Standard Deviation Calculator
Enter your numbers in the box below. Choose Population or Sample mode. Results and step-by-step calculations update as you type.
Results
| Count (n): | 6 |
| Sum: | 108 |
| Mean (average): | 18 |
| Population Std Dev (σ): | 12.3153 |
| Population Variance (σ²): | 151.6667 |
| Min: | 4 |
| Max: | 42 |
| Range: | 38 |
Step 1 — Data values (n = 6): [4, 8, 15, 16, 23, 42] Step 2 — Mean: Mean = Sum / n Mean = (4 + 8 + 15 + 16 + 23 + 42) / 6 Mean = 108 / 6 Mean = 18 Step 3 — Squared deviations (xi − mean)²: x1 = 4: (4 − 18)² = (-14)² = 196 x2 = 8: (8 − 18)² = (-10)² = 100 x3 = 15: (15 − 18)² = (-3)² = 9 x4 = 16: (16 − 18)² = (-2)² = 4 x5 = 23: (23 − 18)² = (5)² = 25 x6 = 42: (42 − 18)² = (24)² = 576 Step 4 — Sum of squared deviations: Σ(xi − mean)² = 196 + 100 + 9 + 4 + 25 + 576 Σ(xi − mean)² = 910 Step 5 — Variance (divide by n = 6): Variance = 910 / 6 Variance = 151.6667 Step 6 — Standard deviation: Std Dev = √Variance Std Dev = √151.6667 Std Dev = 12.3153
What Is Standard Deviation?
Standard deviation is one of the most important measures in statistics. It tells you how much the values in a dataset vary from the average (mean). A small standard deviation means the values are tightly clustered around the mean. A large standard deviation means the values are spread far apart.
The concept was introduced by Karl Pearson in 1893 and has since become central to virtually every field that uses data: finance, engineering, medicine, social science, and manufacturing quality control. Understanding what standard deviation means — not just how to calculate it — is essential for interpreting any statistical report.
Population vs. Sample Standard Deviation
There are two versions of standard deviation, and choosing the wrong one is a common mistake. The difference comes down to what your data represents.
Population standard deviation (σ) — Use this when your dataset contains every member of the group you care about. For example: the test scores of all 30 students in a single classroom, or the heights of all trees in a specific plot of land.
Sample standard deviation (s) — Use this when your dataset is a subset drawn from a larger population. For example: a survey of 500 voters drawn from a city of 2 million, or the weights of 50 boxes sampled from a production run of 10,000.
The formulas are nearly identical, with one difference: sample standard deviation divides by n − 1 instead of n. This adjustment — known as Bessel's correction — accounts for the fact that a sample tends to underestimate the true spread of the population. Using n − 1 produces an unbiased estimate of the population variance.
The Standard Deviation Formula
Both formulas follow the same structure. The variable names differ (σ for population, s for sample; μ for population mean, x̄ for sample mean; N for population size, n for sample size), but the logic is the same.
Population standard deviation:
σ = √( Σ(xi − μ)² / N )
Sample standard deviation:
s = √( Σ(xi − x̅)² / (n − 1) )
Step-by-Step Example
Let's use the default dataset: 4, 8, 15, 16, 23, 42 (the numbers from Lost, if you recognize them). We'll calculate the population standard deviation.
Step 1 — Find the mean:
Sum = 4 + 8 + 15 + 16 + 23 + 42 = 108
n = 6
Mean = 108 / 6 = 18
Step 2 — Find each squared deviation (xi − mean)²:
| Value (xi) | xi − mean | (xi − mean)² |
|---|---|---|
| 4 | 4 − 18 = −14 | 196 |
| 8 | 8 − 18 = −10 | 100 |
| 15 | 15 − 18 = −3 | 9 |
| 16 | 16 − 18 = −2 | 4 |
| 23 | 23 − 18 = 5 | 25 |
| 42 | 42 − 18 = 24 | 576 |
Step 3 — Sum the squared deviations:
196 + 100 + 9 + 4 + 25 + 576 = 910
Step 4 — Divide by N (population) or n−1 (sample):
Population variance = 910 / 6 = 151.6667
Sample variance = 910 / 5 = 182
Step 5 — Take the square root:
Population std dev = √151.6667 = 12.3154
Sample std dev = √182 = 13.4907
What Does the Result Tell You?
For the dataset 4, 8, 15, 16, 23, 42, the mean is 18 and the population standard deviation is about 12.3. This means that, on average, each value is about 12.3 units away from the mean of 18. The dataset is fairly spread out — ranging from 4 to 42.
A useful rule is the 68-95-99.7 rule (also called the empirical rule), which applies to 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.
Real-World Uses of Standard Deviation
Standard deviation appears throughout science, finance, and everyday life:
- Finance: Investment volatility is measured as annualized standard deviation of returns. A stock with higher standard deviation carries more risk.
- Manufacturing: Six Sigma quality control defines defects as outcomes more than 6 standard deviations from the target mean.
- Medicine: Blood test results are reported with reference ranges typically set at mean ± 2 standard deviations for a healthy population.
- Education: Standardized test scores (SAT, IQ tests) are scaled so the mean is 100 (or 500) and each standard deviation represents 15 (or 100) points.
- Weather: Climate scientists use standard deviation to measure how unusual a temperature or rainfall event is compared to historical norms.
Frequently Asked Questions
What is the difference between population and sample standard deviation?
Population standard deviation (σ) divides the sum of squared deviations by n (the total number of values). Sample standard deviation (s) divides by n−1. The n−1 correction, called Bessel's correction, compensates for the fact that a sample tends to underestimate the spread of the full population. Use population standard deviation when you have data for every member of the group. Use sample standard deviation when your data is a subset drawn from a larger population.
What does standard deviation tell you?
Standard deviation measures how spread out the values in a dataset are around the mean. A low standard deviation means most values are close to the mean. A high standard deviation means values are spread widely. For example, if the average exam score is 70 with a standard deviation of 5, most students scored between 65 and 75. If the standard deviation were 20, scores would range much more widely.
What is the formula for standard deviation?
For population standard deviation: σ = √(Σ(xi − μ)² / N), where μ is the mean and N is the count. For sample standard deviation: s = √(Σ(xi − x̄)² / (n−1)), where x̄ is the sample mean and n is the sample size. The steps are: (1) find the mean, (2) subtract the mean from each value and square the result, (3) sum all squared differences, (4) divide by N or n−1, (5) take the square root.
Can standard deviation be negative?
No. Standard deviation is always zero or positive. It equals zero only if all values in the dataset are identical (there is no spread at all). Because the formula involves squaring differences and taking a square root, the result is always non-negative.
What is variance and how does it relate to standard deviation?
Variance is the square of standard deviation: Variance = σ² (or s²). It represents the average of the squared deviations from the mean. Standard deviation is in the same units as the original data (e.g., dollars, meters), while variance is in squared units (e.g., dollars², meters²). Standard deviation is easier to interpret because it shares units with the data, but variance is useful in many statistical formulas such as analysis of variance (ANOVA).
Related Calculators
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