 # Chebyshev’s Inequality

How you`re supposed to know Chebyshev’s inequality table??

I mean see the exercise:

According to Exhibit 27, the arithmetic mean monthly return and standard deviation of monthly returns on the S&P 500 were 0.95 percent and 5.39 percent, respectively, during the 1926–2017 period, totaling 1,104 monthly observations. Using this information, address the following:

1 Calculate the endpoints of the interval that must contain at least 75 percent of monthly returns according to Chebyshev’s inequality.

How I`m supposed to know that 75 percent is 2 standard deviations??

There`s a table in page 421 that states the percentage or k values of 1,25, 1,5, 2, 2,5 3 4

But how I`m supposed to know how many standard deviation I should put?

WTF

Chebyshev’s inequality says that:

P(|X – μ| / σ ≤ k) ≥ 1 – 1/k2

So, if they give you, say, k = 3, then,

P(|X – μ| / σ ≤ 3) ≥ 1 – 1/32 = 1 – 1/9 = 8/9 = 88.89%

If they give you, say, P(|X – μ| / σ ≤ k) ≥ 75%, then,

75% = 1 – 1/k2

1/k2 = 1 – 75% = 25% = ¼

k2 = 4

k = √4 = 2

Don’t memorize it; understand it.

The proportion of the observations within k standard deviations of the arithmetic mean is at least 1 − 1/k2 for all k > 1.

The book didn`t showed the proportion in the formula ex= x% = 1 - 1/K2

I just saw the formula how was it, but the book actually says it but its was wrote before.

Thanks your explanation clarified it all!

I looked at the 2019 curriculum and it didn’t have it written out as I did.

Stupid.

Also when solving the book didn`t showed the use of this formula:

Solution to 1: According to Chebyshev’s inequality, at least 75 percent of the observations must lie within two standard deviations of the mean, X ± 2s. For the monthly S&P 500 return series, we have 0.95% ± 2(5.39%) = 0.95% ± 10.78%. Thus the lower endpoint of the interval that must contain at least 75 percent of the observations is 0.95% − 10.78% = −9.83%, and the upper endpoint is 0.95% + 10.78% = 11.73%.

It just assumed you knew it by the table, they should have showed the use of the formula you stated above.

Yup.

Stupid.

Exactly they messed up with that they should have showed that!