The table below shows the score distribution of a group of students in a quiz: | Score (x) | 1 | 2 | 3 | 4 | 5 | |---|---|---|---|---|---| | Frequency (f) | 4 | 3 | k | 2 | 1 | If the mean score of the distribution is 2.5, what is the median score of the dataset?

The table below shows the score distribution of a group of students in a quiz: | Score (x)… — JAMB UTME

The table below shows the score distribution of a group of students in a quiz:

Score ( $x$ )	1	2	3	4	5
Frequency ( $f$ )	4	3	$k$	2	1

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

The median score of the dataset is

2.5

To find the median, we first find the missing frequency

k

using the mean formula

\text{Mean} = \frac{\sum fx}{\sum f}

. Setting

\frac{23 + 3k}{10 + k} = 2.5

gives

k = 4

. With

k = 4

, the total number of observations is

N = 14

. The median is the average of the 7th and 8th scores in the ordered dataset. From cumulative frequencies, the 7th score is

2

and the 8th score is

3

. Thus, the median is

\frac{2 + 3}{2} = 2.5

Set up the equation for the mean to find the missing frequency

k

\text{Mean} = \frac{\sum fx}{\sum f} = \frac{1(4) + 2(3) + 3(k) + 4(2) + 5(1)}{4 + 3 + k + 2 + 1} = \frac{23 + 3k}{10 + k} = 2.5

The mean of an ungrouped frequency table is the sum of all values divided by total frequency.

Solve the linear equation for

k

23 + 3k = 2.5(10 + k) \implies 23 + 3k = 25 + 2.5k \implies 0.5k = 2 \implies k = 4

Cross-multiplying and grouping like terms isolates the variable

k

Determine the total frequency

N

and median positions

N = 4 + 3 + 4 + 2 + 1 = 14

. Median position = average of

\frac{14}{2}\text{th}

(

7\text{th}

) and

(\frac{14}{2} + 1)\text{th}

(

8\text{th}

) values.

For an even total number of observations

N

, the median is the arithmetic mean of the two central numbers.

Find the

7\text{th}

and

8\text{th}

scores using cumulative frequency and calculate median

Cumulative frequencies: Score 1 (1–4), Score 2 (5–7), Score 3 (8–11).

7\text{th}\text{ term} = 2

8\text{th}\text{ term} = 3

\text{Median} = \frac{2 + 3}{2} = 2.5

The 7th score is 2 and the 8th score is 3, making their average 2.5.

Calculating the median of ungrouped frequency data after finding a missing frequency using the mean

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