Thursday, January 10, 2008

Lecture 2 - Ch 3 - Using Standard Deviation

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/z2)*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

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