To find the standard deviation using the direct method, we need to follow these steps:

Step 1: Find the Midpoints ($x$) of Each Class Interval

The midpoint of each class interval is calculated as: $x = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}$ For the given intervals:

• $0-10: \frac{0 + 10}{2} = 5$
• $10-20: \frac{10 + 20}{2} = 15$
• $20-30: \frac{20 + 30}{2} = 25$
• $30-40: \frac{30 + 40}{2} = 35$
• $40-50: \frac{40 + 50}{2} = 45$

Step 2: Set Up the Table

We create a table for $x$, $f$, $fx$, and $(x - \bar{x})^2f$, where:

• $f$: Frequency
• $fx = f \times x$
• $(x - \bar{x})^2f$: Squared deviation multiplied by the frequency

Class Interval	$x$	$f$	$fx$	$(x - \bar{x})^2f$
$0-10$	$5$	$10$	$50$
$10-20$	$15$	$3$	$45$
$20-30$	$25$	$2$	$50$
$30-40$	$35$	$1$	$35$
$40-50$	$45$	$4$	$180$

Step 3: Find $\bar{x}$ (Mean)

$\bar{x} = \frac{\Sigma fx}{\Sigma f}$ $\Sigma f = 10 + 3 + 2 + 1 + 4 = 20$ $\Sigma fx = 50 + 45 + 50 + 35 + 180 = 360$ $\bar{x} = \frac{360}{20} = 18$

Step 4: Calculate $(x - \bar{x})^2f$

Now compute $(x - \bar{x})^2$ and $(x - \bar{x})^2f$ for each class.

Class Interval	$x$	$f$	$fx$	$x - \bar{x}$	$(x - \bar{x})^2$	$(x - \bar{x})^2f$
$0-10$	$5$	$10$	$50$	$-13$	$169$	$1690$
$10-20$	$15$	$3$	$45$	$-3$	$9$	$27$
$20-30$	$25$	$2$	$50$	$7$	$49$	$98$
$30-40$	$35$	$1$	$35$	$17$	$289$	$289$
$40-50$	$45$	$4$	$180$	$27$	$729$	$2916$

$\Sigma \left((x - \bar{x})^2f\right) = 1690 + 27 + 98 + 289 + 2916 = 5020$

Step 5: Calculate Standard Deviation

The formula for standard deviation is: $\sigma = \sqrt{\frac{\Sigma (x - \bar{x})^2f}{\Sigma f}}$ $\sigma = \sqrt{\frac{5020}{20}}$ $\sigma = \sqrt{251} \approx 15.84$

Final Answer:

The standard deviation is approximately 15.84.

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Would you like detailed explanations or assistance with related concepts?

Relative Questions:

1. How is the direct method for standard deviation different from the assumed mean method?
2. What are the advantages of calculating midpoints in grouped data?
3. How do changes in class intervals affect standard deviation?
4. Can the direct method be applied to continuous data as well?
5. How does the frequency distribution shape affect the standard deviation?

Tip:

Always ensure to use consistent class intervals to avoid bias in statistical calculations.