Question 51·Hard·One-Variable Data Distributions; Measures of Center and Spread
The list below shows the number of minutes each of 10 runners spent exercising on a certain day:
18, 22, 23, 25, 25, 27, 29, 31, 35, 90
The coach later realizes that the last value in the list should have been 28 minutes instead of 90 minutes. Consider the following statistics for the data set.
I. The mean
II. The median
III. The standard deviation
Which of these statistics will decrease when the incorrect value of 90 is replaced with the correct value of 28?
For questions about how changing one value affects statistics, reason conceptually instead of doing full calculations. Remember: the mean uses every value, so changing any data point will change the mean in the same direction as the change in the sum. The median for an even number of values depends only on the two middle values in the sorted list—if those don’t change, the median doesn’t change. The standard deviation reflects spread and is very sensitive to outliers; making an extreme value more typical shrinks the spread and lowers the standard deviation. Quickly check which of these conditions apply and then match them to the answer choices.
Hints
Think about each statistic separately
Ask yourself: Which statistics use all the data values, and which depend mainly on the middle values? Which one measures how spread out the data are?
Focus on the median
You have 10 numbers. After sorting, the median is the average of the 5th and 6th numbers. Identify those two positions before and after replacing 90 with 28. Do those middle values change?
Compare the effect on mean and spread
Replacing 90 with 28 changes the total sum. Does that make the average bigger or smaller? Also, think about whether the data become more or less spread out without the very large value 90.
Desmos Guide
Compute statistics for the original data
In one expression line, enter L1 = [18,22,23,25,25,27,29,31,35,90]. On new lines, type mean(L1), median(L1), and stdev(L1) to see the original mean, median, and standard deviation.
Compute statistics for the corrected data and compare
Enter L2 = [18,22,23,25,25,27,28,29,31,35]. Then type mean(L2), median(L2), and stdev(L2). Compare each statistic from L1 to L2 to see which ones became smaller and which stayed the same; use this comparison to decide which statistics decreased.
Step-by-step Explanation
Organize the data and recall the definitions
Original data (already in increasing order):
18, 22, 23, 25, 25, 27, 29, 31, 35, 90
Corrected data (replace 90 with 28 and then sort):
18, 22, 23, 25, 25, 27, 28, 29, 31, 35
Key ideas:
- Mean: sum of all values divided by 10.
- Median (for 10 values): average of the 5th and 6th values in the ordered list.
- Standard deviation: measures how spread out the data are from the mean (larger spread → larger standard deviation).
Check what happens to the median
For 10 values, the median is the average of the 5th and 6th numbers in the ordered list.
- Original ordered list: 18, 22, 23, 25, 25, 27, 29, 31, 35, 90
5th and 6th values: 25 and 27, so the median is
- Corrected ordered list: 18, 22, 23, 25, 25, 27, 28, 29, 31, 35
5th and 6th values are still 25 and 27, so the median is again
The median does not change, so it does not decrease.
Check what happens to the mean
The mean is .
The only change is replacing 90 with 28. That reduces the total sum by
Since the number of data points (10) stays the same, the new mean is
So the mean becomes smaller; it definitely decreases.
Check what happens to the standard deviation and conclude
Standard deviation looks at how far the numbers are from the mean. The value 90 is far above the rest of the data, so it makes the spread large. Replacing 90 with 28 (which is near the center of the data) makes the spread smaller.
When the data are less spread out, the standard deviation decreases.
- Mean: decreases (I).
- Median: stays the same (II does not decrease).
- Standard deviation: decreases (III).
Therefore, the statistics that decrease are I and III only, which corresponds to answer choice C) I and III only.