Question 47·Medium·One-Variable Data Distributions; Measures of Center and Spread
Which of the following data sets has a median less than its mean?
For questions comparing mean and median, quickly sort each list (if needed), then find the median first—this is usually fast, especially when the number of data points is odd. Next, look for symmetry or outliers to estimate the mean: symmetric sets about a central value have mean equal to that central value, and extreme high or low values pull the mean toward them more than they affect the median. Often you can determine whether the mean is larger or smaller than the median without computing exact decimals, saving time by focusing only on the direction of the difference the question asks about.
Hints
Review what mean and median are
Ask yourself: How do you compute the mean of a list of numbers? How do you find the median when there are 5 numbers in order?
Use the fact that there are 5 numbers in each set
When there are 5 numbers, once they are sorted, the median is always the 3rd number. That makes the median quick to find in each choice.
Look for the effect of extreme values (outliers)
Which sets have a number that is much larger or much smaller than the rest? Think about whether that extreme value pulls the mean up or down compared to the median.
Compare mean and median signs and sizes
After finding the median, estimate or compute the mean. Is the average pulled higher or lower than the middle number by the other values in the list?
Desmos Guide
Enter each data set as a list
In Desmos, type each list using curly braces, for example:
A = {3,3,3,3,3}B = {-10,-5,0,5,10}C = {1,2,2,2,50}D = {-100,-2,-2,-2,-1}
Use Desmos to find medians
On new lines, type median(A), median(B), median(C), and median(D) to see the median of each data set. Note each value Desmos gives you.
Use Desmos to find means
Then type mean(A), mean(B), mean(C), and mean(D) to see the mean of each data set. Again, note each value.
Compare mean and median for each list
For each of A, B, C, and D, compare the median value from step 2 with the mean value from step 3. Identify the one list where the median is less than the mean.
Step-by-step Explanation
Recall the definitions of mean and median
- Mean (average): Add all the numbers and divide by how many numbers there are.
- Median (middle value): When the numbers are in order, the median is the middle number.
Each data set here has 5 numbers, so the median is always the 3rd number when the list is in order.
Check the symmetric sets (A and B)
Choice A: 3, 3, 3, 3, 3
- In order: 3, 3, 3, 3, 3
- Median is the 3rd number: .
- Mean: .
- Mean and median are equal, so the median is not less than the mean.
Choice B: -10, -5, 0, 5, 10
- In order: -10, -5, 0, 5, 10 (already ordered)
- Median is the 3rd number: .
- Mean: .
- Again, mean and median are equal, so the median is not less than the mean.
Examine the set with a large positive outlier
Choice C: 1, 2, 2, 2, 50
- In order: 1, 2, 2, 2, 50 (already ordered)
- Median is the 3rd number: .
- Mean:
- Sum:
- Mean:
- Compare: median and mean .
- Here, the median is less than the mean.
Check the set with a large negative outlier and decide
Choice D: -100, -2, -2, -2, -1
- In order: -100, -2, -2, -2, -1 (already ordered)
- Median is the 3rd number: .
- Mean:
- Sum:
- Mean:
- On the number line, is to the left of , so .
- That means the mean is less than the median here, not the other way around.
Only one data set has its median less than its mean, and that is 1, 2, 2, 2, 50.