Mean Variate Given Variate 'x' With Recurrence Interval For Practical Use Calculator

Formula Used:

\[ \text{Mean of the Variate X} = \text{Variate 'X' with a Recurrence Interval} - (\text{Frequency Factor} \times \text{Standard Deviation of the Sample of Size N}) \] \[ x_m = x_T - (K_z \times \sigma_{n-1}) \]

Mean of the Variate X:

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1. What is Mean Variate Given Variate 'x' With Recurrence Interval?

The Mean of the Variate X represents the average value of a random hydrologic series with a return period. It is calculated by adjusting the variate 'X' with a recurrence interval using the frequency factor and standard deviation of the sample.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ x_m = x_T - (K_z \times \sigma_{n-1}) \]

Where:

\( x_m \) — Mean of the Variate X
\( x_T \) — Variate 'X' with a Recurrence Interval
\( K_z \) — Frequency Factor (varies between 5 to 30)
\( \sigma_{n-1} \) — Standard Deviation of the Sample of Size N

Explanation: This formula calculates the mean value by subtracting the product of frequency factor and standard deviation from the variate with recurrence interval.

3. Importance of Mean Variate Calculation

Details: Accurate calculation of mean variate is crucial for hydrologic analysis, flood frequency studies, and water resource management. It helps in understanding the central tendency of hydrologic data series with specific return periods.

4. Using the Calculator

Tips: Enter the variate 'X' with recurrence interval, frequency factor, and standard deviation of the sample. All values must be valid numerical values.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of frequency factor values?
A: The frequency factor typically varies between 5 to 30 depending on rainfall duration and is a function of recurrence interval and coefficient of skew.

Q2: How is standard deviation calculated for hydrologic samples?
A: Standard deviation is calculated as the square root of variance, measuring how much individual data points differ from the mean value of the sample.

Q3: What is the significance of recurrence interval in hydrology?
A: Recurrence interval represents the average time between occurrences of a specific hydrologic event of given magnitude, crucial for flood risk assessment and infrastructure design.

Q4: Are there limitations to this calculation method?
A: This method assumes normal distribution and may require adjustments for skewed distributions or extreme hydrologic events.

Q5: How is this calculation used in practical applications?
A: It's used in flood frequency analysis, dam design, bridge construction, and other water resource management projects where understanding extreme hydrologic events is critical.