Question 9·Hard·One-Variable Data Distributions; Measures of Center and Spread
The table shows the number of employees at each of the 17 restaurants in a town.
| Number of employees | Number of restaurants |
|---|---|
| 2 to 7 | 2 |
| 8 to 13 | 4 |
| 14 to 19 | 2 |
| 20 to 25 | 7 |
| 26 to 31 | 2 |
Which of the following could be the median number of employees for the restaurants in this town?
For grouped data with frequencies, first add the counts to confirm the total number of data points. For an odd total , the median is the value in position when ordered. Use cumulative frequencies (running totals of the "Number of restaurants") to see which interval contains that position. The median can be any value in that interval, so on the SAT, simply pick the answer choice that falls inside that interval instead of trying to pinpoint an exact value.
Hints
Find the median’s position
First, figure out which position in the ordered list of 17 restaurants will be the median. For an odd number of items, how do you find that position?
Track cumulative counts from the table
Once you know the median’s position, use the "Number of restaurants" column to add up how many restaurants are in each employee range (2–7, 8–13, etc.) until you pass the median’s position.
Connect the position to a range, then to an answer
Identify which employee range contains the restaurant at the median position. Then look at the answer choices and see which number of employees falls within that range.
Desmos Guide
Use Desmos to confirm the median’s position
In Desmos, type (17+1)/2 and press Enter. The result tells you the position of the median in an ordered list of 17 values. Then, using the table, determine which employee range contains the restaurant at that position by adding the counts in the "Number of restaurants" column until you reach or pass that position; finally, select the answer choice that lies in that range.
Step-by-step Explanation
Identify which position the median is in
There are 17 restaurants total.
For an odd number of data values, the median is the value in position when the data are listed in order.
Here, , so the median is the value in position . That means we need to find the number of employees for the 9th restaurant in order from smallest to largest.
Use the table to find which group contains the 9th restaurant
Use the "Number of restaurants" column to see how many restaurants are in each employee range, and keep a running total.
- 2 to 7 employees: 2 restaurants (positions 1–2)
- 8 to 13 employees: 4 more restaurants, for a total of (positions 3–6)
- 14 to 19 employees: 2 more, total (positions 7–8)
- 20 to 25 employees: 7 more, total (positions 9–15)
The 9th restaurant falls in the group that covers positions 9–15, which is the 20 to 25 employees range.
Match the range to the answer choices
We now know the median must be some number of employees between 20 and 25, inclusive.
Check the answer choices:
- A) 2 is in the 2–7 range
- B) 9 is in the 8–13 range
- C) 15 is in the 14–19 range
- D) 21 is in the 20–25 range
Only 21 lies in the 20–25 range, so the median number of employees could be 21.