- CFA Exams
- 2021 Level I
- Study Session 2. Quantitative Methods (1)
- Reading 7. Statistical Concepts and Market Returns
- Subject 7. Chebyshev's Inequality

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##### Subject 7. Chebyshev's Inequality PDF Download

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.

**Learning Outcome Statements**

CFA® 2021 Level I Curriculum, 2021, Volume 1, Reading 7

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**User Contributed Comments**
10

User |
Comment |
---|---|

sarath |
Important concept. |

itconcepts |
why does the empirical rule differ from the "other useful values to know" |

achu |
empirical rule shows the dispersion of a normal distribution (within 1 sd =68%, 2sds 95%). As you can see, the normal distribution dispersion is 'tighter' than that shown by Cheby's inequality, but this is fine. At least 75% must be within 2sd's, but in a normal dist it goes up to 95%. |

Cesarjr20 |
What would be a good book to read about this topic? |

timmak |
I would say any entry level university textbook on stats and probability should cover this topic in more detail. |

Nuta |
You can find good practical examples for understanding if you refer to 6 sigma rule. |

Sophorior |
Empirical rule: 68% within 1 sd;, 95% within 2 sd;, and 99% within 3 sd. this is valid for a normal distribution, not for any distribution (so it shoudn't be mentioned here...) |

tschorsch |
is this not a limit of maximum dispersion? i.e. regardless of the distribution, no more than 1/k² of the values can be more than k std deviations from the mean |

Ryan1234 |
That is correct. |

mydogo |
I found this to a much simpler explanation to remember and apply, thus good for retention. |

I am using your study notes and I know of at least 5 other friends of mine who used it and passed the exam last Dec. Keep up your great work!