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Find the mean of this question..... By the help of step deviation method

To calculate the mean using the step-deviation method from the given data, we follow these steps:

### Step 1: Organize the Data
We have the following table:

| Size (X) | Frequency (f) |
|----------|---------------|
| 18       | 12            |
| 20       | 18            |
| 21       | 16            |
| 19       | 15            |
| 23       | 6             |
| 22       | 10            |
| 24       | 2             |

### Step 2: Choose an Assumed Mean (A)
Choose a value of size (X) from the data as the assumed mean (\(A\)). Let’s choose \(A = 21\).

### Step 3: Calculate the Step-Deviation
We compute the deviation from the assumed mean and the step-deviation as:

\[
d_i = X_i - A
\]

Then, we define a constant value \(h\) (a step size) that simplifies the calculations. Typically, \(h\) is the difference between two consecutive sizes. Here, \(h = 1\).

We will then calculate \(u_i = \frac{d_i}{h}\), which in this case is just \(d_i\).

| Size (X) | Frequency (f) | \(d_i = X_i - 21\) | \(u_i = \frac{d_i}{h}\) |
|----------|---------------|-------------------|------------------------|
| 18       | 12            | -3                | -3                     |
| 20       | 18            | -1                | -1                     |
| 21       | 16            | 0                 | 0                      |
| 19       | 15            | -2                | -2                     |
| 23       | 6             | 2                 | 2                      |
| 22       | 10            | 1                 | 1                      |
| 24       | 2             | 3                 | 3                      |

### Step 4: Calculate \( f_i u_i \)
Multiply the frequency by \(u_i\):

| Size (X) | Frequency (f) | \(u_i\) | \(f_i u_i\) |
|----------|---------------|---------|-------------|
| 18       | 12            | -3      | -36         |
| 20       | 18            | -1      | -18         |
| 21       | 16            | 0       | 0           |
| 19       | 15            | -2      | -30         |
| 23       | 6             | 2       | 12          |
| 22       | 10            | 1       | 10          |
| 24       | 2             | 3       | 6           |

### Step 5: Apply the Step-Deviation Formula for Mean
The formula for mean in the step-deviation method is:

\[
\bar{X} = A + h \times \frac{\sum f_i u_i}{\sum f_i}
\]

Where:
- \(A = 21\) (assumed mean),
- \(h = 1\) (step size),
- \(\sum f_i u_i = -56\) (sum of the products \(f_i u_i\)),
- \(\sum f_i = 79\) (sum of the frequencies).

Now, applying the values:

\[
\bar{X} = 21 + 1 \times \frac{-56}{79}
\]

\[
\bar{X} = 21 - 0.7089
\]

\[
\bar{X} \approx 20.29
\]

### Final Answer:
The mean of the given distribution is approximately \( \bar{X} = 20.29 \).

Would you like any further details or clarifications?

Here are some follow-up questions to enhance understanding:
1. How is the assumed mean method different from the direct method for calculating the mean?
2. What advantages does the step-deviation method offer when handling large datasets?
3. Can the choice of assumed mean affect the final result in any way?
4. How does the step size \(h\) simplify the computation?
5. What happens if you use a different value of \(h\)?

**Tip:** Always pick a value near the center of the data as the assumed mean to minimize the deviations and simplify calculations.

Find the median from the following distribution: Size - [18, 20, 21, 19, 23, 22, 24], Frequency - [12, 18, 16, 15, 6, 10, 2]

This example shows how to find the median from a frequency distribution with size values [18, 20, 21, 19, 23, 22, 24] and frequencies [12, 18, 16, 15, 6, 10, 2]. The solution involves applying the formula for the median in statistics.

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