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.
