Question 20·Medium·One-Variable Data Distributions; Measures of Center and Spread
Each dot plot shows the length of each of 8 metal rods. The top plot shows the lengths measured by Lab C in centimeters, and the bottom plot shows the lengths of the same rods measured by Lab M in millimeters.
Note: centimeter equals millimeters.
Which choice best compares the standard deviations of the rod lengths measured by Lab C and Lab M?
When comparing standard deviations from dot plots, look for transformations between the two sets. Adding/subtracting a constant shifts all values and keeps the standard deviation the same, but multiplying/dividing by a constant stretches or compresses the data and scales the standard deviation by that factor.
Hints
Notice the units
The top plot is in centimeters and the bottom plot is in millimeters, and both plots represent the same rods.
Connect centimeters to millimeters
Use the fact that to describe how each data value changes when converting from cm to mm.
Think about what happens to distances from the mean
If every data value is multiplied by the same positive number, all distances from the mean scale by that same factor.
Desmos Guide
Enter the centimeter data
From the top dot plot, enter the list C=[12,12,13,14,14,15,16,16].
Convert to millimeters in Desmos
Create the millimeter list with M=10C (or enter M=[120,120,130,140,140,150,160,160]).
Compare the standard deviations
Compute stdev(C) and stdev(M). Observe that stdev(M) is 10 times stdev(C), and select the matching choice.
Step-by-step Explanation
Relate the two data sets
The rods are the same in both plots, but Lab C reports in centimeters and Lab M reports in millimeters. Since , each millimeter value is times the corresponding centimeter value.
Use how scaling affects spread
Standard deviation measures typical distance from the mean. If every data value is multiplied by , then every distance from the mean is also multiplied by , so the standard deviation is multiplied by .
Select the correct comparison
Therefore, the standard deviation of the lengths measured by Lab M is 10 times the standard deviation of the lengths measured by Lab C.