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.