Calculate the mean of the distribution, given below, using the short cut method:

Answer

To calculate the mean of the given distribution using the shortcut method, follow these steps:

### Step 1: Create a Frequency Distribution Table
First, we need to set up a table with the class intervals and their corresponding frequencies.

| Class Interval | Frequency (F) |
|----------------|---------------|
| 11 - 20 | 2 |
| 21 - 30 | 6 |
| 31 - 40 | 10 |
| 41 - 50 | 12 |
| 51 - 60 | 9 |
| 61 - 70 | 7 |
| 71 - 80 | 4 |

### Step 2: Calculate the Mid Values (X)
Next, we calculate the mid value for each class interval using the formula:
\[ \text{Mid Value} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \]

| Class Interval | Mid Value (X) |
|----------------|----------------|
| 11 - 20 | 15.5 |
| 21 - 30 | 25.5 |
| 31 - 40 | 35.5 |
| 41 - 50 | 45.5 |
| 51 - 60 | 55.5 |
| 61 - 70 | 65.5 |
| 71 - 80 | 75.5 |

### Step 3: Assume a Mean (A)
We need to assume a mean (A). The middle class interval is 41 - 50, so we take the mid value of this class as our assumed mean:
\[ A = 45.5 \]

### Step 4: Calculate the Deviation (D)
Now, we calculate the deviation (D) for each mid value using the formula:
\[ D = X - A \]

| Mid Value (X) | Deviation (D = X - A) |
|----------------|------------------------|
| 15.5 | -30 |
| 25.5 | -20 |
| 35.5 | -10 |
| 45.5 | 0 |
| 55.5 | 10 |
| 65.5 | 20 |
| 75.5 | 30 |

### Step 5: Calculate FD (Frequency × Deviation)
Next, we calculate the product of frequency (F) and deviation (D) for each class.

| Frequency (F) | Deviation (D) | FD (F × D) |
|----------------|----------------|------------|
| 2 | -30 | -60 |
| 6 | -20 | -120 |
| 10 | -10 | -100 |
| 12 | 0 | 0 |
| 9 | 10 | 90 |
| 7 | 20 | 140 |
| 4 | 30 | 120 |

### Step 6: Sum of FD and Sum of Frequencies
Now, we sum up the FD column and the frequency column.

- Sum of FD = -60 - 120 - 100 + 0 + 90 + 140 + 120 = 70
- Sum of Frequencies (ΣF) = 2 + 6 + 10 + 12 + 9 + 7 + 4 = 50

### Step 7: Calculate the Mean
Finally, we can calculate the mean using the shortcut formula:
\[ \text{Mean} = A + \frac{\Sigma FD}{\Sigma F} \]

Substituting the values we have:
\[ \text{Mean} = 45.5 + \frac{70}{50} \]
\[ \text{Mean} = 45.5 + 1.4 = 46.9 \]

### Final Answer
The mean of the distribution is **46.9**.

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