17. | Critical Thinking: Data Transformation In this problem, we explore the sam

Question

17. | Critical Thinking: Data Transformation In this problem, we explore the same number. Consider the data set 2, 2, 3, 6, 10. (a) Compute the mode, median, and mean. (b) Multiply each data value by 5. Compute the mode, median, and mean. (c) Compare the results of parts (a) and (b). In general, how do you think the mode, median, and mean are affected when each data value in a set is multiplied by the same constant? (d) Suppose you have information about average heights of a random sample of airplane passengers. The mode is 70 inches, the median is 68 inches, and the mean is 71 inches. To convert the data into centimeters, multiply each data value by 2.54. What are the values of the mode, median, and mean in centimeters?

Explanation / Answer

1) mode is the data value which occurs most frequently in our data set. That is, which has the higher frequency.

Hence here, 2 is the mode.

Median is the middle most observation. If n is the no. Of observation then data at (n+1)/2 gives median if n is odd and if n is even then the mean of n/2 and (n/2)+1 gives median

Here, n=5,hence (5+1)/2=3rd observation is median

Hence 3 is median.

Mean=sum of observations/total number of observation.

=2+2+3+6+10/5

=23/5

B) now multiplying each data by 5,we get the following data sets:-

10,10,15,30,50

Preceeding in similar way as in part a) we get,

Mode=10

Median=15

Mean=10+10+15+30+50/5=23

C)comparing tge results of part a) and part b), we see that if we are multiplying the each value of data sets by 5.

Then, mean(or median or mode) of the new datasets are equal to 5*initial mean(or median or mode).

That means, if the each value of data is multiplied by any constant then the value of mean median and mode is obtained by just multiplying the initial mean median mode with that constant.

D) here, we are multiplying each value heights of the passengers with the same constant 2.54

So the mean median mode of the resulted data in centimeters will be obtained by multiplying the initial mean median mode by 2.54

So, mode in centimeters is initial mode*2.54= 70*2.54=177.8

Median in cm is initial median*2.54=68*2.54=172.72

Mean in cm is initial mean*2.54=71*2.54=180.34

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