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**Subject 7. Chebyshev's Inequality**

The Russian mathematician, Pafnuty Chebyshev, developed a useful theorem of inequality dealing with standard deviation as a measure of dispersion.

Chebyshev's Inequality: 1-(1/c

Let c be any number greater than 1. For any sample or population of data, the proportion of observations that lie FEWER than c standard deviations from the mean is at least:

- This applies to both populations and samples, and to discrete and continuous data, regardless of the shape of the distribution.
- It gives a conservative estimate of the proportion of observations in an interval around the mean.
- It highlights the importance of σ and places lower and upper limits.
- Empirical rule: 68% within 1σ, 95% within 2σ, and 99% within 3σ.

Chebyshev's Inequality theorem is useful in that if we know the standard deviation, we can use it to measure the

*minimum*amount of dispersion.

*Example*What approximate percent of a distribution will lie within ± two standard deviations of the mean?

Chebyshev's Inequality: 1-(1/c

^{2}) = 1 - (1/2^{2}) = 0.75 or 75%Several other values are also useful to know.

- A minimum of 36% of observations lie within 1.25 σ of the mean.
- A minimum of 56% of observations lie within 1.5 σ of the mean.
- A minimum of 75% of observations lie within 2 σ of the mean.
- A minimum of 84% of observations lie within 2.5 σ of the mean.
- A minimum of 89% of observations lie within 3 σ of the mean.
- A minimum of 94% of observations lie within 4 σ of the mean.

#### Practice Question 1

It has been determined that the mean return rate for tax-exempt municipal bonds is 9.2% with a standard deviation of 3%. What is the minimum percentage of return rates for tax-exempt municipal bonds with rates between 4.7% and 13.7%?A. 56%

B. 67%

C. 75%Correct Answer: A

Since 13.7% is 1.5 standard deviations above the mean of 9.2% and 4.7% is 1.5 standard deviations below the mean of 9.2%, we would expect at least 1 - 1/1.5

^{2}= 0.56 of the rates to fall within this range (Chebyshev's Inequality).

#### Practice Question 2

What is the difference between Chebyshev's Inequality and the Empirical Rule?A. Chebyshev's Inequality applies only to distributions in which 68% of the data points fall within one standard deviation of the mean; the Empirical Rule applies only to mound or symmetrical distributions.

B. Chebyshev's Inequality applies to any probability distribution; the Empirical Rule applies only to distributions in which 68%/95%/99% of the data points fall within one/two/three standard deviations of the mean.

C. Chebyshev's Inequality applies to any mound-shaped probability distribution; the Empirical Rule applies only to distributions in which 95% of the data points fall within two standard deviation of the mean.Correct Answer: B

#### Practice Question 3

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?A. 66%

B. 75%

C. 85%Correct Answer: B

Chebyshev's inequality holds for any distribution, regardless of shape, and states that the minimum proportion of observations located within k standard deviations of the mean is equal to 1 - 1/k

^{2}. In this case, k = 2 and 1 - 1/4 = 0.75 or 75%.

#### Practice Question 4

An investment has a mean return of 15% with a standard deviation of 4.5%. You expect that 75% of the rates will fall within which values?A. 5.9%; 24.9%

B. 6.0%; 24.0%

C. 6.2%; 24.0%Correct Answer: B

75% corresponds to k=2 because 0.75 = 1 - 1/2^{2}. Therefore, the extremes for the interval are 15% +- 2 * (4.5%) or 6% and 24% (Chebyshev's Inequality).

#### Practice Question 5

In 2011, a portfolio with 6 stocks had the following total return rates in percentages:27.98%, 44.94%, 54.53%, -52.68%, 10.21%, 0.50%

The average return rate for this portfolio was 4.92% and the standard deviation was 37.66%. How confident are you that the return rates will fall within -51.57% and 61.41%?

A. 56%

B. 75%

C. 85%Correct Answer: A

-51.57% is 1.5 standard deviations below the mean and 61.41% is 1.5 standard deviations above the mean. Therefore, we are confident that 1 - 1/1.5^{2} = 56% of the return rates will fall within the specified range (Chebyshev's inequality).

### Study notes from a previous year's CFA exam:

7. Chebyshev's Inequality