The table below shows the distribution of the number of books read by a group of students during a summer reading program. | Number of Books Read | Number of Students | | :---: | :---: | | 1 | 1 | | 2 | 4 | | 3 | 8 | | 4 | x | | 5 | 4 | If the median number of books read by the students is 3, what i

The table below shows the distribution of the number of books read by a group of students … — ACT

The table below shows the distribution of the number of books read by a group of students during a summer reading program.

If the median number of books read by the students is 3, what is the maximum possible value for the mean number of books read by these students?

The maximum possible value for the mean number of books read is 3.4.

To maximize the mean, we want to maximize the number of students who read 4 books, which is

x

. However,

x

is constrained by the requirement that the median must remain 3. The total number of students is

17 + x

. The 3s occupy sorted positions 6 through 13. For the median to be 3, the middle position(s) of the sorted list must not exceed position 13. When

N = 17+x

is odd (so

x

is even), the median position is

\frac{18+x}{2}

. Setting

\frac{18+x}{2} \le 13

yields

x \le 8

. For

x = 8

, the total number of students is 25, and the 13th student is the last one who read 3 books, making the median 3. If

x = 9

, the total is 26, and the median is the average of the 13th (3) and 14th (4) values, which is 3.5. Therefore, the maximum integer value of

x

is 8. The mean with

x=8

is the sum of all books read divided by the total number of students:

(1 \times 1 + 2 \times 4 + 3 \times 8 + 4 \times 8 + 5 \times 4) / 25 = 85 / 25 = 3.4

Express the total number of students,

N

, in terms of

x

N = 1 + 4 + 8 + x + 4 = 17 + x

To calculate the mean and locate the median, we need the total count of data points.

Determine the range of positions that the value 3 occupies in the sorted dataset.

Positions 6 through 13.

Since the ratings are sorted, the single 1 is at position 1, the four 2s are at positions 2–5, and the eight 3s are at positions 6–13.

Find the maximum integer value of

x

that keeps the median at 3.

x = 8

x = 8

, the total number of students is

25

. The median is the 13th value, which is 3. If

x = 9

, the total is

26

, and the median is the average of the 13th (3) and 14th (4) values, which is 3.5.

Calculate the mean of the dataset when

x = 8

Mean =

85 / 25 = 3.4

The sum of the books read is

1(1) + 2(4) + 3(8) + 4(8) + 5(4) = 85

. Dividing this by the total of 25 students gives the maximum mean.

Finding the maximum mean of a frequency distribution given a median constraint by setting up inequalities for the median position.