Question 23·Medium·One-Variable Data Distributions; Measures of Center and Spread
The table below shows the number of text messages each of the nine employees at a small company sent on an average workday in both January and February.
| Employee | January | February |
|---|---|---|
| A | 45 | 50 |
| B | 60 | 58 |
| C | 39 | 41 |
| D | 52 | 55 |
| E | 48 | 47 |
| F | 63 | 67 |
| G | 59 | 62 |
| H | 42 | 45 |
| I | 56 | 54 |
By how many messages was the median number of messages sent per day in February greater than the median number sent per day in January?
(Express the answer as an integer)
For median questions with a small number of data points, skip formulas and quickly list each set of numbers in order from smallest to largest. With an odd number of values, go straight to the single middle value (no averaging needed), then carefully read whether the question wants a comparison like “how much greater,” which tells you to subtract in the correct order (new minus old).
Hints
Recall what the median means
For a set of numbers, the median is the middle value after you put the numbers in order from smallest to largest. With 9 values, you do not average two numbers; you pick the single middle one.
Work with January first
Write all 9 January message counts in increasing order. Then identify which position is the middle value and find that number.
Repeat for February, then compare
Do the same ordering process for the February data to find its median, then subtract the January median from the February median to answer the question.
Desmos Guide
Find the January median with Desmos
In Desmos, type J = {45, 60, 39, 52, 48, 63, 59, 42, 56} on one line. On the next line, type median(J). The output is the median number of January messages.
Find the February median with Desmos
Type F = {50, 58, 41, 55, 47, 67, 62, 45, 54} on a new line. Then type median(F). The output is the median number of February messages.
Compare the medians in Desmos
On a new line, type median(F) - median(J). The value Desmos displays is how many messages greater the February median is than the January median.
Step-by-step Explanation
List and sort the January data
Write down the January numbers from the table:
45, 60, 39, 52, 48, 63, 59, 42, 56
Now put them in increasing order:
39, 42, 45, 48, 52, 56, 59, 60, 63
Find the January median
There are 9 employees, so there are 9 data values. When the number of values is odd, the median is the middle value.
- The middle (5th) value in the ordered January list is the median.
- Count: 1st = 39, 2nd = 42, 3rd = 45, 4th = 48, 5th = 52.
So the median number of messages in January is 52.
List and sort the February data
Write down the February numbers from the table:
50, 58, 41, 55, 47, 67, 62, 45, 54
Now put them in increasing order:
41, 45, 47, 50, 54, 55, 58, 62, 67
Find the February median and compare
Again, there are 9 values for February, so the median is the middle (5th) value of the ordered list.
- Count: 1st = 41, 2nd = 45, 3rd = 47, 4th = 50, 5th = 54.
- So the median number of messages in February is 54.
Now compare the medians:
- January median: 52
- February median: 54
The question asks: By how many messages was the median in February greater than the median in January?
Compute .
So the answer is 2.