Question 507·Hard·Math

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The dot plots show the values in data sets A and B. A new data set X is created by multiplying each value in data set A by $2$ and then subtracting $6$ . A new data set Y is created by adding $4$ to each value in data set B.

Which choice must be true?

A dot plot shows every value, including repeats. When a rule changes every value, compare the new mean and standard deviation separately: enter the lists in Desmos, apply each rule, then check both statistics. The mean follows the entire rule, but adding a constant doesn't change spread. Don't assume that doubling one set makes it more spread out than the other.

Hints

Hint 1
On a dot plot, each dot represents one value. A stack of three dots means the same value appears three times. Write one entry for every dot before applying either rule.
Hint 2
The mean is the total divided by the number of values. Apply both parts of X's rule to its whole list before comparing it with Y's mean.
Hint 3
Standard deviation measures how spread out values are around their own mean. Adding the same amount to every value leaves that spread unchanged; multiplying changes it. Compare the two new sets, not only their multipliers.

Step-by-step

Transform the lists in Desmos
Step 1
Read every dot into a list
Type $a=[4,5,5,5,6]$ for A and $b=[2,4,5,6,8]$ for B. Desmos stores each as a five-value list. The three dots above $5$ in A mean three entries of $5$ , not one.
Step 2
Make data set X
Type $c=2a-6$ . Desmos applies that rule to every entry of $a$ , making $c=[2,4,4,4,6]$ , the list for X. The subtraction matters: multiplying by $2$ alone isn't X's rule.
Step 3
Make data set Y
Type $d=b+4$ . Desmos makes $d=[6,8,9,10,12]$ , the list for Y. Every value goes up by $4$ , so the whole set moves without changing the gaps between its values.
Step 4
Compare the means
The mean is the total divided by the number of values. Type $\operatorname{mean}(c)$ and $\operatorname{mean}(d)$ . Desmos shows $4$ and $9$ , so X has the smaller mean.
Step 5
Compare the standard deviations
Type $\operatorname{stdevp}(c)$ and $\operatorname{stdevp}(d)$ . Desmos shows about $1.265$ and $2$ . Standard deviation measures spread around each set's own mean: three of X's values sit at its mean, while only one of Y's does, and Y's other values reach farther out. Adding shifts a set without changing its spread; multiplying stretches its distances. So X has both the smaller mean and the smaller standard deviation. Choice B.

Question 507·Hard·Math

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