Using this data

Interval.               Frequency

1-2.                         40

3-4.                         56

5-6.                         68

7-8.                         75

9-10.                       51

11-12.                     39

13-14.                     26

 Calculate

a- the standard deviation

b- Variance

c- Mean deviation

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.

Using this data

Interval.               Frequency

1-2.                         40

3-4.                         56

5-6.                         68

7-8.                         75

9-10.                       51

11-12.                     39

13-14.                     26

Calculate:
a- the standard deviation
b- Variance
c- Mean deviation

Learn how to calculate the standard deviation, variance, and mean deviation from a frequency distribution with intervals 1-14. This problem covers key statistical concepts and includes step-by-step solutions. Suitable for high school students in Grades 10-12.

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