At the end of the football season, a team analyzes the number of goals they conceded in each game. The table shows their information.

a) Find the median number of goals conceded.
b) Work out the mean number of goals conceded.

\begin{tabular}{|c|c|}
\hline
Conceded goals & Frequency \\
\hline
0 & 2 \\
\hline
1 & 8 \\
\hline
2 & 4 \\
\hline
3 & 10 \\
\hline
4 & 6 \\
\hline
\end{tabular}

a) Median:



Answer :

To find the median number of goals conceded, follow these steps:

1. List All Goals Based on Frequency:
We first create a list where each number of goals is repeated according to its frequency. The table provides the following information:

- 0 goals: frequency 2
- 1 goal: frequency 8
- 2 goals: frequency 4
- 3 goals: frequency 10
- 4 goals: frequency 6

So, the expanded list of goals based on the frequencies is:

[tex]\( 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4 \)[/tex]

2. Sort the List:
The list is already sorted:

[tex]\( 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4 \)[/tex]

3. Find the Middle of the List:
Since there are [tex]\( 30 \)[/tex] numbers in the list (an even number), the median will be the average of the 15th and 16th numbers in the sorted list.

4. Locate the Middle Numbers:
The 15th number in the list is 3.
The 16th number in the list is 3.

5. Calculate the Median:
The median is the average of these two middle numbers:
[tex]\[ \text{Median} = \frac{3 + 3}{2} = 3.0 \][/tex]

Therefore, the median number of goals conceded is 3.0.

b) mean:

To find the mean number of goals conceded, follow these steps:

1. Calculate the Total Goals Conceded:
Multiply each number of goals by its frequency, then sum these values:

[tex]\[ (0 \times 2) + (1 \times 8) + (2 \times 4) + (3 \times 10) + (4 \times 6) = 0 + 8 + 8 + 30 + 24 = 70 \][/tex]

2. Calculate the Total Number of Matches:
Add up all the frequencies to get the total number of matches:

[tex]\[ 2 + 8 + 4 + 10 + 6 = 30 \][/tex]

3. Calculate the Mean:
Divide the total goals conceded by the total number of matches:

[tex]\[ \text{Mean} = \frac{70}{30} \approx 2.3333333333333335 \][/tex]

Therefore, the mean number of goals conceded is approximately 2.3333333333333335.