The mean deviation of the data 3,10,10,4,7,10,5 from the mean is

Options

A. 2
B. 2.57
C. 3
D. 3.75

Explanation

To find the mean deviation of the data set \(3, 10, 10, 4, 7, 10, 5\) from the mean, we will follow these steps:

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

- Here, the data points are \(3, 10, 10, 4, 7, 10, 5\).
- The number of observations \( n = 7 \).
- The sum of the data points:
\[ 3 + 10 + 10 + 4 + 7 + 10 + 5 = 49 \]
- Now, substituting into the formula:
\[ \bar{x} = \frac{49}{7} = 7 \]

### Step 2: Calculate the Deviations (\( d_i \))
Next, we need to find the absolute deviations from the mean for each data point:
\[ d_i = |x_i - \bar{x}| \]

- For \( x_1 = 3 \): \( d_1 = |3 - 7| = 4 \)
- For \( x_2 = 10 \): \( d_2 = |10 - 7| = 3 \)
- For \( x_3 = 10 \): \( d_3 = |10 - 7| = 3 \)
- For \( x_4 = 4 \): \( d_4 = |4 - 7| = 3 \)
- For \( x_5 = 7 \): \( d_5 = |7 - 7| = 0 \)
- For \( x_6 = 10 \): \( d_6 = |10 - 7| = 3 \)
- For \( x_7 = 5 \): \( d_7 = |5 - 7| = 2 \)

### Step 3: Calculate the Mean Deviation
The mean deviation is calculated using the formula:
\[ \text{Mean Deviation} = \frac{\sum_{i=1}^{n} d_i}{n} \]

- Now, we sum the deviations:
\[ \sum d_i = 4 + 3 + 3 + 3 + 0 + 3 + 2 = 18 \]
- Now, substituting into the mean deviation formula:
\[ \text{Mean Deviation} = \frac{18}{7} \approx 2.57 \]

### Final Answer
The mean deviation of the data set \(3, 10, 10, 4, 7, 10, 5\) from the mean is approximately \(2.57\).

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