Given that the total of 16 values is 528 and the sum of the squares of deviation from 33 is 9158. The variance is
Show Hint
Always double check if the numerical value given for deviations matches the actual mean of the dataset. Since $\frac{528}{16} = 33$, you are dealing with standard deviations about the mean, allowing you to bypass any complex correction formulas and solve the problem using a single division step: $\frac{9158}{16} = 572.375$.
Step 1: Understanding the Question:
We are given a statistical dataset with $n = 16$ data values. The sum of these values is $\sum x_i = 528$, and the sum of the squared deviations from a specific value (33) is 9158. We need to calculate the variance of this dataset.
Step 2: Key Formula or Approach:
The mathematical formula for variance ($\sigma^2$) is:
$$\sigma^2 = \frac{1}{n} \sum (x_i - \bar{x})^2$$
where $\bar{x}$ is the arithmetic mean. First, calculate the mean $\bar{x} = \frac{\sum x_i}{n}$. If the value 33 given in the prompt matches the calculated mean $\bar{x}$, we can compute the variance directly by dividing the given sum of squares by $n$.
Step 3: Detailed Explanation:
1. Find the arithmetic mean of the dataset:
$$\bar{x} = \frac{\sum x_i}{n} = \frac{528}{16} = 33$$
2. Since the calculated mean $\bar{x}$ is exactly 33, the given "sum of squares of deviation from 33" is exactly equal to the standard term $\sum (x_i - \bar{x})^2$:
$$\sum (x_i - \bar{x})^2 = 9158$$
3. Substitute these values directly into the variance formula:
$$\text{Variance} = \frac{9158}{16}$$
4. Perform the division step-by-step:
$$\text{Variance} = 572.375$$
Step 4: Final Answer:
The variance of the data values is 572.375, which corresponds to option (D).
