1. **Problem Statement:** Calculate the mean, median, mode, variance, and standard deviation for the given frequency distribution of passengers (in millions) handled by 20 airports, and interpret these statistics for strategic planning. 2. **Given Data:** Passengers (millions) | Number of Airports (Frequency, $f$) 10–20 | 6 20–30 | 8 30–40 | 4 40–50 | 2 3. **Step 1: Calculate the midpoints ($x$) of each class interval:** - 10–20: $\frac{10+20}{2} = 15$ - 20–30: $\frac{20+30}{2} = 25$ - 30–40: $\frac{30+40}{2} = 35$ - 40–50: $\frac{40+50}{2} = 45$ 4. **Step 2: Calculate the mean ($\bar{x}$):** $$\bar{x} = \frac{\sum f x}{\sum f} = \frac{6\times15 + 8\times25 + 4\times35 + 2\times45}{6+8+4+2} = \frac{90 + 200 + 140 + 90}{20} = \frac{520}{20} = 26$$ 5. **Step 3: Calculate the median:** - Total frequency $N=20$ - Median class is where cumulative frequency reaches $\frac{N}{2} = 10$ - Cumulative frequencies: 6, 14, 18, 20 - Median class is 20–30 (since cumulative frequency 14 > 10) - Use median formula: $$\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f_m}\right) \times h$$ Where: $L=20$ (lower boundary of median class), $F=6$ (cumulative frequency before median class), $f_m=8$ (frequency of median class), $h=10$ (class width) $$\text{Median} = 20 + \left(\frac{10 - 6}{8}\right) \times 10 = 20 + \frac{4}{8} \times 10 = 20 + 5 = 25$$ 6. **Step 4: Calculate the mode:** - Mode class is the class with highest frequency: 20–30 with frequency 8 - Use mode formula: $$\text{Mode} = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$$ Where: $L=20$ (lower boundary of modal class), $f_1=8$ (frequency of modal class), $f_0=6$ (frequency before modal class), $f_2=4$ (frequency after modal class), $h=10$ $$\text{Mode} = 20 + \frac{8 - 6}{2\times8 - 6 - 4} \times 10 = 20 + \frac{2}{16 - 10} \times 10 = 20 + \frac{2}{6} \times 10 = 20 + 3.33 = 23.33$$ 7. **Step 5: Interpret mean, median, and mode:** - Mean (26 million) represents the average passenger volume. - Median (25 million) shows the middle value, indicating half the airports handle fewer than 25 million passengers. - Mode (23.33 million) indicates the most common passenger volume range. - For strategic planning, these suggest focusing resources around 20–30 million passengers range. 8. **Step 6: Calculate variance ($\sigma^2$) and standard deviation ($\sigma$):** - Calculate $x^2$ for each midpoint: $15^2=225$, $25^2=625$, $35^2=1225$, $45^2=2025$ - Calculate $\sum f x^2$: $$6\times225 + 8\times625 + 4\times1225 + 2\times2025 = 1350 + 5000 + 4900 + 4050 = 15300$$ - Variance formula: $$\sigma^2 = \frac{\sum f x^2}{N} - \bar{x}^2 = \frac{15300}{20} - 26^2 = 765 - 676 = 89$$ - Standard deviation: $$\sigma = \sqrt{89} \approx 9.43$$ 9. **Step 7: Interpret variance and standard deviation:** - Variance and standard deviation measure the spread of passenger volumes. - A standard deviation of 9.43 million indicates moderate variability. - For risk assessment, this variability suggests some airports have significantly different passenger volumes, affecting resource allocation and operational risk. **Final answers:** - Mean = 26 million - Median = 25 million - Mode = 23.33 million - Variance = 89 - Standard deviation $\approx$ 9.43 million These statistics help the airline plan resource allocation and expansion by focusing on the most common passenger volume ranges and understanding variability for risk management.