# 7. Chebyshev's Inequality

## h. calculate and interpret the proportion of observations falling within a specified number of standard deviations of the mean using Chebyshev's inequality;

What is the name of the Russian Scientist who created the standard deviation theorem?

Pafnuty Chebyshev. I'm sure I don't need this info for the test, but it's just fun to say!

What is the standard deviation math formula to calculate the proportion of observations that lie FEWER than `c`

deviations from the mean?

$$1 - \frac{1}{c^2}$$

When and why can Chebyshev's inequality be used?

Chebyshev's inequality can be used when you know the mean and standard deviation for any shape of distribution. It is used to find the percent of data points that fall within the given values.

Using Chebyshev's Inequality, what is the minimum proportion of observations from a population of 500 that must lie within two standard deviations of the mean, regardless of the shape of the distribution?

Chebyshev says that for any distribution shape, the minimum proportion of observations that lie within `k`

standard deviations is equal to `1 - 1/k^2`

. So in this case it would be `1 - 1/4`

= 75%

What is the empirical rule?

68/95/99.7 - In a normal distribution, the amount of data that lies within 1 std dev is 68%, it is 95% within 2 std dev, and 99.7% within 3 std dev.

##### Source:

- CFA