Question 18·Hard·One-Variable Data Distributions; Measures of Center and Spread
For a certain computer game, individuals receive an integer score that ranges from 2 through 10. The table below shows the frequency distribution of the scores of the 9 players in group A and the 11 players in group B.
| Score | Group A | Group B |
|---|---|---|
| 2 | 1 | 0 |
| 3 | 1 | 0 |
| 4 | 2 | 0 |
| 5 | 1 | 4 |
| 6 | 3 | 2 |
| 7 | 0 | 0 |
| 8 | 0 | 2 |
| 9 | 1 | 1 |
| 10 | 0 | 2 |
| Total | 9 | 11 |
The median of the scores for group B is how much greater than the median of the scores for group A?
(Express the answer as an integer)
When you see a frequency table and a question about median, first find how many data values are in each group and use to locate the position of the median for an odd-sized set. Then, instead of writing every value, run a cumulative count down the scores (adding the frequencies) until you reach the median position; the score at that position is the median. Repeat for the other group and finally subtract the two medians to get the requested difference, being careful not to confuse the median itself with the difference between medians.
Hints
Think about the position of the median
First, note how many players are in each group. For a group with an odd number of players, which position (1st, 2nd, 3rd, etc.) is the median in?
Use the frequency table for Group A carefully
For Group A, use the counts in the table to track cumulative positions (1st, 2nd, 3rd, etc.) as you move from score 2 up to score 10. Which score lands in the middle position for the 9 players?
Repeat for Group B, then compare
Do the same running count of positions for Group B and identify which score is in the middle position for the 11 players. Once you have both medians, subtract the smaller from the larger.
Desmos Guide
Enter the score lists for each group
From the table, type the full list of scores (repeating each score according to its frequency) for each group:
- For Group A, enter something like:
A = [2,3,4,4,5,6,6,6,9]. - For Group B, enter something like:
B = [5,5,5,5,6,6,8,8,9,10,10].
Use Desmos to find each median
On a new line, type median(A) to see the median score for Group A, and on another line type median(B) to see the median score for Group B. Note both values.
Have Desmos compute the difference
On a new line, type median(B) - median(A). The value Desmos displays is how much greater Group B’s median is than Group A’s median; use that as your final answer.
Step-by-step Explanation
Identify how to find the median for each group
The median of a data set with an odd number of values is the value in position when the data are ordered from least to greatest.
- Group A has players, so the median is the value in position
- Group B has players, so the median is the value in position
Now we need to figure out what the 5th value is in Group A and what the 6th value is in Group B using the frequency table.
Find the median score for Group A
Use the frequencies for Group A to see which score is in the 5th position when the scores are listed in order.
Group A frequencies:
- Score 2: 1 player (positions 1)
- Score 3: 1 player (positions 2)
- Score 4: 2 players (positions 3–4)
- Score 5: 1 player (position 5)
So the 5th value is a score of 5. Therefore, the median score for Group A is 5.
Find the median score for Group B
Now do the same for Group B to find which score is in the 6th position.
Group B frequencies:
- Score 5: 4 players (positions 1–4)
- Score 6: 2 players (positions 5–6)
- Score 8: 2 players (positions 7–8)
- Score 9: 1 player (position 9)
- Score 10: 2 players (positions 10–11)
The 6th value is a score of 6. Therefore, the median score for Group B is 6.
Compare the medians to answer the question
The question asks: The median of the scores for group B is how much greater than the median of the scores for group A?
- Median of Group A: 5
- Median of Group B: 6
Compute the difference:
So, the median of Group B is 1 point greater than the median of Group A.