What is Standard Deviation?

In simple terms, it measures how spread out your data is. A low SD means most numbers are close to the average (mean). A high SD means the numbers are spread out over a wider range.

What is the difference between Sample and Population?

Population (σ) is used when you have data for the ENTIRE group (e.g., ages of every US President). Sample (s) is used when you only have data for a subset (e.g., ages of 5 random US Presidents). The math differs slightly (dividing by N vs N-1).

Variance is simply the Standard Deviation squared. It is used in more complex statistics, but SD is generally more useful because it is in the same units as the original data.

Standard Deviation: Measuring the Spread

The "mean" (average) doesn't tell the whole story. Standard deviation tells you if the result was a fluke or a pattern.

Analyze Your Data

Enter a list of numbers to find the Mean, Variance, and SD.

Standard Deviation Calculator

Step-by-Step Calculation

Let's find the SD of the heights of 5 dogs: 600mm, 470mm, 170mm, 430mm, and 300mm.

Step 1: Find the Mean (Average)

(600 + 470 + 170 + 430 + 300) / 5 = 394mm

Step 2: Find the Differences

Subtract the mean from each number and square the result.
(600 - 394)^2 = 42,436
(470 - 394)^2 = 5,776... and so on.

Step 3: Average the Squared Differences

Add up all the squared results from Step 2 and divide by (N-1). This is your Variance.

Step 4: The Square Root

Take the square root of the Variance. This is your Standard Deviation.

Why N-1?

This is called "Bessel's Correction." When working with a Sample (small group) instead of a Population (everyone), dividing by N-1 makes the result larger, correcting for the fact that small samples tend to underestimate the true spread of the population.

The Bell Curve

In a normal distribution:

68% of data falls within 1 Standard Deviation.
95% of data falls within 2 Standard Deviations.
99.7% of data falls within 3 Standard Deviations.