From the following data, using mean, calculate mean deviation and the coefficient of mean deviation.
21, 23, 25, 28, 30, 32, 38, 39, 46, 48

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the Mean
The mean (average) is calculated using the formula:

\[ \text{Mean} (X̄) = \frac{\text{Sum of observations}}{\text{Number of observations}} \]

Given data: 21, 23, 25, 28, 30, 32, 38, 39, 46, 48

1. **Sum of observations**:
\[ 21 + 23 + 25 + 28 + 30 + 32 + 38 + 39 + 46 + 48 = 330 \]

2. **Number of observations**:
There are 10 observations.

3. **Calculate the mean**:
\[ X̄ = \frac{330}{10} = 33 \]

### Step 2: Calculate the Mean Deviation
The mean deviation (MD) is calculated using the formula:

\[ \text{Mean Deviation} (MD) = \frac{\sum |X_i - X̄|}{N} \]

Where \(X_i\) are the individual observations, \(X̄\) is the mean, and \(N\) is the number of observations.

1. **Calculate the absolute deviations**:
- For \(X_1 = 21\): \(|21 - 33| = 12\)
- For \(X_2 = 23\): \(|23 - 33| = 10\)
- For \(X_3 = 25\): \(|25 - 33| = 8\)
- For \(X_4 = 28\): \(|28 - 33| = 5\)
- For \(X_5 = 30\): \(|30 - 33| = 3\)
- For \(X_6 = 32\): \(|32 - 33| = 1\)
- For \(X_7 = 38\): \(|38 - 33| = 5\)
- For \(X_8 = 39\): \(|39 - 33| = 6\)
- For \(X_9 = 46\): \(|46 - 33| = 13\)
- For \(X_{10} = 48\): \(|48 - 33| = 15\)

2. **Sum of absolute deviations**:
\[ 12 + 10 + 8 + 5 + 3 + 1 + 5 + 6 + 13 + 15 = 78 \]

3. **Calculate the mean deviation**:
\[ MD = \frac{78}{10} = 7.8 \]

### Step 3: Calculate the Coefficient of Mean Deviation
The coefficient of mean deviation is calculated using the formula:

\[ \text{Coefficient of Mean Deviation} = \frac{MD}{X̄} \]

1. **Substituting the values**:
\[ \text{Coefficient of Mean Deviation} = \frac{7.8}{33} \approx 0.236 \]

### Final Answers:
- Mean: \(33\)
- Mean Deviation: \(7.8\)
- Coefficient of Mean Deviation: \(0.236\)

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