Standard Deviation Calculator Online Free
Standard Deviation Calculator
Enter Your Data
Enter your numbers separated by commas. Example: 5, 10, 15, 20, 25
Sample: A subset of the population (uses N-1 for unbiased estimate)
The standard deviation calculator measures how spread out a set of numbers is from their mean (average). A small standard deviation means values are clustered close to the average. A large standard deviation means they are more scattered.
Standard Deviation Formula
Population SD (σ) = √[Σ(x - μ)² / N] Sample SD (s) = √[Σ(x - x̅)² / (N-1)]
μ = population mean. x̅ = sample mean. N = number of values. Σ = sum of. Use population SD when you have all data points. Use sample SD when your data is a subset of a larger group.
Step-by-Step Calculation
Dataset: 4, 8, 6, 5, 7. Step 1: Find the mean. (4+8+6+5+7)/5 = 6. Step 2: Subtract mean from each value and square: (4-6)^2=4, (8-6)^2=4, (6-6)^2=0, (5-6)^2=1, (7-6)^2=1. Step 3: Sum the squared differences: 4+4+0+1+1=10. Step 4: Divide by N (population) or N-1 (sample). Population: 10/5=2. Sample: 10/4=2.5. Step 5: Take the square root. Population SD: √2 = 1.41. Sample SD: √2.5 = 1.58.
When to Use Population vs. Sample SD
| Situation | Use | Denominator |
|---|---|---|
| All values in the group are included | Population (σ) | N |
| Data is a sample from a larger group | Sample (s) | N - 1 |
| Exam scores for one class (full group) | Population (σ) | N |
| Survey sample from a larger population | Sample (s) | N - 1 |
The Empirical Rule (68-95-99.7 Rule)
In a normal distribution: about 68% of values fall within 1 standard deviation of the mean. About 95% fall within 2 standard deviations. About 99.7% fall within 3 standard deviations. A value more than 2 SDs from the mean is unusual. More than 3 SDs is rare.
Frequently Asked Questions
What does standard deviation tell you?⌄
It tells you how spread out the data values are around the mean. A small SD means most values cluster near the average. A large SD means values vary widely. It is one of the most used measures of variability in statistics, finance, and science.
What is the difference between population and sample standard deviation?⌄
Population SD (symbol: σ) is used when you have every data point in the group. Sample SD (symbol: s) is used when your data is a subset. Sample SD divides by N-1 instead of N to correct for the tendency of samples to underestimate the true variability of the population.
How do I calculate standard deviation by hand?⌄
Find the mean. Subtract the mean from each value and square the result. Sum all squared differences. Divide by N (population) or N-1 (sample). Take the square root of the result.
Is a higher or lower standard deviation better?⌄
Neither is inherently better; it depends on context. In quality control, low SD means consistent output. In investment returns, low SD means lower volatility. In some research, high variability is informative. The goal is to match the SD to what the context requires.
What is the relationship between variance and standard deviation?⌄
Variance is the average of squared differences from the mean. Standard deviation is the square root of variance. SD is preferred in most contexts because it is in the same units as the original data, making it easier to interpret and communicate.