Now that we understand how to calculate the standard deviation, we can define three statistical concepts that apply the standard deviation in practice. Those three concepts are the z-score, the empirical rule and Chebyshev's rule.

Z-score - The z-score of an observed value is the difference between the observed value and the mean, divided by the standard deviation. The z-score is used to express how many standard deviations an observation is away from the mean.

Empirical Rule - In a bell-shaped distribution:

1 standard deviation (in either direction from the mean) has about 68% of the data.

2 standard deviations have about 95% of the data.

3 standard deviations have about 99.7% of the data.

This "bell-shaped" distribution is called the standard normal distribution and we'll learn more about it in a later lecture/chapter.

Chebyshev Rule - In any distribution:

At least (1-1/z^{2})*100% of the values will be within z standard deviations.

A few interesting things were noted briefly about Chebyshev's Rule:

1. It works with any, yes any, shaped distribution. That's really amazing.

2. Outside of the z standard deviations, we don't know anything about the data. Specifically, it may not be symmetric, so we can't assume that the data outside the z std devs is evenly distributed between them.

3. Chebyshev's Rule doesn't tell us anything about data within 1 standard deviation.

Did you know? There's a crater on the moon named after Chebyshev. Source: USGS

## Thursday, January 10, 2008

### Lecture 2 - Ch 3 - Using Standard Deviation

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