Question 24·Hard·One-Variable Data Distributions; Measures of Center and Spread
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.
For dot-plot questions about center and spread, first rewrite the plot as actual data values with frequencies. To find the mean quickly, look for symmetry or “balance” around a central value instead of doing a long sum. For standard deviation comparisons, don’t look only at the single most extreme point; compare how the full set of distances from the mean differs between the two data sets (more values away from the mean and/or values farther away indicates larger standard deviation).
Hints
Translate each dot plot into a list
Write down the values in each data set, including repeats (or record each value with its frequency).
Check the mean by balancing
To see whether the mean is without doing a full calculation, compare the total amount above to the total amount below . If they balance, the mean is .
Compare spread using distances from the mean
Standard deviation depends on how far the values are from the mean and how many values are at each distance (like distance , distance , distance , etc.). Look for which data set has more values away from the mean and/or values farther away.
Desmos Guide
Enter each data set as a list
In Desmos, create two lists using the dot plot counts:
A=[2,4,4,5,5,5,5,5,6,6,8]B=[0,4,5,5,5,6,6,6,6,6,6]
Compute and compare the means
Compute mean(A) and mean(B) and compare the results to decide whether statement I must be true.
Compute and compare the standard deviations
Compute stdev(A) and stdev(B) and compare the results to decide whether statement II must be true.
Match your conclusions to an answer choice
Use your results for statements I and II to select the answer choice.
Step-by-step Explanation
Read the values from each dot plot
From the dot plots:
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Data set A has values: (1 time), (2 times), (5 times), (2 times), (1 time).
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Data set B has values: (1 time), (1 time), (3 times), (6 times).
Show the means are equal
For data set A, the values pair symmetrically around 5:
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and average to .
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Each and each average to .
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The five 's are already at .
So the mean of A is .
For data set B, compare values to :
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The is below .
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The six 's are each above , for a total of above.
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The is below .
Overall, the amount below is , which balances the above , and the three 's don’t change the balance. So the mean of B is also .
Compare the standard deviations using how many values are away from the mean
Standard deviation measures spread around the mean, taking into account how far all values are from the mean.
With mean :
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In data set A, there are values at (distance ), values that are away (two ’s and two ’s), and values that are away ( and ).
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In data set B, there are only values at (distance ), values that are away (one and six ’s), and value that is away (the ).
Compared with A, B has fewer values at the mean (so more values contribute to spread) and it includes a value very far from the mean. Therefore, data set B has the greater standard deviation, so the standard deviation of A is less than the standard deviation of B.
Select the choice that matches both conclusions
Statement I is true (the means are equal), and statement II is true (the standard deviation of A is less than the standard deviation of B).
Therefore, the correct choice is I and II.