Question 26·Medium·One-Variable Data Distributions; Measures of Center and Spread
A city health department surveyed the number of customers each of 37 independent coffee shops served in a typical day. Seven of the shops served fewer than 80 customers. Nineteen of the shops served at least 80 but fewer than 140 customers. The remaining shops served 140 or more customers.
Which of the following numbers could be the median of the 37 daily customer counts?
For grouped-data median questions, first compute the position of the median using for data points when is odd. Then, mentally place the data in order using the group counts (for example, first 7 values in the lowest range, next 19 in the middle range, etc.) and see which group includes the median position. Finally, translate that group into a numeric interval and pick the answer choice that lies in that interval, paying close attention to phrases like "at least" and "fewer than" so you handle the endpoints correctly.
Hints
Think about what "median" means
The median is the middle value when the data are arranged from smallest to largest. For 37 values, where in the list is that middle value located?
Use the counts of shops in each group
You know how many shops fall into each customer-count range. If you list the shops in increasing order of customers served, which positions (1st, 2nd, 3rd, etc.) belong to each range?
Locate the median in the grouped data
Once you know which positions belong to each group, check which group contains the median position. The median must come from that group's range of customer counts.
Match the range to the choices
After you know the range of possible values for the median, see which answer choice lies in that range.
Desmos Guide
Confirm the median position with Desmos
In Desmos, type (37+1)/2 and press Enter. The output is the position of the median in the ordered list (the 19th value). Then, use the problem’s group counts (7 shops in the first group, 19 in the second, the rest in the third) to reason which group contains that position and what range the median must fall in.
Step-by-step Explanation
Find the position of the median
For a data set with values, where is odd, the median is the value in position when the data are ordered from least to greatest.
Here, , so the median is in position
So the median is the 19th value in the ordered list of customer counts.
Determine which group contains the 19th value
We are told:
- 7 shops served fewer than 80 customers.
- 19 shops served at least 80 but fewer than 140 customers.
- The remaining shops (the rest) served 140 or more customers.
When ordered from smallest to largest:
- Positions 1 through 7 are the shops with fewer than 80 customers.
- The next 19 shops (positions 8 through 26) have between 80 and 139 customers.
Since the median is the 19th value, it lies within positions 8 to 26, which are all between 80 and 139 customers.
Translate this to a numerical range for the median
From step 2, the median must be in the group of shops that served at least 80 but fewer than 140 customers.
That means the median must be a number that is and .
Look at the answer choices and identify which one is in this range.
Select the answer that fits this range
Among the options 75, 90, 140, and 200:
- 75 is less than 80.
- 140 and 200 are 140 or more.
- Only 90 is at least 80 but less than 140.
Therefore, the number that could be the median of the 37 daily customer counts is 90.