The table below shows the frequency distribution of marks obtained by 25 students in a mathematics quiz: | Score (x) | 1 | 3 | 5 | 7 | 9 | 11 | |---|---|---|---|---|---|---| | Frequency (f) | 2 | p | 6 | q | 3 | 2 | If the mean score of the distribution is 5.48, what is the median score?

The table below shows the frequency distribution of marks obtained by 25 students in a mat… — JAMB UTME

The table below shows the frequency distribution of marks obtained by $25$ students in a mathematics quiz:

Score ( $x$ )	1	3	5	7	9	11
Frequency ( $f$ )	2	$p$	6	$q$	3	2

If the mean score of the distribution is $5.48$ , what is the median score?

5Answer
B
6
C
3
D
7

Answer

The median score is 5.

By using the total student count of 25 and the mean formula, we obtain the simultaneous equations

p + q = 12

and

3p + 7q = 56

, which yield

p = 7

and

q = 5

. Computing cumulative frequencies shows that items 1 to 2 have score 1, items 3 to 9 have score 3, and items 10 to 15 have score 5. The 13th item lies in this third group, so the median score is 5.

Step-by-Step Solution

Set up an equation for total frequency

2 + p + 6 + q + 3 + 2 = 25 \implies p + q + 13 = 25 \implies p + q = 12

The sum of all frequencies equals the total number of students (

25

Set up an equation for the mean score

\sum fx = 1(2) + 3(p) + 5(6) + 7(q) + 9(3) + 11(2) = 3p + 7q + 81.
\text{Mean} = \frac{3p + 7q + 81}{25} = 5.48 \implies 3p + 7q + 81 = 137 \implies 3p + 7q = 56.

The mean of an ungrouped frequency distribution is calculated using \bar{x} = \frac{\sum fx}{N}.

Solve the system of linear equations for

p

and

q

Substitute

p = 12 - q

into

3p + 7q = 56

:
3(12 - q) + 7q = 56 \implies 36 + 4q = 56 \implies 4q = 20 \implies q = 5.
Then

p = 12 - 5 = 7

Finding the missing frequencies is necessary to determine cumulative frequencies.

Determine the position and value of the median score

Position of median = \frac{N + 1}{2} = \frac{25 + 1}{2} = 13\text{th position}.
Cumulative frequencies:
- Score 1: 2
- Score 3: 2 + 7 = 9
- Score 5: 9 + 6 = 15
Since the 13th value lies in the cumulative frequency interval up to 15, the median score is 5.

The median of

N=25

items is the score corresponding to the

\frac{N+1}{2}

th item when ordered.

Key Concept

Measures of Central Tendency for Ungrouped Data

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