Find the mean deviation from the mean for the following data:
38, 70, 48, 40, 42, 55, 63, 46, 54, 44

Answer

To find the mean deviation from the mean for the given data set: 38, 70, 48, 40, 42, 55, 63, 46, 54, 44, we will follow these steps:

### Step 1: Calculate the Mean

The mean (μ) is calculated using the formula:

\[ \text{Mean} = \frac{\sum x_i}{n} \]

Where \(x_i\) are the data points and \(n\) is the number of data points.

1. **Sum of the data points**:
\[ 38 + 70 + 48 + 40 + 42 + 55 + 63 + 46 + 54 + 44 = 500 \]

2. **Number of data points (n)**:
\[ n = 10 \]

3. **Calculate the mean**:
\[ \text{Mean} = \frac{500}{10} = 50 \]

### Step 2: Calculate the Deviations from the Mean

Next, we calculate the absolute deviations from the mean for each data point:

\[ d_i = |x_i - \mu| \]

Where \(d_i\) is the deviation for each data point.

1. **Calculating deviations**:
- For 38: \(d_1 = |38 - 50| = 12\)
- For 70: \(d_2 = |70 - 50| = 20\)
- For 48: \(d_3 = |48 - 50| = 2\)
- For 40: \(d_4 = |40 - 50| = 10\)
- For 42: \(d_5 = |42 - 50| = 8\)
- For 55: \(d_6 = |55 - 50| = 5\)
- For 63: \(d_7 = |63 - 50| = 13\)
- For 46: \(d_8 = |46 - 50| = 4\)
- For 54: \(d_9 = |54 - 50| = 4\)
- For 44: \(d_{10} = |44 - 50| = 6\)

### Step 3: Calculate the Mean Deviation

The mean deviation (MD) is calculated using the formula:

\[ \text{Mean Deviation} = \frac{\sum d_i}{n} \]

1. **Sum of the deviations**:
\[ \sum d_i = 12 + 20 + 2 + 10 + 8 + 5 + 13 + 4 + 4 + 6 = 84 \]

2. **Calculate the mean deviation**:
\[ \text{Mean Deviation} = \frac{84}{10} = 8.4 \]

### Final Answer

The mean deviation from the mean for the given data is **8.4**.

---