Chebyshevs theorem

Question

Consider a data set that has a mean of 54.9 and a standard deviation of 13.2. According to chebyshevs theorem, give an interval that has at least 85.76% or that data.

Answer 1

Chebyshev's Theorem:

(1 - 1/k^2)* 100% of the data lies within k standard deviations of the mean.

solve (1 - 1/k^2) * 100% = 85.76% to get number of standard deviations within 85.76% of the mean.

(1 - 1/k^2) * 1 = 0.8576

1 - 1/k^2 = 0.8576

-1/k^2 = -0.1424

-1 = -0.1424 * k^2

7.02 = k^2

k = 2.64

Since standard deviation is 13.2 and mean is 54.9, the interval containing 85.76% of the data is:

54.9 +/- 2.64 * 13.2 = 20.05, 89.75