Question 611·Medium·Math

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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?

\text{(See dot plot)}

A mean tracks the total per value, while standard deviation tracks how spread out the values are around their mean. A dot plot with one corrected entry calls for two checks: divide the entry’s change by the unchanged dot count, then compare the before-and-after lists in Desmos. Don’t mistake the change in the total for the change in the mean.

Hints

Hint 1
In a dot plot, each dot represents one student. A stack of dots over one number means that number occurs more than once. Count every dot: does correcting one student’s reading change the number of students?
Hint 2
The mean is the total divided by the number of values. Replacing one reading changes the total by the difference between the old and new readings. Divide that difference by the unchanged number of students.
Hint 3
Standard deviation measures spread around each data set’s own mean. Because the correction also moves the mean, compare the complete lists before and after the correction rather than judging only the dot that moves.

Step-by-step

Track the mean change, then compare spread
Step 1
Read the original dot plot
Each dot stands for one student, so the two dots above $1$ give two entries of $1$ . Read every stack and type $a=[1,1,2,2,2,3,3,3,4,10]$ in Desmos. It shows a list of $10$ entries, matching the $10$ dots.
Step 2
Make the corrected list
The rightmost dot changes from $10$ to $6$ , so type $b=[1,1,2,2,2,3,3,3,4,6]$ . Desmos shows $10$ entries again. Replace the $10$ ; keeping it and adding a $6$ would count an extra student.
Step 3
Find how much the mean falls
The mean is the total divided by the number of values. Replacing $10$ with $6$ lowers the total, but there are still $10$ students. Divide the change in the total by the unchanged count to get the change in the mean. Type $\frac{10-6}{10}$ ; Desmos shows $0.4$ . So the mean decreases by $0.4$ book, not by the full change in the total.
Step 4
Compare the standard deviations
Standard deviation measures spread around each list’s own mean. Since the mean moves too, don’t judge the spread from the changed dot alone. Type $\operatorname{stdevp}(a)$ and $\operatorname{stdevp}(b)$ ; Desmos shows about $2.47$ and $1.42$ . The corrected list has the smaller standard deviation. The mean decreases by $0.4$ book, and the standard deviation decreases. Choice A.

Question 611·Medium·Math

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