Statistics - Chebyshev's Theorem

Statistics – Chebyshev’s Theorem

The fraction of any set of numbers mendacity inside okay normal deviations of these numbers of the imply of these numbers is a minimum of

11okay21−1k2

The place −

  • okay=the within numberthe standard deviationokay=the inside numberthe normal deviation

and okayokay should be better than 1

Instance

Drawback Assertion:

Use Chebyshev’s theorem to seek out what % of the values will fall between 123 and 179 for a knowledge set with imply of 151 and normal deviation of 14.

Answer:

  • We subtract 151-123 and get 28, which tells us that 123 is 28 items under the imply.
  • We subtract 179-151 and likewise get 28, which tells us that 151 is 28 items above the imply.
  • These two collectively inform us that the values between 123 and 179 are all inside 28 items of the imply. Due to this fact the “inside quantity” is 28.
  • So we discover the variety of normal deviations, okay, which the “inside quantity”, 28, quantities to by dividing it by the usual deviation:
okay=the within numberthe standard deviation=2814=2okay=the inside numberthe normal deviation=2814=2

So now we all know that the values between 123 and 179 are all inside 28 items of the imply, which is similar as inside okay=2 normal deviations of the imply. Now, since okay > 1 we are able to use Chebyshev’s method to seek out the fraction of the information which might be inside okay=2 normal deviations of the imply. Substituting okay=2 now we have:

11okay2=1122=114=341−1k2=1−122=1−14=34

So 3434 of the information lie between 123 and 179. And since 34=7534=75% that suggests that 75% of the information values are between 123 and 179.