Question

In this problem, we explore the effect on the standard deviation of multiplying each data value in a data set by the same constant. Consider the data set 8, 7, 16, 8, 16.

(a) Use the defining formula, the computation formula, or a calculator to compute s. (Round your answer to one decimal place.)
s =  

(b) Multiply each data value by 2 to obtain the new data set 16, 14, 32, 16, 32. Compute s. (Round your answer to one decimal place.)
s =  

(c) Compare the results of parts (a) and (b). In general, how does the standard deviation change if each data value is multiplied by a constant c?

Multiplying each data value by the same constant c results in the standard deviation remaining the same.Multiplying each data value by the same constant c results in the standard deviation increasing by c units.    Multiplying each data value by the same constant c results in the standard deviation being |c| times smaller.Multiplying each data value by the same constant c results in the standard deviation being |c| times as large.

(d) You recorded the weekly distances you bicycled in miles and computed the standard deviation to be s = 2.6 miles. Your friend wants to know the standard deviation in kilometers. Do you need to redo all the calculations?

YesNo    

Given 1 mile ? 1.6 kilometers, what is the standard deviation in kilometers? (Enter your answer to two decimal places.)
s =  km

Answer 1

a) mean = 11

variance = So, variance = 21, sd =

b) Now, mean = 22

Now, by the same calculation, variance = 84 = 4 * 21 = 2² * 21

So, sd =

c) Actually, mutiplying 2 makes the variance increased by 2²=4 times, which results in making the standard deviation increased by 2 times. Understand that, if we would have multiplied by -2, then the variance would have increased by (-2)² = 4 times and then standard deviation would increase by 2 times, because standard deviation is never negative.

So, in general, if we multiply each number by c times(c can be positive or negative), then the standard deviation would increase by |c| times, because standard deviation is non-negative.

d) Because 1 mile = 1.6 km, so to get the standard deviation in km, we should multiply each data value by 1.6, and then the standard deviation of those numbers will be 1.6* standard deviation in mile (by the formula we derived in (c))

So, s = 1.6*2.6 km = 4.16km