Calculating Optimum Number of Raingauge Stations
This analysis determines the minimum number of raingauge stations needed to estimate the mean annual rainfall accurately within a 10% error margin, using data from existing stations.
Input Data Summary
- Number of existing stations: $N = 8$
- Annual rainfall (cm): 93.8, 106.5, 170.6, 138.7, 87.8, 156.2, 180.9, 110.3
- Allowable percentage error: $p = 10\% = 0.10$
Step 1: Calculate Mean Rainfall
Compute the average rainfall from the existing stations.
$ \bar{x} = \frac{93.8 + 106.5 + 170.6 + 138.7 + 87.8 + 156.2 + 180.9 + 110.3}{8} $The mean rainfall is:
$ \bar{x} = \frac{944.8}{8} = 118.1 \text{ cm} $Step 2: Calculate Standard Deviation
Determine the standard deviation ($\sigma$) of the rainfall data to understand its variability. Using the sample standard deviation formula:
$ \sigma = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \bar{x})^2}{N-1}} $First, calculate the sum of squared differences from the mean:
$ \sum_{i=1}^{8}(x_i - 118.1)^2 \approx 10280 $Calculate the variance ($\sigma^2$):
$ \sigma^2 = \frac{10280}{8-1} = \frac{10280}{7} \approx 1468.57 $Calculate the standard deviation ($\sigma$):
$ \sigma = \sqrt{1468.57} \approx 38.32 \text{ cm} $Step 3: Determine Optimum Number of Stations
Use the formula to find the minimum number of stations ($n_{opt}$) required for the desired error tolerance ($p$):
$ n_{opt} = \left( \frac{\sigma}{\bar{x} p} \right)^2 $Substitute the values:
$ n_{opt} = \left( \frac{38.32}{118.1 \times 0.10} \right)^2 $Calculate the value:
$ n_{opt} = \left( \frac{38.32}{11.81} \right)^2 \approx (3.245)^2 \approx 10.53 $Since the number of stations must be a whole number, round up:
$ n_{opt} = \lceil 10.53 \rceil = 11 $Result Interpretation
The calculation indicates that 11 raingauge stations are needed for a 10% error tolerance. Based on this calculation, the existing 8 stations are insufficient.
Note: The calculated value (11) is not among the provided options (4, 6, 8, 10). The standard method yields 11 stations required.
Important Questions from Precipitation Analysis
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