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

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/c²) = 1 - (1/2²) = 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 - 1/1.5² = 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/k². 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 - 1/2². 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 - 1/1.5² = 56% of the return rates will fall within the specified range (Chebyshev's inequality).