1. **Problem Statement:** We are given a frequency polygon showing the number of goals scored by a football team in 25 matches. We need to complete a frequency table, find the mode, median, and mean number of goals scored. 2. **Given Data from the Polygon:** Points: (0,5), (1,7), (2,3), (3,3), (4,4), (5,2), (6,1) Here, $x$ = number of goals, $f$ = number of matches. 3. **(i) Complete the Table:** | Number of goals scored ($x$) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |------------------------------|---|---|---|---|---|---|---| | Number of matches ($f$) | 5 | 7 | 3 | 3 | 4 | 2 | 1 | 4. **(ii) Modal Number of Goals:** The mode is the value of $x$ with the highest frequency $f$. From the table, highest $f$ is 7 at $x=1$. **Mode = 1 goal** 5. **(iii) Median Number of Goals:** Total matches = 25. Median position = $\frac{25+1}{2} = 13^{th}$ value when data is ordered. Cumulative frequencies: - $x=0$: 5 - $x=1$: 5 + 7 = 12 - $x=2$: 12 + 3 = 15 The 13th value lies in the group where $x=2$ (since cumulative frequency reaches 15). **Median = 2 goals** 6. **(iv) Mean Number of Goals:** Formula: $$\text{Mean} = \frac{\sum f x}{\sum f}$$ Calculate $\sum f x$: $$5\times0 + 7\times1 + 3\times2 + 3\times3 + 4\times4 + 2\times5 + 1\times6 = 0 + 7 + 6 + 9 + 16 + 10 + 6 = 54$$ Total matches $\sum f = 25$ Mean: $$\frac{54}{25} = 2.16$$ **Mean number of goals = 2.16** --- **Final answers:** - Completed table as above. - Mode = 1 - Median = 2 - Mean = 2.16