Each of the following data sets has a mean of x = 10. (i) 8 9 10 11 12 (ii) 7 9

Question

Each of the following data sets has a mean of x = 10.

(i) 8 9 10 11 12 (ii) 7 9 10 11 13 (iii) 7 8 10 12 13

(a) Without doing any computations, order the data sets according to increasing value of standard deviations. (i), (iii), (ii)

(ii), (i), (iii)

(iii), (i), (ii)

(iii), (ii), (i)

(i), (ii), (iii)

(ii), (iii), (i)

(b) Why do you expect the difference in standard deviations between data sets (i) and (ii) to be greater than the difference in standard deviations between data sets (ii) and (iii)? Hint: Consider how much the data in the respective sets differ from the mean.

The data change between data sets (i) and (ii) increased the squared difference ?(x - x)2 by more than data sets (ii) and (iii).

The data change between data sets (ii) and (iii) increased the squared difference ?(x - x)2 by more than data sets (i) and (ii).

The data change between data sets (i) and (ii) decreased the squared difference ?(x - x)2 by more than data sets (ii) and (iii).

none of the above

Explanation / Answer

(a)

The first data set has the least deviation because its values are more closer to the mean value of 10.

In second and third data sets, the extreme values are the same but in second data set, the values on either side of 10 are more closer as compared to the third set. So third set has higher deviation than second.

So the correct order of increasing standard deviations is:

(i) < (ii) < (iii)

(b)

The correct answer is:

The data change between data sets (i) and (ii) increased the squared difference sum (x - x)2 by more than data sets (ii) and (iii)

When you shift from set (i) to (ii), see that the error values change from 1,1,2,2 to 1,1,3,3.

When you shift from set (ii) to (iii), see that the error values change from 1,1,3,3 to 2,2,3,3.

A change of 2->3 is more heavy as compared to 1->2 because this error term will be squared, so the difference gets increased.

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