Question 38·Medium·One-Variable Data Distributions; Measures of Center and Spread
The test scores of 15 students on a science quiz are summarized in the frequency table below.
| Score | Frequency |
|---|---|
| 72 | 2 |
| 74 | 3 |
| 75 | 4 |
| 78 | 3 |
| 80 | 2 |
| 96 | 1 |
Which of the following statements correctly compares the mean, median, and mode of the scores?
For questions comparing mean, median, and mode from a frequency table, first translate the table into a clear mental (or written) picture of the data: note how many values there are and where the “middle” lies. Quickly identify the mode by the highest frequency, then find the median by locating the middle position in the ordered data (or using the index formula for odd/even counts). Compute the mean using a weighted sum (score × frequency) divided by the total number of data points; you usually only need to know whether it is larger or smaller than the other measures, not an exact decimal. Finally, compare the three measures and match the inequality pattern to the answer choices, keeping in mind that extreme values pull the mean more than the median or mode.
Hints
Turn the table into a list
Use the frequency column to write out each score as many times as it appears, in ascending order, so you can see all 15 values clearly.
Start with mode and median
From your ordered list, identify which score appears most often (mode), and then locate the middle value (8th value) to find the median.
Estimate the mean carefully
Use the table to multiply each score by its frequency, add these products to get the total of all scores, then divide by 15; you only need to know whether this average is less than, greater than, or equal to the mode and median.
Compare and match to a choice
Once you know the numerical values of the mean, median, and mode, decide which is smallest, which is largest, and whether any are the same, then pick the choice that matches that ordering.
Desmos Guide
Enter the list of scores
In Desmos, type a list of all 15 scores based on the table, for example:
scores = [72,72,74,74,74,75,75,75,75,78,78,78,80,80,96]
Use Desmos to find mean and median
On new lines, type mean(scores) and median(scores). Desmos will display the mean and median values; compare these numbers to see which is larger and whether they are equal.
Determine the mode from the table
Look back at the original frequency table (or your list) to see which score occurs most often; that is the mode. Now compare this mode to the mean and median you saw in Desmos to decide their correct ordering.
Step-by-step Explanation
Rewrite the data from the table
Use the frequency column to write out all 15 scores in order.
- Score 72 appears 2 times:
- Score 74 appears 3 times:
- Score 75 appears 4 times:
- Score 78 appears 3 times:
- Score 80 appears 2 times:
- Score 96 appears 1 time:
So the full ordered list is:
.
There are 15 scores total.
Find the mode and median
Mode: The mode is the value that appears most often.
- 72 appears 2 times
- 74 appears 3 times
- 75 appears 4 times
- 78 appears 3 times
- 80 appears 2 times
- 96 appears 1 time
The most frequent score is 75, so the mode is .
Median: With 15 scores (an odd number), the median is the 8th value in the ordered list.
Index the scores:
1: 72
2: 72
3: 74
4: 74
5: 74
6: 75
7: 75
8: 75
9: 75
10: 78
11: 78
12: 78
13: 80
14: 80
15: 96
The 8th value is 75, so the median is also .
Compute the mean
Use the table to find the total sum of all scores.
Now add these:
The sum of all 15 scores is 1156. The mean is
So the mean is about , which is greater than 75.
Compare mean, median, and mode and choose the statement
From the previous steps:
- Mode
- Median
- Mean
So we have
Among the choices, this matches C) mode = median < mean.