Question

The variance of the discrete data 3, 4, 5, 6, 7, 8, 10, 13 is

• A. 7.5
• B. 8
• C. 9.5
• D. 9

Hint:

When calculating variance, you have two main formulas. The definitional formula $\frac{\sum (x_i - \mu)^2}{N}$ is good if the mean is a nice integer. The computational formula $\frac{\sum x_i^2}{N} - \mu^2$ is generally faster if you're using a calculator but can be prone to rounding errors if the mean is not exact. Always double-check your arithmetic, as it's easy to make a small error in summing or squaring.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Variance is a measure of the spread or dispersion of a set of data. It is the average of the squared differences from the mean.
Step 2: Key Formula or Approach
1. Calculate the mean ($\mu$) of the data. $\mu = \frac{\sum x_i}{N}$. 2. Calculate the variance ($\sigma^2$). The formula for variance is $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$ or the shortcut formula $\sigma^2 = \frac{\sum x_i^2}{N} - \mu^2$. We will use the shortcut formula as it's often computationally easier.
Step 3: Detailed Explanation
The given data points are $x_i$: 3, 4, 5, 6, 7, 8, 10, 13. The number of data points is $N=8$. 1. Calculate the mean ($\mu$): Sum of the data points: \[ \sum x_i = 3+4+5+6+7+8+10+13 = 56 \] Mean: \[ \mu = \frac{\sum x_i}{N} = \frac{56}{8} = 7 \] 2. Calculate the sum of the squares of the data points ($\sum x_i^2$): \[ \sum x_i^2 = 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + 8^2 + 10^2 + 13^2 \] \[ \sum x_i^2 = 9 + 16 + 25 + 36 + 49 + 64 + 100 + 169 \] \[ \sum x_i^2 = (9+16) + (25+36) + (49+64) + (100+169) \] \[ \sum x_i^2 = 25 + 61 + 113 + 269 = 468 \] 3. Calculate the variance ($\sigma^2$): Using the formula $\sigma^2 = \frac{\sum x_i^2}{N} - \mu^2$: \[ \sigma^2 = \frac{468}{8} - (7)^2 \] \[ \sigma^2 = 58.5 - 49 \] \[ \sigma^2 = 9.5 \] Let me re-check the calculations. Sum: 3+4+5+6+7+8+10+13 = 56. Correct. Mean: 56/8 = 7. Correct. Sum of squares: 9+16+25+36+49+64+100+169 = 468. Correct. 468/8 = 234/4 = 117/2 = 58.5. Correct. Variance = 58.5 - 49 = 9.5. Correct. My calculated value is 9.5, which is option (C). The provided answer key indicates (D) 9. Let's re-calculate using the definition $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$. Mean $\mu=7$. Deviations $(x_i - \mu)$: (3-7), (4-7), (5-7), (6-7), (7-7), (8-7), (10-7), (13-7) = -4, -3, -2, -1, 0, 1, 3, 6 Squared deviations $(x_i - \mu)^2$: 16, 9, 4, 1, 0, 1, 9, 36 Sum of squared deviations: \[ \sum (x_i - \mu)^2 = 16+9+4+1+0+1+9+36 = 76 \] Variance: \[ \sigma^2 = \frac{76}{8} = \frac{19}{2} = 9.5 \] Both methods give a variance of 9.5. The provided answer of 9 is incorrect. The correct answer is 9.5. Step 4: Final Answer
The mean of the data is $\mu=7$. The sum of the squared deviations from the mean is 76. The variance is $\sigma^2 = \frac{76}{8} = 9.5$. The correct option is (C), but the provided solution states (D). The calculation clearly shows the variance is 9.5.