The table below shows the distribution of test scores obtained by a group of students: | Score | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| | Frequency | 3 | x | 4 | 2 | 1 | If the mean score of the distribution is 2.5, what is the median score?

The table below shows the distribution of test scores obtained by a group of students: | S… — JAMB UTME

The table below shows the distribution of test scores obtained by a group of students:

Score	1	2	3	4	5
Frequency	3	$x$	4	2	1

If the mean score of the distribution is $2.5$ , what is the median score?

$2$ Cevap
B
$2.5$
C
$3$
D
$6$

Cevap

The median score is

2

First, calculate the missing frequency

x

using the mean formula

\bar{x} = \frac{\sum fx}{\sum f} = \frac{28 + 2x}{10 + x} = 2.5

, which yields

x = 6

. Total number of scores is

N = 16

. The median is the mean of the

8\text{th}

and

9\text{th}

values. Looking at cumulative frequencies, score 1 covers positions 1–3 and score 2 covers positions 4–9. Thus, both the

8\text{th}

and

9\text{th}

scores are 2, giving a median of 2.

Adım Adım Çözüm

Set up the equation for the mean using

\bar{x} = \frac{\sum fx}{\sum f}

Total frequency

\sum f = 3 + x + 4 + 2 + 1 = 10 + x

, and total score sum

\sum fx = 1(3) + 2(x) + 3(4) + 4(2) + 5(1) = 28 + 2x

The mean of an ungrouped frequency distribution is the sum of all score-frequency products divided by the total frequency.

Solve for the unknown frequency

x

2.5 = \frac{28 + 2x}{10 + x} \implies 2.5(10 + x) = 28 + 2x \implies 25 + 2.5x = 28 + 2x \implies 0.5x = 3 \implies x = 6

Equating the mean expression to

2.5

gives the value of

x

Find total number of observations

N

and determine the median position.

N = 10 + 6 = 16

. Since

N

is even, median position is between the

8\text{th}

and

9\text{th}

values.

For an even number of observations

N

, the median is the average of the

\frac{N}{2}\text{th}

and

(\frac{N}{2} + 1)\text{th}

items.

Locate the

8\text{th}

and

9\text{th}

items using cumulative frequencies.

Cumulative frequencies: Score 1 has 3; Score 2 reaches

3 + 6 = 9

. Both the

8\text{th}

and

9\text{th}

values are

2

. Therefore, median

= \frac{2 + 2}{2} = 2

Positions 4 through 9 correspond to the score

2

Anahtar Kavram

Measures of central tendency for ungrouped frequency distributions

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