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

Answer

To find the mean deviation about 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 (\( \bar{X} \)) is calculated using the formula:
\[ \bar{X} = \frac{X_1 + X_2 + ... + X_n}{n} \]
where \( n \) is the number of observations.

For our data:
\[ \bar{X} = \frac{38 + 70 + 48 + 40 + 42 + 55 + 63 + 46 + 54 + 44}{10} \]

Calculating the sum:
\[ 38 + 70 + 48 + 40 + 42 + 55 + 63 + 46 + 54 + 44 = 500 \]

Now, substituting back into the mean formula:
\[ \bar{X} = \frac{500}{10} = 50 \]

### Step 2: Calculate the Mean Deviation
The mean deviation is given by the formula:
\[ \text{Mean Deviation} = \frac{\sum_{i=1}^{n} |X_i - \bar{X}|}{n} \]

Now, we will calculate \( |X_i - \bar{X}| \) for each observation:
- For \( 38 \): \( |38 - 50| = 12 \)
- For \( 70 \): \( |70 - 50| = 20 \)
- For \( 48 \): \( |48 - 50| = 2 \)
- For \( 40 \): \( |40 - 50| = 10 \)
- For \( 42 \): \( |42 - 50| = 8 \)
- For \( 55 \): \( |55 - 50| = 5 \)
- For \( 63 \): \( |63 - 50| = 13 \)
- For \( 46 \): \( |46 - 50| = 4 \)
- For \( 54 \): \( |54 - 50| = 4 \)
- For \( 44 \): \( |44 - 50| = 6 \)

Now, summing these absolute deviations:
\[ 12 + 20 + 2 + 10 + 8 + 5 + 13 + 4 + 4 + 6 = 86 \]

Now, substituting into the mean deviation formula:
\[ \text{Mean Deviation} = \frac{86}{10} = 8.6 \]

### Final Answer
The mean deviation about the mean for the given data is \( 8.6 \).

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