Question

We have seen that the standard deviation σ measures the spread of a data set about the mean μ. Chebyshev’s inequality gives an estimate of how well the standard deviation measures that spread. One consequence of this inequality is that for every data set at least 75% of the data points lie within two standard deviations of the mean, that is, between μ−2σ and μ+2σ (inclusive). For example, if μ = 20 and σ = 5, then at least 75% of the data are at least 20−2×5 = 10 and at most 20+2×5 = 30. Our statement of this result says at least 75%, not exactly 75%. Consider the following data: 5, 10, 10, 10, 10, 10, 10, 15. The percentage of data points that actually lie within two standard deviation of the mean is ______%

Answer 1

Answer:

100

Step-by-step explanation:

First we find the mean of this set of data.  We find the mean by finding the sum of data values and dividing it by the number of data values:

5+10+10+10+10+10+10+15 = 80

There are 8 data values; 80/8 = 10.  The mean is 10.

To find the standard deviation, subtract the mean from each value; square the difference; add the squares; divide by the number of data values; and take the square root:

5-10 = -5; 10-10 = 0; 10-10 = 0; 10-10 = 0; 10-10 = 0; 10-10 = 0; 10-10 = 0; 15-10 = 5

(-5)^2 = 25; 0^2 = 0; 0^2 = 0; 0^2 = 0; 0^2 = 0; 0^2 = 0; 0^2 = 0; 5^2 = 25

25+0+0+0+0+0+0+25 = 50

50/8 = 6.25

√6.25 = 2.5

The standard deviation is 2.5.

This means two standard deviations, or 2σ, is 2(2.5) = 5.

This gives us μ-2σ = 10-5 = 5 and μ+2σ = 10+5 = 15.

Since all of our data is between 5 and 15, 100% is within two standard deviations of the mean.