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Multiplying by a constant

If we multiply the list 1, 3, 4, 4 by a constant, say 5, how will the average and SD change? The new list is 5, 15, 20, 20. The stacked dotplots follow:

                          O
                    O   O O
         Old:   --+---------+---------+---------+---------+--
                  0         5        10        15        20


                                                          O
                            O                   O         O
         New:   --+---------+---------+---------+---------+--
                  0         5        10        15        20

This time, the old histogram looks like it has been stretched out (over an interval that is 5 times longer) instead of shifted. Phrased differently, all distances have been multiplied by 5. Therefore, the new average should be 5 times the old average, and the new SD should be 5 times the old SD. It should be easy to verify that the list 5, 15, 20, 20 has average 15 and SD equal to 7.07.

$\fbox{ \parbox{5.5in}{ \vspace*{1ex} If we multiply a list by a positive constan... ... , the average and SD will be multiplied by the same constant. \vspace*{1ex} }}$

If we multiplied by -5 instead of 5, the new histogram will still be stretched out by the same factor, but reflected over to the other side of zero. Thus, the average will change signs, but the SD remains positive.

$\fbox{ \parbox{5.5in}{ \vspace*{1ex} If we multiply a list by a negative constan... ... $-c$ , the SD will be multiplied by the positive number $c$ . \vspace*{1ex} }}$

2003-09-08