Standard Deviation Calculator

The Standard Deviation Calculator helps you quickly measure how spread out numbers are in a dataset. Whether you’re analyzing grades, sales figures, or scientific data, this tool makes it easy to understand how much your values vary from the average.

Enter numbers separated by commas (e.g., 5, 7, 3, 8):

How to Use the Calculator

1. Enter your numbers into the input field, separated by commas or spaces (e.g., 5, 10, 15, 20).
2. Click “Calculate”.
3. The calculator will show you:
   • The mean (average)
   • The variance
   • The standard deviation

You can usually choose between:

• Sample Standard Deviation (used when data is a sample of a larger population)
• Population Standard Deviation (used when data represents the entire population)

Formulas Used:

1. Mean (Average)

$$\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i$$

Where $x_i$ are your values and $n$ is the total number of values.

2. Standard Deviation (Population)

$$\sigma = \sqrt{ \frac{1}{n} \sum_{i=1}^{n} (x_i – \bar{x})^2 }$$

3. Standard Deviation (Sample)

$$s = \sqrt{ \frac{1}{n – 1} \sum_{i=1}^{n} (x_i – \bar{x})^2 }$$

Example:

Let’s say your dataset is: 4, 8, 6, 5, 3

1. Mean = (4 + 8 + 6 + 5 + 3) / 5 = 5.2

2. Find squared differences from the mean:

   $$(4 – 5.2)^2 = 1.44,\ (8 – 5.2)^2 = 7.84,\ (6 – 5.2)^2 = 0.64,\ …$$

3. Add them up: 11.2

4. For sample standard deviation:

   $$\sqrt{11.2\mathrm{/}(5-1)}=\sqrt{2.8}\approx 1.67$$