Chebyshev’s inequality

I am having problems understanding two questions from the CFA book, could anyone help me to understand?

Q: Over the past 240 months, an investor’s portfolio had a mean monthly return of 0.79%, with a standard deviation of monthly returns of 1.16%. According to Chebyshev’s inequality, the minimum number of the 240 monthly returns that fall into the range of −0.95% to 2.53% is closest to: A 80. B 107. C 133.

Q: For a distribution of 2,000 observations with finite variance, sample mean of 10.0%, and standard deviation of 4.0%, what is the minimum number of observations that will lie within 8.0% around the mean according to Chebyshev’s Inequality? A 720 B 1,500 C 1,680

In each question, what is k?

In each case, we need to find the "K’? I got it on the first question, which is:

Range=Mean+/Standard deviation (K)

-0.95%=0.79% +/- 1.16% (k)

But I can’t solve in the 2d question.

8% is how many standard deviations?

4% standard deviation is given.

x = 1-1/k2

x = 1-1/22

x = 1-1/4

x = 0,75

.75*2000 = 1500

For the first question first discover how far is the mean from the lowest and highest range:

Range = 1,74 from the mean

1,74/1,16 = 1,5 standard deviations

x = 1-1/k2

x = 1-1/1,52

x = 1-1/2,25

x = 1 - 0,444444444444

x = 0,555555555555

55,55%*240 = 133

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I’m aware of that.

So . . . 8% is _ how many _ standard deviations?

I am not sure why we have to calculate the standard deviation when 4%is given. Could you please help me to understand and how to calculate it?

You don’t have to calculate the standard deviation; it’s given, as you say.

I’m not asking you to calculate the standard deviation. I’m asking _ how many _ (4%) standard deviations is 8%? Is it one standard deviation? Three? 6.5? 0.8? How many?

2 standard deviation.

Magician can really be persuasive.

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Thanks, that means “2” is the value of K, but I am still trying to wrap my head around!

I shouldn’t think that there should be much difficulty there; that’s the definition of k: how many standard deviations away from the mean.

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that makes sense:) Thanks so much!

You’re welcome.