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

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**CFA Practice Question**

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%?

B. 67%

C. 75%

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).###
**User Contributed Comments**
10

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

chuong |
9.2% + 1.5x3%=13.7% 9.2% -1.5x3% = 4.7% |

Khadria |
NOTE ---- Chebyshev's Inequality defines how many values are within C standard deviation. So here it means that at least 56% of the total data set values are lying between interest rates 4.7% and 13.7%. |

TammTamm |
Did the 1.5 come from 3/2? |

Nuta |
1.5 = (13.7-9.2)/3 |

Sabs |
or 1.5 = (9.2-4.7)/3 |

skath |
If the questions mentions the word "between" then you can SD/2. |

johntan1979 |
I know you can't determine the 1.5 sd from the empirical rule, but what if it is 2 sd? When do we use the empirical rule (95%) vs Chebyshev's (75%)? |

johntan1979 |
Think I found the answer myself, which is to examine the answer options. Tricky, if you are like me, automatically apply the empirical rule. |

sgossett86 |
1 sd is 66%, so why would 1.5sd be less than 66%? couldn't we do something like 1.5 * .66 to solve ? I am not relying on you guys for the answer. I'm trying to make this make sense to me better. I guess if the empirical rule is used only for normal distributions, which basically never occur, we're using this law to evaluate all other distributions, so i guess it makes sense in that regard. |

msk500 |
Simply, the mean return = 9.2. Anything below or above this mean deviates from the mean. So, if a return is 4.7, then it is 4.5 less than the mean of 9.2. If every (ie 1) deviation is 3, then 4.5 deviations = 1.5 standard deviations. Similarly, if a return is 13.7, then this is 4.5 deviations above the mean, and a standard deviation of 1.5. Applying Chebychev's inequality, you get 1-(1/1.5^2) = 0.555556 or 56%. |