For the following, calculate mean deviation and coefficient of mean deviation.
3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21

Answer

To calculate the mean deviation and the coefficient of mean deviation for the given data set \(3, 9, 5, 3, 12, 10, 18, 4, 7, 19, 21\), we can follow these steps:

### Step 1: Calculate the Mean
The mean (average) is calculated using the formula:
\[ \text{Mean} (A) = \frac{\Sigma x}{n} \]
where \(\Sigma x\) is the sum of all data points and \(n\) is the number of data points.

1. **Sum the data points**:
\[ \Sigma x = 3 + 9 + 5 + 3 + 12 + 10 + 18 + 4 + 7 + 19 + 21 = 111 \]
2. **Count the number of data points**:
\[ n = 11 \]
3. **Calculate the mean**:
\[ A = \frac{111}{11} = 10.09 \]

### Step 2: Calculate the Deviation from the Mean
The deviation for each data point is calculated as:
\[ D = |x - A| \]
where \(x\) is each individual data point.

1. Calculate the absolute deviations:
- For \(3\): \( |3 - 10.09| = 7.09 \)
- For \(9\): \( |9 - 10.09| = 1.09 \)
- For \(5\): \( |5 - 10.09| = 5.09 \)
- For \(3\): \( |3 - 10.09| = 7.09 \)
- For \(12\): \( |12 - 10.09| = 1.91 \)
- For \(10\): \( |10 - 10.09| = 0.09 \)
- For \(18\): \( |18 - 10.09| = 7.91 \)
- For \(4\): \( |4 - 10.09| = 6.09 \)
- For \(7\): \( |7 - 10.09| = 3.09 \)
- For \(19\): \( |19 - 10.09| = 8.91 \)
- For \(21\): \( |21 - 10.09| = 10.91 \)

### Step 3: Calculate the Mean Deviation
The mean deviation is calculated using the formula:
\[ \text{Mean Deviation} = \frac{\Sigma D}{n} \]
where \(\Sigma D\) is the sum of the absolute deviations.

1. **Sum the absolute deviations**:
\[ \Sigma D = 7.09 + 1.09 + 5.09 + 7.09 + 1.91 + 0.09 + 7.91 + 6.09 + 3.09 + 8.91 + 10.91 = 59.27 \]
2. **Calculate the mean deviation**:
\[ \text{Mean Deviation} = \frac{59.27}{11} \approx 5.39 \]

### Step 4: Calculate the Coefficient of Mean Deviation
The coefficient of mean deviation is calculated using the formula:
\[ \text{Coefficient of Mean Deviation} = \frac{\text{Mean Deviation}}{\text{Mean}} \]
1. **Calculate the coefficient**:
\[ \text{Coefficient of Mean Deviation} = \frac{5.39}{10.09} \approx 0.53 \]

### Final Results:
- Mean Deviation: \( \approx 5.39 \)
- Coefficient of Mean Deviation: \( \approx 0.53 \)