The mean deviation about the mean for the values
18, 20, 12, 14, 19, 22, 26, 16, 19, 24 is

Options

A. 3.1
B. 3.4
C. 3.2
D. 3.3

Answer

To find the mean deviation about the mean for the given values \(18, 20, 12, 14, 19, 22, 26, 16, 19, 24\), we will follow these steps:

### Step 1: Calculate the Mean (\( \bar{x} \))
The mean is calculated using the formula:
\[ \bar{x} = \frac{\sum x_i}{n} \]
where \(x_i\) are the values and \(n\) is the number of observations.

1. **Sum the values**:
\[ 18 + 20 + 12 + 14 + 19 + 22 + 26 + 16 + 19 + 24 = 18 + 20 + 12 + 14 + 19 + 22 + 26 + 16 + 19 + 24 = 19 + 20 + 12 + 14 + 19 + 22 + 26 + 16 + 19 + 24 = 190 \]

2. **Count the number of observations**:
\[ n = 10 \]

3. **Calculate the mean**:
\[ \bar{x} = \frac{190}{10} = 19 \]

### Step 2: Calculate the Absolute Deviations from the Mean
Next, we calculate the absolute deviations from the mean for each value:
\[ |x_i - \bar{x}| \]

1. **Calculate each deviation**:
- For \(x_1 = 18\): \(|18 - 19| = 1\)
- For \(x_2 = 20\): \(|20 - 19| = 1\)
- For \(x_3 = 12\): \(|12 - 19| = 7\)
- For \(x_4 = 14\): \(|14 - 19| = 5\)
- For \(x_5 = 19\): \(|19 - 19| = 0\)
- For \(x_6 = 22\): \(|22 - 19| = 3\)
- For \(x_7 = 26\): \(|26 - 19| = 7\)
- For \(x_8 = 16\): \(|16 - 19| = 3\)
- For \(x_9 = 19\): \(|19 - 19| = 0\)
- For \(x_{10} = 24\): \(|24 - 19| = 5\)

2. **List of absolute deviations**:
\[ 1, 1, 7, 5, 0, 3, 7, 3, 0, 5 \]

### Step 3: Calculate the Mean Deviation
Now, we calculate the mean deviation using the formula:
\[ \text{Mean Deviation} = \frac{\sum |x_i - \bar{x}|}{n} \]

1. **Sum the absolute deviations**:
\[ 1 + 1 + 7 + 5 + 0 + 3 + 7 + 3 + 0 + 5 = 32 \]

2. **Calculate the mean deviation**:
\[ \text{Mean Deviation} = \frac{32}{10} = 3.2 \]

### Final Answer
The mean deviation about the mean for the values is \(3.2\).

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