Chebyshev/Confidence Interval

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hopeful231

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Can somebody please help me my telling me the diference in terms of usage.
This topic may seem old but I am so totally confused
 
The bottomline is, Chebyshev’s rule applies regardless of the distribution. If I am not wrong, it is an estimation and 0 % of the observations will fall within 1 s.d. of the mean, 75% will fall between 2 s.d. of the mean.
 
So the first thing is that C.I.’s and Chebyshev are very different animals. A C.I. is a statistical estimate based on data. Chebyshev is a probabilistic bound not used in any data analysis that I have ever seen. Why they are presenting a probabilistic bound in CFA I is beyond me, except that it is a neat result.
So start with some probability distribution and suppose that it has a mean and a standard deviation (not true of all distributions). Chebyshev says you don’t need to know anymore than that to know that at least 75% of the “observations” are within 2 s.d.’s of the mean (and more generally that at least 1 - 1/k^2 are within k s.d.’s of the mean). Note that Chebyshev has nothing to do with normality or any other shape assumption.
A C.I. is an estimator of some parameter, often a mean of a distribution. Usually these are based on some assumption of normality or the central limit theorem. So if I ask you to estimate the mean of a distribution, you sample from the distribution, calculate X-bar and tell me that your belief is that the mean of the distribution is “about X-bar”. That’s just as satisfying to me as when I would ask my Mom when dinner was coming and she would say “about 6 PM”. I would start emptying the contents of the refrigerator onto the kitchen floor until my mother gave me a more precise answer like “With 95% confidence, dinner will be between 5:45 PM and 6:15 PM”. That’s a C.I..
 
Ur answer was very adequate especially the bit “Note that Chebyshev has nothing to do with normality or any other shape assumption” was very helpful.
Thanks alot.
Does the CFAI keep to their LOS’s
 
Chebyshev’s inequality gives a lower bound on the % of the population lying within a given number of standard deviations from the mean and this regardless of the distribution.
Chebyshev’s inequality tells us that at least 75% of the population lies within 2 standard deviations from the mean. If the population is normally distributed, then 95% of the population lies within that interval. If the population is uniformly distributed, then the entire population lies within 2 standard deviations from the mean.
There was a question on my Level 1 exam that required the use of Chebyshev’s inequality .
 
JoeyDVivre Wrote:
——————————————————-
> So the first thing is that C.I.’s and Chebyshev
> are very different animals. A C.I. is a
> statistical estimate based on data. Chebyshev is
> a probabilistic bound not used in any data
> analysis that I have ever seen. Why they are
> presenting a probabilistic bound in CFA I is
> beyond me, except that it is a neat result.
>
> So start with some probability distribution and
> suppose that it has a mean and a standard
> deviation (not true of all distributions).
> Chebyshev says you don’t need to know anymore than
> that to know that at least 75% of the
> “observations” are within 2 s.d.’s of the mean
> (and more generally that at least 1 - 1/k^2 are
> within k s.d.’s of the mean). Note that Chebyshev
> has nothing to do with normality or any other
> shape assumption.
>
> A C.I. is an estimator of some parameter, often a
> mean of a distribution. Usually these are based
> on some assumption of normality or the central
> limit theorem. So if I ask you to estimate the
> mean of a distribution, you sample from the
> distribution, calculate X-bar and tell me that
> your belief is that the mean of the distribution
> is “about X-bar”. That’s just as satisfying to me
> as when I would ask my Mom when dinner was coming
> and she would say “about 6 PM”. I would start
> emptying the contents of the refrigerator onto the
> kitchen floor until my mother gave me a more
> precise answer like “With 95% confidence, dinner
> will be between 5:45 PM and 6:15 PM”. That’s a
> C.I..
Joey, can you please explain in greater detail what a probabilistic bound is? Is it related to the proof of the Chebyshev inequality or something?
 
I would say that Chebshev is just a math thing that has nothing to do with anything in the real data-analysis world. It’s one of those things you use for proofs like Holder’s inequality or something.
 
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