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Standard Deviation Calculator — Step-by-Step Solutions

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The Standard Deviation Calculator computes key statistical measures for any dataset. It calculates both population and sample standard deviation, variance, mean, median, range, and sum.

Enter your data values separated by commas and get comprehensive statistics instantly. The calculator clearly shows the difference between population and sample calculations.

How to Use the Standard Deviation Calculator

  1. Enter your data values separated by commas (e.g. 4, 8, 6, 5, 3, 7, 2, 9).
  2. The calculator computes both population and sample standard deviation, variance, mean, median, range, and sum.
  3. Population standard deviation divides by N; sample standard deviation divides by N-1 (Bessel's correction).

Formula and Theory

Standard deviation measures how spread out values are from the mean. A low standard deviation means data points cluster near the mean; a high standard deviation means they are more dispersed.

The calculation involves: (1) find the mean, (2) subtract the mean from each value and square the result, (3) find the average of those squared differences (variance), (4) take the square root. For a sample, divide by N-1 instead of N to correct for bias.

Worked Examples

Example 1: Population standard deviation

Problem: Find the standard deviation of {2, 4, 4, 4, 5, 5, 7, 9}

Solution: Mean = 40/8 = 5. Variance = ((2-5)^2 + (4-5)^2 + ... + (9-5)^2) / 8 = 4. SD = sqrt(4)

Answer: SD = 2

Example 2: Sample vs population

Problem: For the sample {3, 7, 7, 19}, find the sample standard deviation

Solution: Mean = 36/4 = 9. Sum of squared deviations = 36+4+4+100 = 144. Sample variance = 144/3 = 48

Answer: Sample SD = 6.93

Frequently Asked Questions

When should I use population vs sample standard deviation?
Use population SD when you have data for the entire group. Use sample SD when your data is a subset of a larger population, which is the more common scenario in research.
What is variance?
Variance is the square of the standard deviation. It measures the average squared deviation from the mean.
What does a standard deviation of 0 mean?
A standard deviation of 0 means all values in the dataset are identical. There is no spread at all.
What is Bessel's correction?
Bessel's correction uses N-1 instead of N in the denominator when calculating sample variance. This corrects the bias that would underestimate the population variance from a sample.

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