Question Bank • QB_SWCE_EXP_021 NAT • 2 Marks

Section 4: Soil and Water Conservation Engineering

Hydrology

Subtopic: Hydrological cycle and measurement of its components

Question Statement:

A catchment has 5 rain gauge stations. The annual rainfall recorded at the stations are $80, 100, 120, 140,$ and $160\text{ cm}$. If an allowable error in the estimation of mean annual rainfall is $10\%$, the optimum number of rain gauge stations required in the catchment is ________ (answer in integer).

Answer Key:

7

Step-by-Step Solution & Explanation

1. Mean rainfall $\bar{P}$: $\bar{P} = \frac{80 + 100 + 120 + 140 + 160}{5} = \frac{600}{5} = 120\text{ cm}$ 2. Standard deviation $\sigma$: $\sigma = \sqrt{\frac{\sum (P_i - \bar{P})^2}{n - 1}} = \sqrt{\frac{(-40)^2 + (-20)^2 + 0^2 + 20^2 + 40^2}{5 - 1}} = \sqrt{\frac{1600 + 400 + 0 + 400 + 1600}{4}} = \sqrt{\frac{4000}{4}} = \sqrt{1000} \approx 31.623\text{ cm}$ 3. Coefficient of variation $C_v$: $C_v = \frac{\sigma}{\bar{P}} \times 100 = \frac{31.623}{120} \times 100 = 26.35\%$ 4. Optimum number of stations $N$ for allowable error $\epsilon = 10\%$: $N = \left(\frac{C_v}{\epsilon}\right)^2 = \left(\frac{26.35}{10}\right)^2 = (2.635)^2 \approx 6.94$ Rounding up to the next integer yields $N = 7$.
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