# Sample Standard Deviation

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### What is the Formula for Sample Standard Deviation ?

Standard Deviation of Sample s = \sqrt{ \frac{1}{n-1} \sum_{i=1}^n (x - \overline{x})^2}

### What is Sample Standard Deviation in Statistics?

Sample Standard Deviation gives the average distance of your numbers to the mean of those numbers.

Example 1: A Standard Deviation of 0 means that the given set of numbers are the same since they don’t differ from their mean.

Example 2: A Standard Deviation of 1 means that the given set of numbers differ – on average – by 1 from their mean.

Bowling Example: A consistent bowler that bowls 110 and 90 games has a lower Standard Deviation than a bowler who bowls 50 and 150. While they each have a mean of 100 the 2. bowler scores varied much more from 100 than the first bowler.

### What is the Difference between Sample and Population Standard Deviation?

Their Difference lies in the Denominators of their Formulas (for technical reasons):

When computing the Sample Standard Deviation we divide by n-1.

When computing the Population Standard Deviation we divide by n instead.

This is the formula for the Population Standard Deviation:

\sigma = \sqrt{ \frac{1}{n} \sum_{i=1}^n (x - \overline{x})^2}

### What is the 1 Standard Deviation Rule? What is the 2 Standard Deviation Rule?

The 1 Standard Deviation rule refers to the Empirical Rule of Normal Distributions (aka Bell Curves). See image.

About 68% of the Data fall within 1 Standard Deviation of the Mean.

About 95% of the Data fall within 2 Standard Deviations of the Mean.

About 99.7% of the Data fall within 3 Standard Deviations of the Mean.

Example: About 95% of the US Population have an IQ between 80 and 120.
Reason: Mean IQ=100 and Standard Deviation=10. Thus, 95% Americans fall within 2 standard deviations, 2*10 = 20 of 100.
Note: This means that about 2.5% have an IQ higher than 120.