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?

5Cevap
B
6
C
3
D
7

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Measures of Central Tendency for Ungrouped Data

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