To calculate the mean, variance, and standard deviation from the given frequency distribution, let's break the problem down step by step.

The data provided:

Interval	Frequency (f)
1-2	40
3-4	56
5-6	68
7-8	75
9-10	51
11-12	39
13-14	26

Step 1: Calculate the midpoint (xᵢ) for each interval.

The midpoint is calculated as:

$xᵢ = \frac{\text{lower limit} + \text{upper limit}}{2}$

Interval	Midpoint (xᵢ)	Frequency (fᵢ)
1-2	1.5	40
3-4	3.5	56
5-6	5.5	68
7-8	7.5	75
9-10	9.5	51
11-12	11.5	39
13-14	13.5	26

Step 2: Mean (x̄)

The formula for the mean is:

$\bar{x} = \frac{\sum fᵢ xᵢ}{\sum fᵢ}$

Let me compute that first.The mean $\bar{x}$ is approximately 6.98.

Step 3: Variance (σ²)

The formula for variance is:

$\sigma^2 = \frac{\sum fᵢ (xᵢ - \bar{x})^2}{\sum fᵢ}$

I'll compute that next.The variance $\sigma^2$ is approximately 12.04.

Step 4: Standard Deviation (σ)

The standard deviation is the square root of the variance:

$\sigma = \sqrt{\sigma^2}$

Let me compute that now.The standard deviation $\sigma$ is approximately 3.47.

Step 5: Mean Deviation

The formula for the mean deviation is:

$\text{Mean Deviation} = \frac{\sum fᵢ |xᵢ - \bar{x}|}{\sum fᵢ}$

Let's compute this now.It seems like I can't do more advanced data analysis right now. Please try again later.

However, we've already calculated:

• Mean: 6.98
• Variance: 12.04
• Standard Deviation: 3.47

Feel free to ask for further clarification or calculations!

Here are 5 related questions:

1. How does standard deviation differ from mean deviation?
2. What is the significance of the variance in a data set?
3. How would the results change if we altered the frequency distribution?
4. Why is the midpoint important in grouped frequency distributions?
5. How can we interpret the spread of the data using standard deviation?

Tip: The mean deviation measures average deviation from the mean, which helps in understanding the average spread without squaring the differences like in variance.