Question 38·Medium·One-Variable Data Distributions; Measures of Center and Spread
The dot plot shows the number of books read last month by each student in a reading club.
The rightmost dot at 10 books was recorded incorrectly and should be 6 books.
Which choice describes how the mean and the standard deviation of the data set change after this correction?
For “change one data value” questions, avoid recomputing everything from scratch. First, count the number of data values from the graph. Then use “mean changes by (change in total)/.” For standard deviation, use a conceptual check: moving an extreme value closer to the center reduces spread, so the standard deviation decreases.
Hints
Use the dots to find
Count how many dots are in the dot plot. That is the number of data values.
Track the total change
Only one value changes (from to ). How much does the total sum change?
Think about spread
After the correction, the largest value is smaller and closer to the rest of the data. Would that make the data more spread out or less spread out?
Desmos Guide
Enter the original data as a list
In Desmos, enter the list of values from the dot plot:
L=[1,1,2,2,2,3,3,3,4,10]
Compute the original mean and standard deviation
Enter mean(L) and stdev(L) to see the original mean and standard deviation.
Enter the corrected data and compare
Create the corrected list by replacing with :
C=[1,1,2,2,2,3,3,3,4,6]
Then enter mean(C) and stdev(C).
Compare mean(L) to mean(C) and compare stdev(L) to stdev(C) to determine the direction and size of the changes.
Step-by-step Explanation
Count the number of data values
Count the dots in the plot to find the number of students: there are dots, so .
Find how the mean changes
Changing one value from to decreases the total sum by .
So the mean decreases by
Decide how the standard deviation changes
The corrected value () is closer to the rest of the data than was, so the data become less spread out around the mean. Therefore, the standard deviation decreases.
So the correct choice is: The mean decreases by , and the standard deviation decreases.