Question 55·Easy·One-Variable Data Distributions; Measures of Center and Spread
The data set below is written in ascending order.
3, 5, 8, 10, 14
Which of the following lists represents a data set that has the same mean as the data set shown?
For questions asking which data set has the same mean as a given one, first quickly find the original mean (or, if all sets have the same number of values, just find the original sum). Then use the fact that equal means with the same count of values require equal sums, so you only need to add the numbers in each choice and see which total matches; this avoids repeatedly dividing and speeds up your work while reducing arithmetic mistakes.
Hints
Recall how to find the mean
The mean (average) of a data set is the sum of all the numbers divided by how many numbers there are. Start by doing this for the original data set.
Use the fact that each list has 5 numbers
Each answer choice also has 5 values. If two 5-number data sets have the same mean, what must be true about the total sum of their values?
Compare sums instead of full means
Add the numbers in the original data set, then add the numbers in each answer choice. Which choice has the same total as the original set?
Desmos Guide
Find the mean of the original data set
In Desmos, type (3+5+8+10+14)/5 and press Enter. Note the numerical result; this is the mean of the original data set.
Compute and compare the means of the answer choices
On new lines in Desmos, type (2+6+8+12+14)/5, (5+7+9+11+13)/5, (1+2+4+8+15)/5, and (0+8+8+8+16)/5. Compare each output to the mean you found for the original set and identify which choice gives the same value.
Step-by-step Explanation
Find the mean of the original data set
The mean (average) is the sum of the values divided by the number of values.
Add the numbers:
There are 5 numbers, so the mean is
So the original data set has a mean of 8.
Relate equal means to equal sums
Each answer choice is also a set of 5 numbers.
If two data sets have the same number of values and the same mean, then their sums must be the same.
So we only need to find which answer choice has a sum of 40, like the original set.
Check the sum of each answer choice
Compute the sum of the numbers in each choice:
- Choice A: (too large)
- Choice B: (too large)
- Choice C: (too small)
- Choice D: (matches the original sum)
Only choice D has a sum of 40, so it has the same mean of 8 as the original data set. Therefore, the correct answer is D) 0, 8, 8, 8, 16.