Introduction to Statistics

Question

The mean upload speed for a particular Internet provider is 500 KBps with a standard deviation of 60KBps. Use Chebyshev's Theorem to estimate the percentage of customers who should be able to upload data at speeds between 320 and 680 KBps.

Answer 1

Chebyshev's Theorem is often a victim of poor presentation. In its abstract form, it looks dreadful, but in practice it can be quite handy. It means that, no matter how the data is distributed, you can still know where certain percentages of the data must be. So if n is the number of standard deviations above and below the mean, then the formula 1 - 1/n² is the minimum proportion of the data that is within that interval.

So if the mean is 500, and the standard deviation is 60, then 680 and 320 are 180 above and below the mean. That 180 is 3 standard deviations (3 · 60 = 180). So that 3 is the n in the theorem. 1 - 1/3² = 8/9 = 0.8889, or 88.89%.