Question 9·Hard·One-Variable Data Distributions; Measures of Center and Spread
The dot plot above represents a data set of the number of cups of soup sold at a café on 30 different days.
The café removes the record for the day with the smallest number of cups sold and replaces it with a new day on which 7 cups were sold, creating a new data set of 30 days.
I. The median number of cups sold
II. The mean number of cups sold
III. The range of the numbers of cups sold
Which choice states the quantities above that are greater for the new data set than for the original data set?
For “replace one data point” questions, treat each measure separately: (1) median depends only on the middle position(s), so locate the middle index (or pair) and track how the ordered positions shift; (2) mean changes by the change in total divided by the (unchanged) number of data points, so use added − removed; (3) range depends only on the minimum and maximum, so check whether either endpoint changes.
Hints
Median position(s) matter
There are 30 days, so the median is the average of the 15th and 16th values in the ordered list.
Mean changes without recomputing
Replacing one value changes the total by (new value) minus (old value). The number of days stays the same.
Range depends only on endpoints
Find the original minimum and maximum, then see what the new minimum and maximum become after removing the smallest value and adding 7.
Desmos Guide
Enter the original data as a list
In an expression line, enter a list with repeats matching the dot plot:
L=[2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6]
Compute the original median, mean, and range
Enter:
median(L)mean(L)max(L)-min(L)
Record these three values.
Create the new list and compare
Create a new list by removing one 2 and adding 7, for example:
N=[3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,7]
Then enter median(N), mean(N), and max(N)-min(N) and compare each to the original results to pick the matching answer choice.
Step-by-step Explanation
Read the original data from the dot plot
From the dot plot, the values and frequencies are:
- occurs time
- occurs times
- occurs times
- occurs times
- occurs times
There are values total.
Compare the medians (even number of values)
With values, the median is the average of the th and th values in the ordered list.
Cumulative positions in the original data:
- Position :
- Positions –: (5 values)
- Positions –: (10 values)
So the th and th values are both , and the original median is .
Now remove the only and add a (which will be the new largest value). In the new ordered list:
- Positions –:
- Positions –:
- Positions –:
So the th value is and the th value is , making the new median .
Therefore, the median increases.
Compare the means using the total change
Only one value changes: is replaced by .
So the total increases by . Since the number of days stays , the mean increases (by ).
Compare the ranges
Original minimum is and original maximum is , so the original range is .
In the new data set, the minimum becomes (since the only was removed) and the maximum becomes (the added value), so the new range is .
The range does not increase.
Choose the correct option
Only quantities I and II are greater for the new data set than for the original data set.
Answer: I and II only