of variance. It is in the same unit as the original data, making it far more interpretable. When someone says "the data is spread ±2 units from the mean," they mean ±1 standard deviation.
Interpreting Standard Deviation — The 68-95-99.7 Rule
For data that follows a normal (bell-curve) distribution, the empirical rule tells you how much data falls within each standard deviation of the mean:
Example: if the mean test score is 70 and σ = 10, then ~68% of students scored between 60 and 80, and ~95% scored between 50 and 90.
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