A middle school librarian recorded the number of books checked out by 15 students during a reading program, as summarized in the table below: | Number of Books | Number of Students | | :--- | :--- | | 1 | 3 | | 2 | 4 | | 3 | 5 | | 4 | 2 | | 5 | 1 | What is the positive difference between the median

A middle school librarian recorded the number of books checked out by 15 students during a… — ACT

A middle school librarian recorded the number of books checked out by $15$ students during a reading program, as summarized in the table below:

What is the positive difference between the median and the mean of the number of books checked out per student?

The positive difference between the median and the mean is

0.4

To find the mean, multiply each number of books by the corresponding number of students to find the total sum of books:

(1)(3) + (2)(4) + (3)(5) + (4)(2) + (5)(1) = 39

. Dividing

39

by the total of

15

students yields a mean of

2.6

. To find the median of

15

ordered values, locate the 8th value. The cumulative frequencies show that the 1st through 3rd values are

1

, the 4th through 7th values are

2

, and the 8th through 12th values are

3

. Thus, the median is

3

. The positive difference between the median and mean is

3 - 2.6 = 0.4

Calculate the total number of books checked out by all students.

Total books =

(1 \times 3) + (2 \times 4) + (3 \times 5) + (4 \times 2) + (5 \times 1) = 3 + 8 + 15 + 8 + 5 = 39

Each book quantity must be multiplied by its corresponding student frequency.

Calculate the mean number of books per student.

Mean =

\frac{39}{15} = 2.6

Divide the total number of books by the total number of students.

Determine the median number of books.

The total frequency is

15

, so the median is the

\frac{15 + 1}{2} = 8\text{th}

value. Cumulative frequencies are

3

(for 1 book),

7

(for 2 books), and

12

(for 3 books). The 8th value falls in the 3-book category, so Median =

3

The median represents the middle data point when all 15 student values are listed in non-decreasing order.

Calculate the positive difference between the median and the mean.

Difference =

3 - 2.6 = 0.4

Subtract the mean from the median to find the difference.

Calculating weighted mean and median from a frequency table.

Estimated Time:1m 30s