Question 2·Medium·One-Variable Data Distributions; Measures of Center and Spread
The ages of 20 students enrolled in a college class are summarized below.
| Age | Frequency |
|---|---|
| 18 | 6 |
| 19 | 5 |
| 20 | 4 |
| 21 | 2 |
| 22 | 1 |
| 23 | 1 |
| 30 | 1 |
Which of the following gives the correct order of the mean, median, and mode of the ages?
For questions about mean, median, and mode from a frequency table, first use the frequency column to identify the mode (largest frequency). Then, for the median, determine how many data points there are and use positions: for values, the median is at position if is odd or the average of positions and if is even; use cumulative frequencies to see which value sits in those positions. Finally, compute the mean as a weighted average by multiplying each value by its frequency, summing, and dividing by the total count. If time is tight and an extreme high or low value is present, remember that a high outlier usually makes mean > median > mode, while a low outlier usually makes mean < median < mode, which can help you check your work or even infer the correct order quickly.
Hints
Start with the mode
Look at the frequency column: which age occurs the most times? That value is the mode.
Use positions to find the median
There are 20 students. When the number of data points is even, the median is the average of the 10th and 11th values in the ordered list. Use the running totals of frequency to see which age is in positions 10 and 11.
Compute the mean carefully
Multiply each age by how many times it appears, add all those products, and then divide the total by 20. Pay attention to the single age 30—how might a value that is much larger than the others affect the average compared with the middle value?
Compare the three measures
Once you have the numerical values for the mean, median, and mode, arrange them from smallest to largest and then choose the option whose inequality symbols match that order.
Desmos Guide
Enter the full data set as a list
In Desmos, type something like
L1 = [18,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,22,23,30]
This represents all 20 ages in order.
Have Desmos calculate the mean and median
On a new line, type mean(L1) and note the value Desmos displays. On another line, type median(L1) and note that value as well. These give you the mean and median of the ages.
Identify the mode from the list
Look at list L1 and see which age appears the most times—this is the mode. You can scroll along the list or count the repeats for each age.
Compare and match to an answer choice
Compare the three numbers you obtained for the mode, median, and mean. Arrange them from smallest to largest, then select the answer choice whose inequality signs match that ordering.
Step-by-step Explanation
Translate the table into an ordered list
Use the frequency column to imagine the full list of 20 ages, in order from least to greatest.
- Six 18-year-olds: positions 1–6
- Five 19-year-olds: positions 7–11
- Four 20-year-olds: positions 12–15
- Two 21-year-olds: positions 16–17
- One 22-year-old: position 18
- One 23-year-old: position 19
- One 30-year-old: position 20
This tells you where each age falls in the ordered data set.
Find the mode
The mode is the value that appears most often.
Look at the frequency column in the table:
- 18 appears 6 times
- 19 appears 5 times
- 20 appears 4 times
- 21 appears 2 times
- 22, 23, and 30 each appear once
So the mode is 18.
Find the median
There are 20 students, so the median is the average of the 10th and 11th values in the ordered list.
From Step 1, positions 7–11 are all 19-year-olds.
- The 10th value is 19
- The 11th value is 19
So the median is
Find the mean
The mean is the sum of all ages divided by 20.
Multiply each age by its frequency and add:
Now divide by the total number of students, 20:
So the mean age is 20.
Compare the three values and match the choice
You have:
- Mode
- Median
- Mean
From smallest to largest: , which corresponds to
mode < median < mean, matching answer choice A.