To calculate the standard deviation for grouped data (using classes and their corresponding frequencies), we follow these steps:

Step 1: Determine the Midpoints of Each Class

The midpoints ($x_i$) are the average of the lower and upper bounds of each class:

• Class 0 to 20: $\frac{0 + 20}{2} = 10$
• Class 20 to 40: $\frac{20 + 40}{2} = 30$
• Class 40 to 60: $\frac{40 + 60}{2} = 50$
• Class 60 to 80: $\frac{60 + 80}{2} = 70$

So, the midpoints are $x_i = 10, 30, 50, 70$.

Step 2: Use the Frequencies Given

The frequencies ($f_i$) are: 5, 16, 14, 5.

Step 3: Compute the Mean ($\bar{x}$) of the Data

The formula for the mean is:

$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$

Let's calculate it step by step.

Step 4: Compute the Variance

Once the mean is calculated, the variance ($\sigma^2$) for grouped data is:

$\sigma^2 = \frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}$

Finally, the standard deviation ($\sigma$) is the square root of the variance.

I'll now calculate these values.

Summing the values:

• $\sum f_i = 5 + 16 + 14 + 5 = 40$

Let's calculate the mean and then proceed to find the variance and standard deviation.The mean ($\bar{x}$) of the data is 39.5, and the standard deviation ($\sigma$) is approximately 17.31.

Would you like more details or explanations on the process?

Here are five related questions to explore this topic further:

1. How do we calculate the variance before the standard deviation?
2. What is the importance of standard deviation in statistical analysis?
3. How does the size of class intervals affect the accuracy of standard deviation?
4. Can you explain the difference between standard deviation and mean deviation?
5. How would the standard deviation change if the frequencies were different?

Tip: The standard deviation helps to understand how spread out the values in a dataset are relative to the mean. A higher standard deviation indicates more variability.