Question 46·Easy·One-Variable Data Distributions; Measures of Center and Spread
A climatologist recorded the amount of rainfall, in millimeters, for five consecutive days.
| Day | Rainfall (mm) |
|---|---|
| 1 | 12 |
| 2 | 15 |
| 3 | 17 |
| 4 | 18 |
| 5 | 20 |
On a sixth day, a heavy storm produced 40 millimeters of rainfall. Which of the following correctly compares the medians of the original data set and the new data set that includes the sixth day?
For median questions, always put the data in order (or confirm they are already ordered), then use the definition of median: for an odd number of data points, take the middle one; for an even number, average the two middle ones. When a new value is added, especially an extreme high or low value, quickly recount how many data points there are and recompute the median rather than guessing how it changes—this is faster and more reliable than trying to reason about it in your head.
Hints
Think about how many values you have
First, count how many days of data are in the original set and how many are in the new set after adding the sixth day. How does this affect how you find the median?
Identify the median of the original five days
For the original five days, the data are already in order. Since there are 5 values (an odd number), which position in the list gives you the median?
Figure out how a sixth value changes the median
When you add the sixth day's rainfall, you now have an even number of data points. For an even number of values, how do you compute the median from the two middle numbers?
Compare the two medians
After you compute the median for the original 5 days and for all 6 days, decide whether the first is greater than, less than, or equal to the second.
Desmos Guide
Enter the original data and find its median
Type L1 = [12, 15, 17, 18, 20] into Desmos. Then, on a new line, type median(L1) and note the value Desmos gives for the median of the original data set.
Enter the new data set with the sixth day and find its median
Type L2 = [12, 15, 17, 18, 20, 40] into Desmos. Then, on a new line, type median(L2) and note the value Desmos gives for the median of the new data set.
Compare the two medians from Desmos
Compare the number shown for median(L1) with the number shown for median(L2). Since the original median is smaller than the new median, you can conclude that the median of the original data set is less than the median of the new data set.
Step-by-step Explanation
Recall what the median is
The median of a data set is the middle value when the numbers are arranged in order.
- If there is an odd number of data points, the median is the single middle number.
- If there is an even number of data points, the median is the average of the two middle numbers.
Find the median of the original data set
The original data set is already in order:
12, 15, 17, 18, 20
There are 5 numbers, which is an odd amount. The median is the 3rd number.
So, the median of the original data set is .
Include the sixth day's rainfall and locate the new middle positions
Add the sixth day with 40 mm of rainfall to the list and keep the data in order:
12, 15, 17, 18, 20, 40
Now there are 6 numbers, which is an even amount. The median will be the average of the 3rd and 4th numbers in the ordered list.
The 3rd number is and the 4th number is .
Compute the new median and compare the two medians
Find the median of the new data set by averaging the 3rd and 4th numbers:
.
So:
- Original median:
- New median:
Since , the median of the original data set is less than the median of the new data set. This corresponds to choice B) The median of the original data set is less than the median of the new data set.