Question 39·Hard·One-Variable Data Distributions; Measures of Center and Spread
The dot plots show the number of defective parts found in 12 randomly selected boxes from Factory A and 12 randomly selected boxes from Factory B.
Which choice correctly compares the mean, median, and standard deviation of the two data sets?
For dot-plot comparison questions, use the counts to compute totals quickly (mean = total ÷ number of values), then locate the middle value(s) for the median. For standard deviation, avoid formulas: if the means match, compare how far values sit from the mean—more/extremer distances imply a larger standard deviation.
Hints
Start with the mean
Each dot is one value. Since each factory has 12 dots, comparing means comes down to comparing the total sums of the values.
Median for 12 values
With 12 values, locate the 6th and 7th values in order for each factory, then average them.
Standard deviation is about distance from the mean
If two data sets have the same mean, the one with values that are farther from that mean (especially more extreme values) will have the larger standard deviation.
Desmos Guide
Enter the data values as lists
Enter lists that match the dot plots, for example:
A={2,4,4,5,5,5,6,6,6,7,7,9}
B={1,4,4,5,5,6,6,6,6,7,7,9}
Compute mean and median
Compute mean(A), mean(B), median(A), and median(B).
Compare standard deviations
Compute stdev(A) and stdev(B) to compare spread, then select the choice consistent with all three comparisons.
Step-by-step Explanation
Count values and use frequencies
Each dot plot shows 12 observations (12 boxes) for each factory. To compare mean and median, use the counts at each value shown in the dot plots.
Compare the means by comparing totals
Add the values using the frequencies.
- Factory A total:
- Factory B total:
Since both totals are 66 and each data set has 12 values, both means are .
Find and compare the medians
With 12 values, the median is the average of the 6th and 7th values when ordered.
- Factory A: the 6th value is 5 and the 7th value is 6, so median .
- Factory B: the 6th and 7th values are both 6, so median .
Therefore, Factory B has the greater median.
Compare the standard deviations (spread)
Both data sets have the same mean (), so the one with values that tend to be farther from 5.5 has the larger standard deviation.
Most values occur at 4, 5, 6, 7, and 9 in both plots, and 5 and 6 are equally far from 5.5.
A key difference is the minimum value:
- Factory A has a 2, which is away from 5.5.
- Factory B has a 1, which is away from 5.5.
That larger extreme distance increases the overall spread for Factory B, so Factory B has the greater standard deviation.
Choose the statement that matches all three comparisons
The means are the same, Factory B’s median is greater, and Factory B’s standard deviation is greater.
Answer: The two factories have the same mean, but Factory B has a greater median and a greater standard deviation than Factory A.