To calculate the standard deviation of grouped data, follow these steps:

Formula for Standard Deviation of Grouped Data:

$\sigma = \sqrt{\frac{\sum f(x - \bar{x})^2}{\sum f}}$ Where:

• $f$ = frequency of each group.
• $x$ = mid-point of each group.
• $\bar{x}$ = mean of the data.

Alternatively, if the data is large and a deviation method is preferred: $\sigma = \sqrt{\frac{\sum f d^2}{\sum f} - \left(\frac{\sum f d}{\sum f}\right)^2}$ Where:

• $d = x - A$, the deviation from an assumed mean $A$.

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Steps:

1. Calculate Midpoints ($x$): For each class interval, calculate the midpoint: $x = \frac{\text{Lower Bound} + \text{Upper Bound}}{2}$

2. Find Deviations ($x - \bar{x}$): Compute the deviation of each midpoint from the mean ($x - \bar{x}$) or from an assumed mean.

3. Square the Deviations ($(x - \bar{x})^2$): Square each deviation.

4. Multiply by Frequency ($f \times (x - \bar{x})^2$): Multiply each squared deviation by its corresponding frequency.

5. Compute the Mean of Squared Deviations: Divide the sum of the squared deviations multiplied by frequency by the total frequency.

6. Take the Square Root: The square root of the result is the standard deviation.

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Example:

Class Interval	Frequency ($f$)	Midpoint ($x$)	$x - \bar{x}$	$(x - \bar{x})^2$	$f(x - \bar{x})^2$
0 - 10	5	5	$-5$	25	125
10 - 20	8	15	5	25	200
...	...	...	...	...	...

Computation:

1. Calculate the mean ($\bar{x}$).
2. Use the deviations and proceed with the steps.

Would you like assistance with a specific dataset? Feel free to share! 😊

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Related Questions:

1. What is the significance of standard deviation in data analysis?
2. How does standard deviation differ for grouped vs. ungrouped data?
3. What are the implications of a low vs. high standard deviation?
4. How does assuming a mean simplify grouped standard deviation calculation?
5. What are some common errors in calculating grouped standard deviation?

Tip:

Always double-check the sum of frequencies ($\sum f$) to ensure accurate calculations.