Question

In a certain distribution, the mean is 70 with a standard

deviation of 3. At least what fraction of the numbers are
between the following pair of numbers?
61 and 79

Answer 1

8/9

Explanation:

You want to know the least fraction of the distribution that lies between 61 and 79 if the mean is 70 and the standard deviation is 3.

Chebyshev's Inequality

Chebyshev's inequality tells you that at most 1/k² of the distribution will lie beyond k standard deviations from the mean.

Application

Here, the given limits are (61 -70)/3 = -3 and (79 -70)/3 = 3 standard deviations from the mean. That is, we can use k=3 to find the maximum fraction that is beyond the interval [61, 79]. That fraction is ...

1/k² = 1/3² = 1/9

Since this is the maximum fraction outside the given range, the minimum fraction inside the given range is ...

1 -1/9 = 8/9

At least 8/9 of the number are between 61 and 79.