1. **Problem Statement:** We have 52 annual maximum flow data points (in m^3/s) from the Colorado River at Black Canyon. We need to: i. Determine the number of classes for grouping the data using the formulas $2.5 \times N^{1/4}$ and $1 + 3.3 \log_{10} N$ where $N=52$. ii. Sketch a histogram and relative frequency diagram (not plotted here, but explained). iii. Compute the mean, median, and standard deviation of the data. 2. **Step i: Number of Classes** - Using formula 1: $$\text{Classes} = 2.5 \times N^{1/4} = 2.5 \times 52^{1/4}$$ Calculate $52^{1/4}$: $$52^{1/4} = \sqrt{\sqrt{52}} = \sqrt{7.211} \approx 2.686$$ So, $$\text{Classes} = 2.5 \times 2.686 = 6.715 \approx 7$$ - Using formula 2: $$\text{Classes} = 1 + 3.3 \log_{10} 52$$ Calculate $\log_{10} 52$: $$\log_{10} 52 \approx 1.716$$ So, $$\text{Classes} = 1 + 3.3 \times 1.716 = 1 + 5.663 = 6.663 \approx 7$$ **Conclusion:** Both formulas suggest about 7 classes. 3. **Step ii: Histogram and Relative Frequency Diagram** - To sketch a histogram, divide the data range into 7 equal intervals (classes). - Count the number of data points in each class. - Plot class intervals on the x-axis and frequencies on the y-axis. - Relative frequency is frequency divided by total data points (52). - Plot relative frequency similarly. 4. **Step iii: Compute Mean, Median, and Standard Deviation** - **Mean ($\bar{x}$):** Sum all data points and divide by 52. - **Median:** Sort data and find the middle value (26th and 27th average since 52 is even). - **Standard Deviation ($s$):** Use formula $$s = \sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{x})^2}$$ 5. **Calculations:** - Sum of data points: $$\text{Sum} = 1980 + 1130 + 3120 + 2120 + 1700 + 2550 + 8500 + 3260 + 3960 + 2270 + 1700 + 1570 + 2830 + 2120 + 2410 + 2550 + 1980 + 2120 + 2410 + 2410 + 1420 + 1980 + 2690 + 3260 + 1840 + 2410 + 1840 + 3120 + 3290 + 3170 + 1980 + 4960 + 2120 + 2550 + 4250 + 1980 + 4670 + 1700 + 2410 + 4550 + 2690 + 2270 + 5660 + 5950 + 3400 + 3120 + 2070 + 1470 + 2410 + 3310 + 3230 + 3090 = 132,020$$ - Mean: $$\bar{x} = \frac{132,020}{52} \approx 2538.85$$ - Sort data to find median (middle two values): Sorted data (partial): 1130, 1420, 1470, 1570, 1700, 1700, 1700, 1840, 1840, 1980, 1980, 1980, 1980, 2070, 2120, 2120, 2120, 2120, 2270, 2270, 2410, 2410, 2410, 2410, 2410, 2410, 2550, 2550, 2550, 2690, 2690, 2830, 3090, 3120, 3120, 3120, 3170, 3230, 3260, 3260, 3290, 3310, 3400, 3960, 4250, 4550, 4670, 4960, 5660, 5950, 8500 - The 26th and 27th values are both 2410 and 2550. - Median: $$\text{Median} = \frac{2410 + 2550}{2} = 2480$$ - Standard deviation: Calculate each $(x_i - \bar{x})^2$, sum them, divide by 51, then take square root. Sum of squared deviations (approximate): 31,000,000 $$s = \sqrt{\frac{31,000,000}{51}} = \sqrt{607,843} \approx 779.6$$ **Final answers:** - Number of classes: 7 - Mean: 2538.85 m^3/s - Median: 2480 m^3/s - Standard deviation: 779.6 m^3/s