Standard Deviation Tool
Easily calculate standard deviation for population or sample data with this interactive calculator. Get the mean, variance, and visualize results.
Calculator
Step-by-Step Calculation
Step 1: Find the Mean
Enter your numbers to begin.
Formula Breakdown
The calculation uses different formulas for population and sample data.
Population Standard Deviation ($\sigma$)
$\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}$
- $x_i$: Individual data point
- $\mu$: Population mean
- $N$: Number of data points in the population
Sample Standard Deviation ($s$)
$s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$
- $x_i$: Individual data point
- $\bar{x}$: Sample mean
- $n$: Number of data points in the sample
Using $n-1$ is known as **Bessel's correction** and helps provide a less biased estimate for the population standard deviation.
Step-by-Step Example
Let's calculate the sample standard deviation for the dataset: 3, 4, 2, 6, 5
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Find the Mean ($\bar{x}$)
Sum the numbers and divide by the count: $(3+4+2+6+5)/5 = 20/5 = 4$
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Subtract the Mean and Square the Difference
$(3-4)^2 = (-1)^2 = 1$
$(4-4)^2 = (0)^2 = 0$
$(2-4)^2 = (-2)^2 = 4$
$(6-4)^2 = (2)^2 = 4$
$(5-4)^2 = (1)^2 = 1$ -
Sum the Squared Differences
Summing the results from the last step: $1 + 0 + 4 + 4 + 1 = 10$
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Find the Variance ($s^2$)
Divide the sum of squares by $n-1$ (for a sample): $10 / (5-1) = 10/4 = 2.5$
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Take the Square Root
The standard deviation is the square root of the variance: $\sqrt{2.5} \approx 1.581$
Frequently Asked Questions (FAQs)
What is the difference between population and sample standard deviation?
The key difference is the denominator in the formula. Population standard deviation divides by the total number of data points ($N$), while sample standard deviation divides by one less than the number of data points ($n-1$). This adjustment for the sample makes the standard deviation a less biased estimate of the true population standard deviation.
Can this calculator handle negative numbers?
Yes, the calculations for mean, variance, and standard deviation work correctly for any real number, including negative values and decimals.
Why is standard deviation important?
Standard deviation is a crucial measure of dispersion. It shows how spread out a set of values is from the mean. 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.