To find the mean of the given distribution using the Step Deviation Method, we follow these steps:

Step 1: Organize the data

We are given the following distribution of sizes and frequencies:

Step 2: Choose a suitable assumed mean (A)

Let's select $A = 21$ (since it's near the center of the data).

Step 3: Calculate the deviation (d)

We compute the deviation $d_i = x_i - A$ for each size $x_i$.

Step 4: Calculate the step deviation $u_i = \frac{d_i}{c}$

Choose a common factor $c = 1$ for simplicity (since the deviations are small).

Thus, $u_i = d_i$.

Step 5: Multiply $f_i$ and $u_i$

Now, multiply each frequency $f_i$ by the corresponding $u_i$:

Step 6: Calculate the sum of $f_i u_i$

$\sum f_i u_i = -36 - 18 + 0 - 30 + 12 + 10 + 6 = -56$

Step 7: Calculate the sum of frequencies $\sum f_i$

$\sum f_i = 12 + 18 + 16 + 15 + 6 + 10 + 2 = 79$

Step 8: Apply the Step Deviation Formula

The formula for the mean is: $\text{Mean} = A + c \cdot \frac{\sum f_i u_i}{\sum f_i}$

Substituting the values: $\text{Mean} = 21 + 1 \cdot \frac{-56}{79}$ $\text{Mean} = 21 - 0.709$ $\text{Mean} = 20.291$

Final Answer:

The mean of the distribution is approximately 20.29.

Would you like further clarification or details?

Here are 5 follow-up questions to deepen your understanding:

1. How does the choice of the assumed mean affect the calculations in the step deviation method?
2. What happens if you choose a different common factor $c$?
3. Can the step deviation method be used for data with larger deviations effectively?
4. Why is the step deviation method considered more efficient in some cases?
5. How would the result change if the frequencies were different?

Tip: Always choose the assumed mean close to the central values to minimize manual calculation errors in the step deviation method.