Standard Deviation Calculator
Calculate the standard deviation, variance, mean, and other statistical measures for a set of numbers with this free online calculator.
About Standard Deviation
Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.
Population vs. Sample Standard Deviation
There are two types of standard deviation calculations:
- Population Standard Deviation (σ): Used when data represents the entire population.
- Sample Standard Deviation (s): Used when data is a sample from a larger population.
Formulas
Population Standard Deviation:
σ = √(Σ(xi - μ)² / N)
Sample Standard Deviation:
s = √(Σ(xi - x̄)² / (n-1))
Where:
xi = each value in the dataset
μ or x̄ = mean of the dataset
N or n = number of values in the dataset
How to Calculate Standard Deviation
- Calculate the mean (average) of the dataset.
- Subtract the mean from each data point to find the deviation.
- Square each deviation.
- Sum all the squared deviations.
- Divide by N (population) or N-1 (sample).
- Take the square root of the result.
Example Calculation
Let's calculate the standard deviation for the dataset: 4, 8, 15, 16, 23, 42
Step 1: Calculate the mean: (4 + 8 + 15 + 16 + 23 + 42) / 6 = 18
Step 2: Find deviations from the mean:
4 - 18 = -14, 8 - 18 = -10, 15 - 18 = -3, 16 - 18 = -2, 23 - 18 = 5, 42 - 18 = 24
Step 3: Square the deviations:
(-14)² = 196, (-10)² = 100, (-3)² = 9, (-2)² = 4, (5)² = 25, (24)² = 576
Step 4: Sum the squared deviations:
196 + 100 + 9 + 4 + 25 + 576 = 910
Step 5: Divide by N (population) or N-1 (sample):
Population: 910 / 6 = 151.67
Sample: 910 / 5 = 182
Step 6: Take the square root:
Population: √151.67 ≈ 12.32
Sample: √182 ≈ 13.49
Applications of Standard Deviation
| Field | Application |
|---|---|
| Finance | Measuring investment risk and volatility |
| Quality Control | Monitoring manufacturing processes |
| Weather Forecasting | Analyzing temperature variations |
| Medicine | Evaluating clinical trial results |
| Education | Grading on a curve, standardized testing |
Interpreting Standard Deviation
In a normal distribution:
- Approximately 68% of data falls within 1 standard deviation of the mean
- Approximately 95% of data falls within 2 standard deviations of the mean
- Approximately 99.7% of data falls within 3 standard deviations of the mean
This is known as the empirical rule or the 68-95-99.7 rule.