Standard Deviation Calculator
Calculate standard deviation, variance, and other statistical measures for a dataset.
Enter numbers separated by commas, spaces, or new lines
Parsed values: 8 numbers
10, 12, 23, 23, 16, 23, 21, 16
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What is Standard Deviation?
Standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range.
How it Works
The formula depends on whether you are analyzing a population or a sample:
- Population Standard Deviation (σ): σ = √(Σ(x - μ)² / N)
- Sample Standard Deviation (s): s = √(Σ(x - x̄)² / (n-1))
- Variance: The square of the standard deviation (σ² or s²).
- Mean (μ or x̄): The average of all data points.
Example Calculation
Let's find the Sample Standard Deviation for: 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
- Squared Differences: (2-5)²=9, (4-5)²=1, ..., (9-5)²=16. Sum = 9+1+1+1+0+0+4+16 = 32.
- Sample Variance (s²): 32 / (8-1) = 32 / 7 ≈ 4.57
- Standard Deviation (s): √4.57 ≈ 2.14
Frequently Asked Questions
What is the difference between population and sample standard deviation?
Population SD considers all members of a group (divide by N), while Sample SD considers a subset to estimate the population (divide by n-1).
How to calculate variance?
Variance is the average of the squared differences from the Mean. It is simply the square of the Standard Deviation.
What does a high standard deviation mean?
It means the data points are spread out over a large range of values, far from the mean.
What is the symbol for standard deviation?
The Greek letter sigma (σ) is used for population standard deviation, and the letter 's' is used for sample standard deviation.
Why do we divide by n-1 for sample standard deviation?
Dividing by n-1 (Bessel's correction) provides an unbiased estimator of the population variance.