Chebyshev's Inequality

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jatboy

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An industry consists of 5 companies and their P/E ratios are;-
10,12,14,14,50
The arithmetic mean of these P/E ratios is 20 and the standard deviation is 15.07.At least what %of the population wuill have a P/E ratio that is b/W 1.25 standard deviations on the either of the mean;in other words,B/W 1.16 and 38.84?
I am getting confused with this calculation.
How did we arrive on the range(1.16-38.84)?.
How do we calculate it when we have the standard deviation and the mean?
Please help.
Thanks
 
Chebyshev’s: 1-1/k^2
=1-1/15.07^2
= 1- .0044
= 99.56% of all observations lie within 15.07 standard deviations of the mean.
Interval around the sample mean: mean +/- 1.25s
=20 +/- 1.25(15.07)
= [1.17, 38.83]
99% of all observations fall within the range [1.17, 38.83]
 
AndrewP….can’t thank u enough
Appreciate ur help
 
But 80% of the observations fall within 1.16-38.83 not 99,right?
 
For k= 1.25, at least 36% of the observations must lie within 1.25 standard deviations of the mean. In other words, 36% of the population (5) will fall within the interval [1.17, 38.83].
 
Got it…..but my only concern now is that –y did u mention that 99% of observations fall within the range 1.16-38.83?
Thanks again
 
The arithmetic mean of these P/E ratios is 20 and the standard deviation is 15.07.At least what %of the population wuill have a P/E ratio that is b/W 1.25 standard deviations on the either of the mean;in other words,B/W 1.16 and 38.84?
The 1.16 and 38.84 come from mean +/- std deviation * # of standard deviations above/below
ie. 20+/-1.25*15.07 = 38.84, 1.16
so the result of the Chebyshev inequality tells you the minimum % of solutions that are between 1.16 and 38.84
Hope this helps.
 
haha the answer to the question is easy, but the answer to his question “How did we arrive on the range(1.16-38.84)?.” is worth discussing.
 
AndrewP Wrote:
——————————————————-
> Chebyshev’s: 1-1/k^2
>
> =1-1/15.07^2
>
> = 1- .0044
>
> = 99.56% of all observations lie within 15.07
> standard deviations of the mean.
>
> Interval around the sample mean: mean +/- 1.25s
>
> =20 +/- 1.25(15.07)
>
> = [1.17, 38.83]
>
> 99% of all observations fall within the range
> [1.17, 38.83]
Thank u guys……appreciate it……now that i have understoof this inequality……..i am still stuck on this…….Why 99% of all observations will fall within this range?……….would anyone of u be so kind to answer this?
Thanks again
 
It’s not 99%. It was a mistake. It is indeed at minimum 36%. Obviously from observing such a small population one of the variables falls outside of the interval. It speaks to the limitations posed by the inequality.
 
Thank u AndrewP…..i hope u didn’t took offense of this……..i was just curious.
 
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