This solution will guide you through calculating the unknown frequency, $\alpha$, using the given variance of a frequency distribution.

1. Key Concepts and Formulas

• Mean ($\bar{x}$): For a frequency distribution with data points $x_i$ and corresponding frequencies $f_i$, the mean is given by: $\bar{x} = \frac{\sum_{i=1}^{n} f_i x_i}{\sum_{i=1}^{n} f_i}$
• Variance ($\sigma^2$): The variance for a frequency distribution measures the spread of data around the mean. The direct definition formula is often convenient when the mean is a simple integer: $\sigma^2 = \frac{\sum_{i=1}^{n} f_i (x_i - \bar{x})^2}{\sum_{i=1}^{n} f_i}$ where $N = \sum f_i$ is the total frequency.

2. Step-by-Step Solution

Step 1: Calculate Total Frequency ($N$) and Sum of Products ($\sum f_i x_i$)

First, we need to express the total frequency and the sum of products in terms of $\alpha$.

• Total Frequency ($N$): Sum all the given frequencies. $N = \sum f_i = 3 + 6 + 16 + \alpha + 9 + 5 + 6$ $N = 45 + \alpha$

• Sum of Products ($\sum f_i x_i$): Multiply each data point ($x_i$) by its corresponding frequency ($f_i$) and sum the results. $\sum f_i x_i = (2 \times 3) + (3 \times 6) + (4 \times 16) + (5 \times \alpha) + (6 \times 9) + (7 \times 5) + (8 \times 6)$ $\sum f_i x_i = 6 + 18 + 64 + 5\alpha + 54 + 35 + 48$ $\sum f_i x_i = 225 + 5\alpha$

Step 2: Calculate the Mean ($\bar{x}$)

Now, we calculate the mean using the formula $\bar{x} = \frac{\sum f_i x_i}{N}$. $\bar{x} = \frac{225 + 5\alpha}{45 + \alpha}$ Notice that the numerator $225 + 5\alpha$ can be factored as $5(45 + \alpha)$. $\bar{x} = \frac{5(45 + \alpha)}{45 + \alpha} = 5$ This is a crucial simplification: the mean of the distribution is $\bar{x} = 5$, irrespective of the value of $\alpha$. Since the mean is an integer and one of the $x_i$ values, the direct definition formula for variance will be very efficient.

Step 3: Calculate the Sum of Squared Deviations ($\sum f_i (x_i - \bar{x})^2$)

With $\bar{x} = 5$, we can now calculate the deviations $(x_i - \bar{x})$, their squares $(x_i - \bar{x})^2$, and then $f_i (x_i - \bar{x})^2$ for each data point.

Now, sum the last column to find $\sum f_i (x_i - \bar{x})^2$: $\sum f_i (x_i - \bar{x})^2 = 27 + 24 + 16 + 0 + 9 + 20 + 45$ $\sum f_i (x_i - \bar{x})^2 = 141$ Notice that the term involving $\alpha$ is $\alpha \times 0 = 0$, so the numerator for variance does not depend on $\alpha$.

Step 4: Set Up and Solve the Variance Equation

We are given that the variance ($\sigma^2$) is 3. We use the direct definition formula for variance: $\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{N}$ Substitute the known values: $\sigma^2 = 3$, $\sum f_i (x_i - \bar{x})^2 = 141$, and $N = 45 + \alpha$. $3 = \frac{141}{45 + \alpha}$ Now, we solve for $\alpha$: $3(45 + \alpha) = 141$ Divide both sides by 3: $45 + \alpha = \frac{141}{3}$ $45 + \alpha = 47$ Subtract 45 from both sides: $\alpha = 47 - 45$ $\alpha = 2$

3. Common Mistakes & Tips

• Arithmetic Errors: Statistics problems involve many calculations. Double-check all sums and products to avoid small errors that can lead to an incorrect final answer.
• Choosing the Correct Formula: Both variance formulas are equivalent. However, if the mean is an integer (especially if it's one of the $x_i$ values), the direct definition formula $\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{N}$ can simplify calculations significantly.
• Validity of $\alpha$: Frequencies must always be non-negative integers. If your calculated $\alpha$ is negative or a fraction, recheck your calculations. Our result $\alpha=2$ is a valid frequency.

4. Summary

We systematically calculated the total frequency and the mean of the distribution, finding that the mean is $\bar{x}=5$. Then, we computed the sum of squared deviations from the mean, which resulted in 141. Finally, by applying the variance formula and equating it to the given variance of 3, we solved for the unknown frequency $\alpha$.

The final answer is $\boxed{2}$.