Standard Deviation Calculator

What is 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

Population Standard Deviation

Used when you have data for the entire population. The formula is:

sigma = sqrt(sum((xi - mean)^2) / N)

Where N is the total number of values in the population.

Sample Standard Deviation

Used when you have data from a sample of the population. The formula is:

s = sqrt(sum((xi - mean)^2) / (n - 1))

Where (n - 1) is called Bessel's correction, which provides an unbiased estimate.

How to Calculate Standard Deviation

1. Calculate the mean: Add all values and divide by the count
2. Find deviations: Subtract the mean from each value
3. Square the deviations: Square each deviation
4. Calculate variance: Find the average of squared deviations
5. Take square root: The square root of variance is standard deviation

Example Calculation

For data set: 2, 4, 4, 4, 5, 5, 7, 9

Mean = (2+4+4+4+5+5+7+9) / 8 = 5
Deviations: -3, -1, -1, -1, 0, 0, 2, 4
Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16
Sum of squared deviations = 32
Population variance = 32/8 = 4
Population std dev = sqrt(4) = 2
Sample variance = 32/7 = 4.57
Sample std dev = sqrt(4.57) = 2.14

When to Use Each Type

Use Population Standard Deviation when:

• You have data for every member of the group
• Analyzing test scores for a specific class
• Measuring all products from a production batch

Use Sample Standard Deviation when:

• You have data from only a subset of the population
• Conducting surveys or polls
• Quality control sampling
• Scientific research with sample data

Interpreting Standard Deviation

For normally distributed data:

• About 68% of values fall within 1 standard deviation of the mean
• About 95% of values fall within 2 standard deviations of the mean
• About 99.7% of values fall within 3 standard deviations of the mean

This is known as the 68-95-99.7 rule or the empirical rule.