Standard Deviation Calculator
Enter your data set below and choose sample or population standard deviation.
What is Standard Deviation?
Standard deviation measures how spread out numbers are in a data set. A low standard deviation indicates that the values tend to be close to the mean (average), while a high standard deviation shows that the data points are spread over a wider range. It is widely used in statistics, finance, science, and quality control.
Population vs. Sample Standard Deviation
There are two types of standard deviation, and choosing the right one depends on your data:
- Population Standard Deviation (σ) – used when you have data for the entire population. The formula divides the sum of squared differences by N, the total number of data points.
- Sample Standard Deviation (s) – used when your data represents only a sample of a larger population. It divides by n-1 (Bessel's correction) to provide an unbiased estimate of the population standard deviation.
Standard Deviation Formulas
σ = √[ Σ (xáµ¢ - μ)² / N ]
Sample:
s = √[ Σ (xáµ¢ - x̄)² / (n-1) ]
Where:
xáµ¢ = each data value, μ = population mean, x̄ = sample mean, N = population size, n = sample size.
How to Use This Calculator
- Type or paste your data values into the input box, separated by commas (e.g.,
5, 10, 15, 20). - Select Sample if your data is a sample, or Population if it represents the entire group.
- Click Calculate. The mean, variance, and standard deviation will appear instantly.
- Use the Clear button to reset and try a new data set.
Step-by-Step Example
Let’s calculate the sample standard deviation for the data: 2, 4, 4, 4, 5, 5, 7, 9
- Count (n): 8
- Mean (x̄): (2+4+4+4+5+5+7+9) / 8 = 40 / 8 = 5
- Differences squared: (2-5)²=9, (4-5)²=1 (×3), (5-5)²=0 (×2), (7-5)²=4, (9-5)²=16
- Sum of squared differences: 9 + 1+1+1 + 0+0 + 4 + 16 = 32
- Sample variance (s²): 32 / (8-1) = 32 / 7 ≈ 4.5714
- Sample standard deviation (s): √4.5714 ≈ 2.138
You can verify this result using the calculator above.
When to Use Standard Deviation
Standard deviation is essential in many fields:
- Finance: to measure risk and volatility of investments.
- Education: to understand the spread of test scores.
- Quality control: to monitor product consistency.
- Research: to summarize data variability.
Bookmark this free online standard deviation calculator for quick and accurate calculations anytime.
