Question 555·Hard·Math

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The dot plots represent the distributions of values in data sets A and B. (See dot plots.)

Which of the following statements must be true?

I. The mean of data set A is equal to the mean of data set B.

II. The standard deviation of data set A is less than the standard deviation of data set B.

A dot plot shows every value: each dot is one observation, and a stack shows repeats. To compare the mean and standard deviation, put every dot into a Desmos list, then check center and spread separately. Don't judge spread by the rightmost dot or by the range alone; neither tells you where all the values sit.

Hints

Hint 1
Each dot is one observation, so stacked dots are repeated values. The five dots above $5$ in A mean five entries of $5$ . How would you write both plots as lists without leaving out repeats?
Hint 2
The mean is the total divided by the number of observations. Both plots have $11$ dots, so check the mean of each complete list before deciding whether statement I holds.
Hint 3
A standard deviation measures spread around the mean, using every observation. Compare the standard deviations of the complete lists, not the two rightmost dots.

Step-by-step

Enter every dot, then compare
Step 1
Turn the dot plots into lists
Each dot is one observation, so repeat a value once for every dot above it. Type $A=[2,4,4,5,5,5,5,5,6,6,8]$ and $B=[0,4,5,5,5,6,6,6,6,6,6]$ in Desmos. Each list has $11$ entries. Typing each distinct value only once would lose the stacks.
Step 2
Find A's mean
The mean is the total divided by the number of observations. Type $\operatorname{mean}(A)$ ; Desmos prints $5$ . This calculation includes every repeated value in A.
Step 3
Check whether B's mean matches
Type $\operatorname{mean}(B)$ ; Desmos prints $5$ again, so the means are equal and statement I holds. B has many $6$ s, but its $0$ pulls the mean back down.
Step 4
Measure A's spread
The standard deviation measures how spread out all the values are around their mean. Type $\operatorname{stdevp}(A)$ ; Desmos shows about $1.414$ for A. A range uses only the endpoints, so it can't settle this comparison.
Step 5
Compare B's spread and answer
Type $\operatorname{stdevp}(B)$ ; Desmos shows about $1.706$ . That's greater than A's $1.414$ , so statement II holds too. Equal means don't guarantee equal standard deviations. Both statements must be true. Choice C.

Question 555·Hard·Math

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