Question 40·Easy·One-Variable Data Distributions; Measures of Center and Spread
| Number of books read last month | Frequency |
|---|---|
| 0 | 6 |
| 1 | 9 |
| 2 | 10 |
| 3 | 5 |
| Total | 30 |
Based on the table, what is the median number of books read by the 30 students last month?
For median questions based on a frequency table, avoid writing out all the data points. First, sum the frequencies to confirm how many data values there are and decide whether you need a single middle value (odd ) or the average of two middle values (even ). Then use cumulative frequencies to locate which category contains each middle position (for example, positions 15 and 16 when ). Finally, read off the corresponding data values from the table and, if necessary, average the two middle values to get the median.
Hints
Think about how many data points there are
Use the frequency column to find the total number of students. Is this total odd or even? That determines how you find the median.
Locate the middle positions, not just a single middle value
For an even number of data points, there are two middle positions. For 30 data points, which positions in the ordered list are the middle two?
Use cumulative frequencies to find those positions
Add up the frequencies from the top down to see which value covers positions 1–6, which covers 7–15, which covers 16–25, and so on. Then match the 15th and 16th positions to their values and average those two values.
Desmos Guide
Enter the data as a list
In Desmos, type an expression like:
L = [0,0,0,0,0,0, 1,1,1,1,1,1,1,1,1, 2,2,2,2,2,2,2,2,2,2, 3,3,3,3,3]
This list includes 6 zeros, 9 ones, 10 twos, and 5 threes, matching the table.
Use the median function
On a new line, type median(L). The value Desmos returns is the median number of books read; use that value as your answer.
Step-by-step Explanation
Recall how to find the median for an even number of data points
The median is the middle value when data are ordered from least to greatest.
- If there is an odd number of data points, the median is the single middle value.
- If there is an even number of data points, the median is the average of the two middle values.
Here there are 30 students, so is even. The two middle positions are:
So we need the 15th and 16th values in the ordered list of numbers of books.
Use the frequency table to find where each value falls in the ordered list
List the values in order and track their positions using the frequencies:
- 0 books: 6 students → positions 1 through 6
- 1 book: 9 students → positions 7 through 15 (since )
- 2 books: 10 students → positions 16 through 25 (since )
- 3 books: 5 students → positions 26 through 30
Now we can see which values are in positions 15 and 16.
Identify the 15th and 16th values
From the position ranges above:
- Positions 7–15 are all 1 book, so the 15th value is 1.
- Positions 16–25 are all 2 books, so the 16th value is 2.
So the two middle data values we need to average are 1 and 2.
Compute the median from the two middle values
For an even number of data points, the median is the average of the two middle values.
So the median number of books is
Therefore, the median number of books read by the 30 students last month is .