Section 4: Soil and Water Conservation Engineering
Hydrology
Subtopic: Analysis of precipitation data
Question Statement:
A catchment with 5 rain gauge stations recorded annual rainfall of 90, 100, 80, 120, and 110 cm, respectively. Considering 5 % error in estimation of the mean annual rainfall, additional number of rain gauge stations required is __________. (Answer in integer)
Official Answer Key:
5 to 5
Step-by-Step Solution & Explanation
Step 1 — mean annual rainfall:
x̄ = (90 + 100 + 80 + 120 + 110)/5 = 500/5 = 100 cm
Step 2 — standard deviation (sample, with n − 1 = 4 degrees of freedom):
Deviations: −10, 0, −20, +20, +10
Σd² = 100 + 0 + 400 + 400 + 100 = 1000
σ = √(1000/4) = √250 = 15.811 cm
Step 3 — coefficient of variation:
Cᵥ = (σ/x̄) × 100 = (15.811/100) × 100 = 15.811 %
Step 4 — optimum number of stations (IS 4987 / standard hydrology formula):
N = (Cᵥ/ε)² where ε = permissible error = 5 %
N = (15.811/5)² = (3.1623)² = 10 stations
Step 5 — additional stations required:
= N − existing = 10 − 5 = 5
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