The variance \(\sigma^2\) of the data
Is _______.
Is _______.
Correct Answer: 29
Approach Solution - 1
Calculate sums:
\[ \sum f_i = 22, \quad \sum f_i x_i = 176, \quad \sum f_i x_i^2 = 2048. \]
Calculate the mean \( \bar{x} \):
\[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} = \frac{176}{22} = 8. \]
Calculate the variance \( \sigma^2 \):
\[ \sigma^2 = \frac{1}{N} \sum f_i x_i^2 - \bar{x}^2, \]
where \( N = \sum f_i \).
Plugging in values:
\[ \sigma^2 = \frac{1}{22} \times 2048 - (8)^2 = \frac{2048}{22} - 64. \]
Compute:
\[ \frac{2048}{22} = 93.09090909 \quad \text{and} \quad \sigma^2 = 93.09090909 - 64 = 29.09090909. \]
Thus, the variance is: 29.09090909
Approach Solution -2
To find the variance \(\sigma^2\) of the given data, we will use the formula for the variance of a discrete frequency distribution. We need to compute the total frequency, the mean of the data, and the sum of the squared data values multiplied by their frequencies.
Concept Used:
The variance (\(\sigma^2\)) of a discrete frequency distribution is given by the formula:
\[ \sigma^2 = \frac{\sum_{i=1}^{n} f_i x_i^2}{\sum_{i=1}^{n} f_i} - \left( \frac{\sum_{i=1}^{n} f_i x_i}{\sum_{i=1}^{n} f_i} \right)^2 \]
This can also be written as:
\[ \sigma^2 = \frac{\sum f_i x_i^2}{N} - (\bar{x})^2 \]
where \(N = \sum f_i\) is the total frequency and \(\bar{x}\) is the mean of the data.
Step-by-Step Solution:
Step 1: Construct a calculation table to find \(N = \sum f_i\), \(\sum f_i x_i\), and \(\sum f_i x_i^2\).
| \(x_i\) | \(f_i\) | \(f_i x_i\) | \(x_i^2\) | \(f_i x_i^2\) |
|---|---|---|---|---|
| 0 | 3 | \(3 \times 0 = 0\) | 0 | \(3 \times 0 = 0\) |
| 1 | 2 | \(2 \times 1 = 2\) | 1 | \(2 \times 1 = 2\) |
| 5 | 3 | \(3 \times 5 = 15\) | 25 | \(3 \times 25 = 75\) |
| 6 | 2 | \(2 \times 6 = 12\) | 36 | \(2 \times 36 = 72\) |
| 10 | 6 | \(6 \times 10 = 60\) | 100 | \(6 \times 100 = 600\) |
| 12 | 3 | \(3 \times 12 = 36\) | 144 | \(3 \times 144 = 432\) |
| 17 | 3 | \(3 \times 17 = 51\) | 289 | \(3 \times 289 = 867\) |
| Total | \(\sum f_i = 22\) | \(\sum f_i x_i = 176\) | \(\sum f_i x_i^2 = 2048\) |
Step 2: Calculate the total frequency \(N\).
\[ N = \sum f_i = 3 + 2 + 3 + 2 + 6 + 3 + 3 = 22 \]
Step 3: Calculate the mean \(\bar{x}\).
From the table, we have \(\sum f_i x_i = 176\).
\[ \bar{x} = \frac{\sum f_i x_i}{N} = \frac{176}{22} = 8 \]
Step 4: Use the value of \(\sum f_i x_i^2\) from the table.
From the table, we have \(\sum f_i x_i^2 = 2048\).
Final Computation & Result:
Step 5: Substitute the calculated values into the variance formula.
\[ \sigma^2 = \frac{\sum f_i x_i^2}{N} - (\bar{x})^2 \] \[ \sigma^2 = \frac{2048}{22} - (8)^2 \] \[ \sigma^2 = \frac{1024}{11} - 64 \]
To subtract, we find a common denominator:
\[ \sigma^2 = \frac{1024 - 64 \times 11}{11} = \frac{1024 - 704}{11} \] \[ \sigma^2 = \frac{320}{11} \]
The variance \(\sigma^2\) of the data is \(\frac{320}{11}\) = 29.09.
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