Certainly! Let's break the problem down step by step.
### a) Finding the Median Number of Goals Conceded
The median is the middle value when the data is arranged in ascending order. In this case, we are dealing with a frequency distribution, which means we need to consider the cumulative frequencies to find the median.
Here are the conceded goals and their respective frequencies:
- 0 goals: 2 games
- 1 goal: 8 games
- 2 goals: 4 games
- 3 goals: 10 games
- 4 goals: 6 games
First, calculate the total number of games:
[tex]\[ 2 + 8 + 4 + 10 + 6 = 30 \ \text{games} \][/tex]
To find the median, calculate the cumulative frequencies:
- Cumulative frequency for 0 goals: 2
- Cumulative frequency for 1 goal: 2 + 8 = 10
- Cumulative frequency for 2 goals: 10 + 4 = 14
- Cumulative frequency for 3 goals: 14 + 10 = 24
- Cumulative frequency for 4 goals: 24 + 6 = 30
Since there are 30 games, the median will be the value at the [tex]\((\frac{30 + 1}{2})\)[/tex]th item, which is the 15.5th item or the average of the 15th and 16th items.
Look at the cumulative frequencies to determine where the 15th and 16th items fall:
- By the 14th item, 2 goals have been conceded.
- By the 24th item, 3 goals have been conceded.
Therefore, both the 15th and the 16th items will be 3 goals. Hence, the median number of goals conceded is:
[tex]\[ \text{Median} = 3 \ \text{goals} \][/tex]
### b) Working Out the Mean Number of Goals Conceded
The mean is the sum of all the values divided by the number of values.
To find the mean, we need to calculate the total number of goals conceded and then divide by the total number of games.
First, find the total number of goals conceded by multiplying each number of goals by its frequency and then summing these products:
[tex]\[ (0 \times 2) + (1 \times 8) + (2 \times 4) + (3 \times 10) + (4 \times 6) \][/tex]
Let's work out the math:
[tex]\[ 0 + 8 + 8 + 30 + 24 = 70 \][/tex]
Thus, the total number of goals conceded is 70.
Next, divide this total by the number of games:
[tex]\[ \text{Mean} = \frac{70}{30} = 2.3333333333333335 \ \text{goals} \][/tex]
### Summary:
a) The median number of goals conceded is [tex]\(3\)[/tex] goals.
b) The mean number of goals conceded is approximately [tex]\(2.33\)[/tex] goals.
