Question:
The variance of the data
\[
4,\;7,\;8,\;10,\;13,\;16,\;19
\]
is
The variance of the data
\[
4,\;7,\;8,\;10,\;13,\;16,\;19
\]
is
Show Hint
For discrete data,
\[
\sigma^2=\frac{\sum x^2}{n}-\left(\frac{\sum x}{n}\right)^2.
\]
This formula is usually faster than computing each deviation \((x-\bar{x})^2\) separately.
Updated On: Jul 9, 2026
- \(15\)
- \(24\)
- \(25\)
- \(28\) \bigskip
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The Correct Option is B
Solution and Explanation
Concept:
The variance of a data set is given by
\[
\sigma^2=\frac{\sum x_i^2}{n}-\left(\frac{\sum x_i}{n}\right)^2.
\]
That is,
\[
\text{Variance}
=
\text{Mean of squares}
-
(\text{Mean})^2.
\]
Step 1: Find the mean of the data. Given data: \[ 4,\;7,\;8,\;10,\;13,\;16,\;19 \] Number of observations: \[ n=7. \] Sum of observations: \[ \sum x = 4+7+8+10+13+16+19 = 77. \] Therefore, \[ \bar{x} = \frac{77}{7} = 11. \]
Step 2: Find the sum of squares. \[ \sum x^2 = 4^2+7^2+8^2+10^2+13^2+16^2+19^2. \] \[ = 16+49+64+100+169+256+361. \] \[ = 1015. \] Hence, \[ \frac{\sum x^2}{n} = \frac{1015}{7} = 145. \]
Step 3: Calculate the variance. Using \[ \sigma^2 = \frac{\sum x^2}{n} - \bar{x}^{\,2}, \] we get \[ \sigma^2 = 145-11^2. \] \[ = 145-121. \] \[ = 24. \]
Step 4: Write the final answer. \[ \boxed{24} \]
Step 1: Find the mean of the data. Given data: \[ 4,\;7,\;8,\;10,\;13,\;16,\;19 \] Number of observations: \[ n=7. \] Sum of observations: \[ \sum x = 4+7+8+10+13+16+19 = 77. \] Therefore, \[ \bar{x} = \frac{77}{7} = 11. \]
Step 2: Find the sum of squares. \[ \sum x^2 = 4^2+7^2+8^2+10^2+13^2+16^2+19^2. \] \[ = 16+49+64+100+169+256+361. \] \[ = 1015. \] Hence, \[ \frac{\sum x^2}{n} = \frac{1015}{7} = 145. \]
Step 3: Calculate the variance. Using \[ \sigma^2 = \frac{\sum x^2}{n} - \bar{x}^{\,2}, \] we get \[ \sigma^2 = 145-11^2. \] \[ = 145-121. \] \[ = 24. \]
Step 4: Write the final answer. \[ \boxed{24} \]
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