Chapter -2 : Practice Questions & Materials

Measure of Variability - Variance

1. You tracked your daily commute time (in minutes) for 10 days: 30, 35, 40, 25, 50, 45, 30, 20, 55, and 60. Calculate the variance of your daily commute time. Solution:To find the Variance, use the formula: Variance(σ2) = Mean (μ): Mean(μ) = (30 + 35 + 40 + 25 + 50 + 45 + 30 + 20 + 55 + 60) / 10 = 390 / 10 = 39 Squared Differences (x - μ)²: (30 - 39)² = 81 (35 - 39)² = 16 (40 - 39)² = 1 (25 - 39)² = 196 (50 - 39)² = 121 (45 - 39)² = 36 (30 - 39)² = 81 (20 - 39)² = 361 (55 - 39)² = 256 (60 - 39)² = 441 Variance (σ²): σ² = (81 + 16 + 1 + 196 + 121 + 36 + 81 + 361 + 256 + 441) / 10 = 1590 / 10 = 159 Explanation: First, compute the mean of the commute times. Then, subtract the mean from each commute time to find the deviation for each day. Square these deviations to eliminate negative values and sum them. Finally, divide by the number of data points to get the variance.

2. The annual salaries (in $1000s) of 8 employees in a company are 45, 50, 55, 60, 50, 65, 70, and 55. Calculate the variance of the salaries. Solution: To find the Variance, use the formula: Variance(σ2) = Mean (μ): Mean(μ) = (45 + 50 + 55 + 60 + 50 + 65 + 70 + 55) / 8 = 450 / 8 = 56.25 Squared Differences (x - μ)²: (45 - 56.25)² = 126.56 (50 - 56.25)² = 39.06 (55 - 56.25)² = 1.56 (60 - 56.25)² = 14.06 (50 - 56.25)² = 39.06 (65 - 56.25)² = 76.56 (70 - 56.25)² = 189.06 (55 - 56.25)² = 1.56 Variance (σ²): σ² = (126.56 + 39.06 + 1.56 + 14.06 + 39.06 + 76.56 + 189.06 + 1.56) / 8 = 487.48 / 8 = 60.935 Explanation: Compute the mean of the salaries. Then, subtract the mean from each salary to find the deviation for each employee. Square these deviations, sum them up, and divide by the number of employees to get the variance.

3. The monthly electricity bills (in $) for a household over a year are 100, 120, 110, 105, 130, 125, 115, 140, 135, 125, 120, and 115. Calculate the variance of the monthly electricity bills. Solution: To find the Variance, use the formula: Variance(σ2)=∑(xi−μ)2N Mean (μ): Mean(μ) = (100 + 120 + 110 + 105 + 130 + 125 + 115 + 140 + 135 + 125 + 120 + 115) / 12 = 1440 / 12 = 120 Squared Differences (x - μ)²: (100 - 120)² = 400 (120 - 120)² = 0 (110 - 120)² = 100 (105 - 120)² = 225 (130 - 120)² = 100 (125 - 120)² = 25 (115 - 120)² = 25 (140 - 120)² = 400 (135 - 120)² = 225 (125 - 120)² = 25 (120 - 120)² = 0 (115 - 120)² = 25 Variance (σ²): σ² = (400 + 0 + 100 + 225 + 100 + 25 + 25 + 400 + 225 + 25 + 0 + 25) / 12 = 1550 / 12 = 129.17 Explanation: Calculate the mean of the electricity bills. Then, subtract the mean from each bill to find the deviation for each month. Square these deviations, sum them up, and divide by the number of months to get the variance.

4. The mileage (in km/l) of a car for 7 different trips is recorded as 15, 18, 20, 17, 19, 16, and 21. Calculate the variance of the car mileage. Solution: To find the Variance, use the formula: Variance(σ2)= Mean (μ): Mean(μ) = (15 + 18 + 20 + 17 + 19 + 16 + 21) / 7 = 126 / 7 = 18 Squared Differences (x - μ)²: (15 - 18)² = 9 (18 - 18)² = 0 (20 - 18)² = 4 (17 - 18)² = 1 (19 - 18)² = 1 (16 - 18)² = 4 (21 - 18)² = 9 Variance (σ²): σ² = (9 + 0 + 4 + 1 + 1 + 4 + 9) / 7 = 28 / 7 = 4 Explanation: First, find the mean of the mileage data. Then, calculate the deviation of each trip's mileage from the mean. Square these deviations, sum them up, and divide by the number of trips to get the variance. 5. A student recorded his study hours per day for a week: 2, 3, 5, 1, 4, 3, and 6. Calculate the variance of his study hours. Solution: To find the Variance, use the formula: Variance(σ2) = Mean (μ): Mean(μ) = (2 + 3 + 5 + 1 + 4 + 3 + 6) / 7 = 24 / 7 = 3.43 Squared Differences (x - μ)²: (2 - 3.43)² = 2.05 (3 - 3.43)² = 0.18 (5 - 3.43)² = 2.45 (1 - 3.43)² = 5.90 (4 - 3.43)² = 0.33 (3 - 3.43)² = 0.18 (6 - 3.43)² = 6.63 Variance (σ²): σ² = (2.05 + 0.18 + 2.45 + 5.90 + 0.33 + 0.18 + 6.63) / 7 = 17.72 / 7 = 2.53 Explanation: Compute the mean of the study hours, then find the deviation of each study hour from the mean. Square these deviations, sum them up, and divide by the number of days to get the variance.

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