Question 6·Hard·One-Variable Data Distributions; Measures of Center and Spread
Data set A and data set B each consist of 27 values. The table shows the frequencies of the values for each data set.
| Value | Data set A frequency | Data set B frequency |
|---|---|---|
| 30 | 2 | 9 |
| 34 | 4 | 7 |
| 38 | 5 | 5 |
| 42 | 7 | 4 |
| 46 | 9 | 2 |
Which of the following statements best compares the means of the two data sets?
When comparing means from frequency tables, remember that mean = (sum of values) ÷ (number of values). First, check if the data sets have the same number of values; if they do, you only need to compare their total sums. Quickly compute each total by multiplying each value by its frequency and adding the results. You can stop once you clearly see which total is larger—there’s no need to compute exact decimal means on test day, which saves time and reduces arithmetic errors.
Hints
Think about what the mean represents
The mean is the total of all data values divided by how many values there are. How can you get the total from a frequency table?
Use the frequencies to build each sum
For each row, multiply the value by its frequency to find how much that value contributes to the total, then add these contributions for each data set.
Notice something about the number of values
Both data sets have 27 values. If two fractions have the same denominator, what do you need to compare to know which fraction is larger?
Compare the totals, not necessarily the exact means
You don’t need the decimal values of the means. Once you know the total sums for A and B, decide which sum is larger to determine which mean is larger.
Desmos Guide
Compute the mean of data set A
In Desmos, type the expression (30*2 + 34*4 + 38*5 + 42*7 + 46*9) / 27 and note the numerical result. This is the mean of data set A.
Compute the mean of data set B
On a new line, type (30*9 + 34*7 + 38*5 + 42*4 + 46*2) / 27 and note the numerical result. This is the mean of data set B.
Compare the two Desmos outputs
Look at the two decimal values Desmos gives for the means and decide which one is larger; that tells you which data set has the greater mean.
Step-by-step Explanation
Recall how to find a mean from a frequency table
The mean of a data set is
From a frequency table, the sum of all values is found by multiplying each value by its frequency and then adding these products.
Use frequencies to find the total for data set A
For data set A, compute the total of all values:
- There are 2 values of 30, contributing .
- 4 values of 34: .
- 5 values of 38: .
- 7 values of 42: .
- 9 values of 46: .
Add them:
So the sum of all 27 values in data set A is 1,094.
Use frequencies to find the total for data set B
For data set B, compute the total of all values:
- 9 values of 30: .
- 7 values of 34: .
- 5 values of 38: .
- 4 values of 42: .
- 2 values of 46: .
Add them:
So the sum of all 27 values in data set B is 958.
Compare means using the totals and equal counts
Each data set has 27 values. That means
Because the denominators are the same, the data set with the larger numerator (total sum) has the larger mean. Since , data set A has the larger mean.
Correct statement: The mean of data set A is greater than the mean of data set B.