The frequency distribution table below summarizes the scores of 20 students on a statistics test. | Score | Number of Students | | :---: | :---: | | 5 | 3 | | 6 | 5 | | 7 | x | | 8 | y | | 9 | 4 | If the arithmetic mean score of the 20 students is 7.15, what is the median score of the 20 students?

The frequency distribution table below summarizes the scores of 20 students on a statistic… — GRE General Test

The frequency distribution table below summarizes the scores of 20 students on a statistics test.

Score	Number of Students
5	3
6	5
7	$x$
8	$y$
9	4

If the arithmetic mean score of the 20 students is $7.15$ , what is the median score of the 20 students?

A
6.5
B
7.0
C
7.15
7.5Answer
E
8.0

Answer

The median score of the 20 students is 7.5.

To find the median score of 20 students, we first determine the missing student counts

x

and

y

. From the total count of 20 students,

x + y = 20 - (3 + 5 + 4) = 8

. Using the given mean of 7.15, the sum of all scores is

20 \times 7.15 = 143

. The sum expressed in terms of

x

and

y

3(5) + 5(6) + 7x + 8y + 4(9) = 81 + 7x + 8y

. Setting

81 + 7x + 8y = 143

gives

7x + 8y = 62

. Solving

x + y = 8

and

7x + 8y = 62

yields

x = 2

and

y = 6

. Listing scores in non-decreasing order, the 10th student scored 7 and the 11th student scored 8. Because the total number of students is even, the median is the average of these two middle values:

(7 + 8) / 2 = 7.5

Step-by-Step Solution

Set up an equation for the total number of students to find the sum of

x

and

y

3 + 5 + x + y + 4 = 20 \implies x + y + 12 = 20 \implies x + y = 8

The total frequency of all score categories must equal the total number of students (20).

Set up an equation for the sum of all scores using the given arithmetic mean.

Total sum

= 20 \times 7.15 = 143

. Also, total sum

= 3(5) + 5(6) + 7x + 8y + 4(9) = 81 + 7x + 8y

. Therefore,

81 + 7x + 8y = 143 \implies 7x + 8y = 62

The sum of all data values equals the mean multiplied by the total number of observations.

Solve the system of linear equations for

x

and

y

Multiply

x + y = 8

by 7 to get

7x + 7y = 56

. Subtract this from

7x + 8y = 62

to get

y = 6

. Then

x = 8 - 6 = 2

Determining the exact frequencies

x=2

and

y=6

is required to locate the middle values.

Determine the cumulative frequencies and find the 10th and 11th values.

Score 5: positions 1 to 3 (3 students); Score 6: positions 4 to 8 (5 students); Score 7: positions 9 to 10 (2 students); Score 8: positions 11 to 16 (6 students); Score 9: positions 17 to 20 (4 students). The 10th value is 7 and the 11th value is 8.

For an even number of data points (

N=20

), the median is the arithmetic mean of the

(N/2)

-th and

(N/2 + 1)

-th values.

Calculate the median score.

\text{Median} = \frac{7 + 8}{2} = 7.5

Average the two middle values.

Key Concept

Calculating the median from a frequency distribution table with missing counts by combining system of linear equations with cumulative frequency analysis.

Question

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