Calculate the mean deviation from mean for the following data. 
Class Interval : 2-4 4-6 6-8 8-10 
Frequency : 3 4 2 1

To calculate the mean deviation from the mean, follow these steps:

### Step 1: Determine the Class Midpoints
For each class interval, calculate the midpoint (\(x_i\)). The midpoint is calculated as:
\[
x_i = \frac{\text{Lower limit} + \text{Upper limit}}{2}
\]

\[
\begin{array}{|c|c|c|}
\hline
\text{Class Interval} & \text{Midpoint} (x_i) & \text{Frequency} (f_i) \\
\hline
2-4 & \frac{2+4}{2} = 3 & 3 \\
4-6 & \frac{4+6}{2} = 5 & 4 \\
6-8 & \frac{6+8}{2} = 7 & 2 \\
8-10 & \frac{8+10}{2} = 9 & 1 \\
\hline
\end{array}
\]

### Step 2: Calculate the Mean (\(\overline{x}\))
The mean (\(\overline{x}\)) is calculated using the formula:
\[
\overline{x} = \frac{\sum (f_i \times x_i)}{\sum f_i}
\]

First, calculate \(f_i \times x_i\) for each class:

\[
\begin{array}{|c|c|c|c|}
\hline
\text{Class Interval} & x_i & f_i & f_i \times x_i \\
\hline
2-4 & 3 & 3 & 3 \times 3 = 9 \\
4-6 & 5 & 4 & 5 \times 4 = 20 \\
6-8 & 7 & 2 & 7 \times 2 = 14 \\
8-10 & 9 & 1 & 9 \times 1 = 9 \\
\hline
\end{array}
\]

Now sum the \(f_i \times x_i\) values and the frequencies \(f_i\):

\[
\sum (f_i \times x_i) = 9 + 20 + 14 + 9 = 52
\]
\[
\sum f_i = 3 + 4 + 2 + 1 = 10
\]

Thus, the mean (\(\overline{x}\)) is:
\[
\overline{x} = \frac{52}{10} = 5.2
\]

### Step 3: Calculate the Deviations from the Mean
For each class, calculate the absolute deviation from the mean \(|x_i - \overline{x}|\) and then multiply by the frequency \(f_i\).

\[
\begin{array}{|c|c|c|c|c|}
\hline
\text{Class Interval} & x_i & f_i & |x_i - \overline{x}| & f_i \times |x_i - \overline{x}| \\
\hline
2-4 & 3 & 3 & |3 - 5.2| = 2.2 & 3 \times 2.2 = 6.6 \\
4-6 & 5 & 4 & |5 - 5.2| = 0.2 & 4 \times 0.2 = 0.8 \\
6-8 & 7 & 2 & |7 - 5.2| = 1.8 & 2 \times 1.8 = 3.6 \\
8-10 & 9 & 1 & |9 - 5.2| = 3.8 & 1 \times 3.8 = 3.8 \\
\hline
\end{array}
\]

### Step 4: Calculate the Mean Deviation from the Mean
The mean deviation from the mean is calculated using the formula:
\[
\text{Mean Deviation} = \frac{\sum (f_i \times |x_i - \overline{x}|)}{\sum f_i}
\]

Sum the \(f_i \times |x_i - \overline{x}|\) values:
\[
\sum (f_i \times |x_i - \overline{x}|) = 6.6 + 0.8 + 3.6 + 3.8 = 14.8
\]

Now, calculate the mean deviation:
\[
\text{Mean Deviation} = \frac{14.8}{10} = 1.48
\]

### Final Answer:
The mean deviation from the mean is **1.48**.

---

Do you want more details or have any questions?

Here are five related questions:
1. How would the mean deviation change if the frequencies were doubled?
2. What is the significance of mean deviation in data analysis?
3. Can you calculate the standard deviation for the same data?
4. How would the mean deviation be affected if all data points were shifted by a constant value?
5. What is the difference between mean deviation and median deviation?

**Tip:** The mean deviation gives a sense of the average amount by which data points differ from the mean, offering a simple measure of variability.

Learn how to calculate the mean deviation from the mean for given data with class intervals and frequencies. This problem covers statistical concepts including mean calculation and mean deviation. Suitable for students in Grades 9-12.

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