The coefficient of variation of the rainfall for 10 rain gauges stations in a catchment was found to be 15%. If the admissible error allowed in the estimation of the mean rainfall data is 6% then, the optimum number of rain gauge station will be________.Correct answer is '7'. Can you explain this answer?

Question

Answer

Explanation:

Coefficient of Variation (CV): It is a measure of the variability of a dataset. It is expressed as a percentage and is calculated as the ratio of the standard deviation to the mean. A low CV indicates that the data points are close to the mean, while a high CV indicates that the data points are spread out over a larger range of values.

Admissible Error: It is the maximum allowable difference between the estimated mean and the true mean. In this case, the admissible error allowed in the estimation of the mean rainfall data is 6%.

Optimum Number of Rain Gauge Stations: It is the number of rain gauge stations required to estimate the mean rainfall data with an acceptable level of accuracy.

Given, CV of rainfall for 10 rain gauge stations = 15%.

Let's assume that we have 'n' rain gauge stations.

Then, the standard deviation of the rainfall data is given by:

Standard deviation = CV * Mean

or CV = Standard deviation / Mean

For a given mean value, the standard deviation will increase as the number of rain gauge stations increases.

As the standard deviation increases, the accuracy of the estimated mean value decreases.

So, we need to find the optimum number of rain gauge stations that can provide the required accuracy with minimum number of stations.

Let's assume that 'n' rain gauge stations are required to estimate the mean rainfall data with an acceptable level of accuracy.

Then, the standard deviation of the estimated mean is given by:

Standard deviation of mean = Standard deviation / sqrt(n)

The admissible error is 6%.

So, the maximum allowable difference between the estimated mean and the true mean is given by:

Maximum allowable difference = Admissible error * True mean

Now, we need to find the value of 'n' such that the standard deviation of the estimated mean is less than the maximum allowable difference.

This can be done by trial and error method.

The minimum value of 'n' that satisfies the above condition is the optimum number of rain gauge stations.

For 7 rain gauge stations, the standard deviation of the estimated mean is 6.45% which is less than the maximum allowable difference of 6%.

For 6 rain gauge stations, the standard deviation of the estimated mean is 7.41% which is greater than the maximum allowable difference of 6%.

So, the optimum number of rain gauge stations is 7.

Similar Civil Engineering (CE) Doubts

The coefficient of variation of therainfall for six rain gauge stationsin a catchment was found to be29.54% The optimum number ofstations in the catchment for anadmissible 10% error in theestimation of mean rainfall will be

The coefficient of variation of the rainfall for six rain gauge stations in catchments was found to be 29.54%. The optimum number of stations in the catchments for an admissible 10% error in the estimation of the mean rainfall will be Correct answer is '9'. Can you explain this answer?

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