1. **Problem Statement:** We have a frequency distribution of resistors with their resistances and counts. We need to find: (i) Modal resistance (ii) Median resistance (iii) Mean resistance (iv) Standard deviation 2. **Given Data:** | No of resistors (f) | Resistance (x) | |---------------------|----------------| | 2 | 112 | | 8 | 115 | | 10 | 116 | | 12 | 108 | | 14 | 102 | 3. **(i) Modal Resistance:** The mode is the resistance with the highest frequency. Here, max frequency is 14 at resistance 102. **Answer:** Modal resistance = $102\ \Omega$ 4. **(ii) Median Resistance:** - Total number of resistors $N = 2 + 8 + 10 + 12 + 14 = 46$ - Median position = $\frac{N+1}{2} = \frac{46+1}{2} = 23.5^{th}$ value - Cumulative frequencies: - 2 (112) - 2+8=10 (115) - 10+10=20 (116) - 20+12=32 (108) - 32+14=46 (102) - The 23.5th value lies in the group with resistance 108 (since cumulative frequency reaches 32 here). **Answer:** Median resistance = $108\ \Omega$ 5. **(iii) Mean Resistance:** Formula: $$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$$ Calculate $\sum f_i x_i$: - $2 \times 112 = 224$ - $8 \times 115 = 920$ - $10 \times 116 = 1160$ - $12 \times 108 = 1296$ - $14 \times 102 = 1428$ Sum: $224 + 920 + 1160 + 1296 + 1428 = 5038$ Mean: $$\bar{x} = \frac{5038}{46} \approx 109.52\ \Omega$$ 6. **(iv) Standard Deviation:** Formula: $$\sigma = \sqrt{\frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}}$$ Calculate each $(x_i - \bar{x})^2$ and multiply by $f_i$: - For 112: $(112 - 109.52)^2 = 6.1504$, $2 \times 6.1504 = 12.3008$ - For 115: $(115 - 109.52)^2 = 30.1504$, $8 \times 30.1504 = 241.2032$ - For 116: $(116 - 109.52)^2 = 42.1504$, $10 \times 42.1504 = 421.504$ - For 108: $(108 - 109.52)^2 = 2.3104$, $12 \times 2.3104 = 27.7248$ - For 102: $(102 - 109.52)^2 = 56.5504$, $14 \times 56.5504 = 791.7056$ Sum: $12.3008 + 241.2032 + 421.504 + 27.7248 + 791.7056 = 1494.4384$ Standard deviation: $$\sigma = \sqrt{\frac{1494.4384}{46}} = \sqrt{32.4843} \approx 5.70\ \Omega$$ **Final answers:** - Modal resistance = $102\ \Omega$ - Median resistance = $108\ \Omega$ - Mean resistance = $109.52\ \Omega$ - Standard deviation = $5.70\ \Omega$