Standard Deviation Calculator

Measure the Spread of Your Data

Standard deviation is a key statistical measure of the dispersion or variability of a dataset. A low standard deviation indicates that the values tend to be close to the mean (the average), while a high standard deviation indicates that the values are spread out over a wider range. This calculator computes the mean, variance, and standard deviation for any set of numbers.

The Formulas Explained

1. Mean (μ or x̄): The average of all numbers in the dataset.
2. Variance (σ² or s²): The average of the squared differences from the Mean. The formula differs slightly for a population versus a sample.
   • Population Variance: Divide by the total number of data points (N).
   • Sample Variance: Divide by the number of data points minus one (n-1). This is known as Bessel's correction and provides a better estimate of the population variance.
3. Standard Deviation (σ or s): The square root of the variance.

How to Use the Calculator

1. Enter Numbers: Input your data set as a series of numbers separated by commas.
2. Select Type: Choose whether your data represents an entire Population or just a Sample of a larger population.
3. Calculate: The calculator will display the Mean, Variance, and Standard Deviation.

Real-World Example

You have the test scores for a sample of 5 students: 85, 92, 78, 88, 90.

• Mean: (85+92+78+88+90) / 5 = 86.6
• Sample Variance: The sum of the squared differences from the mean is (85-86.6)² + ... = 114.8. The variance is 114.8 / (5-1) = 28.7.
• Sample Standard Deviation: √28.7 ≈ 5.36. This tells you that, on average, the scores are about 5.36 points away from the mean score of 86.6.

Frequently Asked Questions (FAQ)

• When should I use 'Sample' vs. 'Population'? Use Population if your data set includes every member of the group you are studying (e.g., the test scores of every student in a specific class). Use Sample if your data is a subset of a larger group (e.g., the test scores of some students from that class, used to estimate the performance of all students).
• Why is the sample formula different? Dividing by n-1 for a sample provides an unbiased estimate of the population variance. It corrects for the fact that a sample is likely to have slightly less variability than the full population.
• What is variance? Variance is another measure of spread. It's the standard deviation squared. Standard deviation is often preferred because it is in the same units as the original data, making it easier to interpret.