The mean deviation about the mean of the following distribution is
`{:("Size",20,21,22,23,24),("Frequency",6,4,5,1,4):}`

Options

A. 1
B. 1.25
C. 1.5
D. 1.75

Explanation

To find the mean deviation about the mean for the given distribution, we will follow these steps:

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

| Size (Xi) | Frequency (Fi) |
|-----------|----------------|
| 20 | 6 |
| 21 | 4 |
| 22 | 5 |
| 23 | 1 |
| 24 | 4 |

### Step 2: Calculate Fi * Xi
Next, we calculate the product of frequency and size for each class:

- For Xi = 20: Fi * Xi = 6 * 20 = 120
- For Xi = 21: Fi * Xi = 4 * 21 = 84
- For Xi = 22: Fi * Xi = 5 * 22 = 110
- For Xi = 23: Fi * Xi = 1 * 23 = 23
- For Xi = 24: Fi * Xi = 4 * 24 = 96

Now, we can summarize this in a table:

| Size (Xi) | Frequency (Fi) | Fi * Xi |
|-----------|----------------|---------|
| 20 | 6 | 120 |
| 21 | 4 | 84 |
| 22 | 5 | 110 |
| 23 | 1 | 23 |
| 24 | 4 | 96 |
| **Total** | **20** | **483** |

### Step 3: Calculate the Mean (X̄)
To find the mean (X̄), we use the formula:

\[ X̄ = \frac{\sum (Fi * Xi)}{\sum Fi} \]

Substituting the values:

\[ X̄ = \frac{483}{20} = 24.15 \]

### Step 4: Calculate Deviations (Di)
Now we calculate the deviations from the mean (X̄):

- For Xi = 20: Di = 20 - 21.65 = -1.65
- For Xi = 21: Di = 21 - 21.65 = -0.65
- For Xi = 22: Di = 22 - 21.65 = 0.35
- For Xi = 23: Di = 23 - 21.65 = 1.35
- For Xi = 24: Di = 24 - 21.65 = 2.35

### Step 5: Calculate Fi * |Di|
Next, we calculate the absolute deviations multiplied by frequency:

| Size (Xi) | Frequency (Fi) | |Di| | Fi * |Di| |
|-----------|----------------|-------|---------|
| 20 | 6 | 1.65 | 9.90 |
| 21 | 4 | 0.65 | 2.60 |
| 22 | 5 | 0.35 | 1.75 |
| 23 | 1 | 1.35 | 1.35 |
| 24 | 4 | 2.35 | 9.40 |
| **Total** | **20** | | **25.00** |

### Step 6: Calculate Mean Deviation
Finally, we calculate the mean deviation about the mean using the formula:

\[ \text{Mean Deviation} = \frac{\sum (Fi * |Di|)}{\sum Fi} \]

Substituting the values:

\[ \text{Mean Deviation} = \frac{25.00}{20} = 1.25 \]

### Conclusion
The mean deviation about the mean of the given distribution is **1.25**.

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