Question 15·Hard·One-Variable Data Distributions; Measures of Center and Spread
The frequency table summarizes a data set of the weights, rounded to the nearest pound, of 71 tortoises.
| Weight (pounds) | Frequency |
|---|---|
| 13 | 12 |
| 14 | 8 |
| 15 | 5 |
| 16 | 7 |
| 17 | 9 |
| 18 | 10 |
| 19 | 13 |
| 20 | 7 |
A weight of 39 pounds is added to the original data set, creating a new data set of the weights, rounded to the nearest pound, of 72 tortoises.
Which statement best compares the mean and median of the new data set to the mean and median of the original data set?
For questions about how adding a data point affects mean and median, avoid heavy computation. First, compare the new value to the original mean range (often you can bound the mean between the minimum and maximum); if the new value is above the mean, the mean increases, and if it is below, the mean decreases. For the median, focus on positions: find which position(s) in the ordered list define the median before and after the change and check whether the new data point actually changes which values sit in those positions, especially when the new point is an extreme on one end of the data.
Hints
Separate mean and median in your thinking
First, decide what happens to the mean (average) when a very large new value is added to the data set. Then, separately, decide what happens to the median (middle value).
Compare the new weight to the original data
Look at the original weights: they range from 13 to 20. How does 39 compare to these values? Is it smaller than most, about the same, or much larger? How would that affect the average?
Track the position of the median
There are 71 original tortoises. What position in the ordered list is the median? After adding one more tortoise to make 72, which positions determine the median, and does adding a new largest value change the actual weights at those middle positions?
Desmos Guide
Compute the original mean in Desmos
Enter the expression for the original total weight divided by 71:
Desmos will display the original mean; note that it should be between 13 and 20.
Compute the new mean after adding 39
Enter a second expression for the new mean:
Compare this value to the original mean from Step 1 to see whether the mean has increased or decreased.
Use counts to reason about the medians (with Desmos as a calculator)
If you like, have Desmos help with the position calculations by entering and to confirm the original median is at position 36, and and to confirm the new medians are at positions 36 and 37. Then, use the frequency table to see which weight values occupy those positions before and after adding the 39‑pound value, and decide whether the median changes.
Step-by-step Explanation
Clarify what mean and median measure
- The mean is the average: add all values and divide by how many there are.
- The median is the middle value when the data are ordered.
- With an odd number of data points, it is the single middle value.
- With an even number of data points, it is the average of the two middle values.
We need to see how adding one extra (large) weight affects each of these.
Reason about how the mean changes
All 71 original weights are between 13 and 20 pounds, so the original mean must also be between 13 and 20.
The new weight, 39 pounds, is greater than every original weight, so it is definitely greater than the original mean.
When you add a value greater than the current mean, the overall mean must increase (the average is pulled upward by a large value). So the new mean is greater than the original mean.
Find the original median
There are 71 original data points, so the median is the th value when the data are listed from smallest to largest.
Use cumulative frequencies to locate the 36th value:
- Up to 13 lb: values (positions 1–12)
- Up to 14 lb: values (positions 1–20)
- Up to 15 lb: values (positions 1–25)
- Up to 16 lb: values (positions 1–32)
- Up to 17 lb: values (positions 1–41)
The 36th value falls in the 17‑pound group (between positions 33 and 41). So the original median is 17 pounds.
Find the new median after adding the 39‑pound tortoise
After adding the 39‑pound tortoise, there are 72 data points.
- When there are 72 ordered values, the median is the average of the 36th and 37th values.
Because 39 is larger than all existing weights (which go only up to 20), it will be placed at the very end of the ordered list, in position 72.
- This means the original 71 values still occupy positions 1 through 71 exactly as before.
- So the 36th and 37th values in the new data set are the same as they were in the original data set (both 17 pounds).
- The new median is the average of 17 and 17, which is still 17.
Putting this together: the new mean is greater than the original mean, and the two medians are equal. This matches the choice: “The mean of the new data set is greater than the mean of the original data set, and the medians of the two data sets are equal.”