1. **Problem Statement:** We have a class of 20 students, each recording the number of coins in their pockets. The data is given as: | Number of coins | 0 | 1 | 2 | 3 | 4 | 5 | 6 | | Frequency | 3 | 1 | 7 | 8 | 0 | 0 | 1 | We need to find (a) the median and (b) the mean number of coins. 2. **Median Formula and Explanation:** The median is the middle value when data is arranged in order. For grouped data or frequency data, the median position is given by: $$\text{Median position} = \frac{n+1}{2}$$ where $n$ is the total number of observations. 3. **Calculate the Median:** - Total students $n = 20$ - Median position $= \frac{20+1}{2} = 10.5$ We find the 10.5th value in the ordered data. 4. **Cumulative Frequency Table:** | Number of coins | 0 | 1 | 2 | 3 | 4 | 5 | 6 | |-----------------|---|---|---|---|---|---|---| | Frequency | 3 | 1 | 7 | 8 | 0 | 0 | 1 | | Cumulative Freq | 3 | 4 | 11| 19| 19| 19| 20| The 10.5th value lies in the group where cumulative frequency first reaches or exceeds 10.5, which is at 11 for number of coins = 2. **Therefore, the median number of coins is 2.** 5. **Mean Formula and Explanation:** The mean is the average value calculated by: $$\text{Mean} = \frac{\sum (x \times f)}{n}$$ where $x$ is the number of coins and $f$ is the frequency. 6. **Calculate the Mean:** Calculate $x \times f$ for each number of coins: - $0 \times 3 = 0$ - $1 \times 1 = 1$ - $2 \times 7 = 14$ - $3 \times 8 = 24$ - $4 \times 0 = 0$ - $5 \times 0 = 0$ - $6 \times 1 = 6$ Sum these: $0 + 1 + 14 + 24 + 0 + 0 + 6 = 45$ Mean $= \frac{45}{20} = 2.25$ **Final answers:** - Median = 2 - Mean = 2.25