The mean deviation from the mean for the data : 9,12,18,3,5,3,10,12,21,12,21 is ______

Answer

To find the mean deviation from the mean for the given data set: 9, 12, 18, 3, 5, 3, 10, 12, 21, 12, 21, we will follow these steps:

### Step 1: Calculate the Mean
First, we need to calculate the mean (\( \bar{x} \)) of the data.

1. **Sum of the data**:
\[ 9 + 12 + 18 + 3 + 5 + 3 + 10 + 12 + 21 + 12 + 21 = 126 \]

2. **Number of observations (n)**:
There are 11 data points.

3. **Mean**:
\[ \bar{x} = \frac{\text{Sum of data}}{n} = \frac{126}{11} \approx 11.45 \]

### Step 2: Calculate the Absolute Deviations from the Mean
Next, we calculate the absolute deviations from the mean for each data point.

1. **Calculate \( |x_i - \bar{x}| \)** for each \( x_i \):
- For \( x_1 = 9 \): \( |9 - 11.45| = 2.45 \)
- For \( x_2 = 12 \): \( |12 - 11.45| = 0.55 \)
- For \( x_3 = 18 \): \( |18 - 11.45| = 6.55 \)
- For \( x_4 = 3 \): \( |3 - 11.45| = 8.45 \)
- For \( x_5 = 5 \): \( |5 - 11.45| = 6.45 \)
- For \( x_6 = 3 \): \( |3 - 11.45| = 8.45 \)
- For \( x_7 = 10 \): \( |10 - 11.45| = 1.45 \)
- For \( x_8 = 12 \): \( |12 - 11.45| = 0.55 \)
- For \( x_9 = 21 \): \( |21 - 11.45| = 9.55 \)
- For \( x_{10} = 12 \): \( |12 - 11.45| = 0.55 \)
- For \( x_{11} = 21 \): \( |21 - 11.45| = 9.55 \)

### Step 3: Sum of Absolute Deviations
Now, we sum all the absolute deviations calculated in the previous step.

\[ \text{Total Absolute Deviations} = 2.45 + 0.55 + 6.55 + 8.45 + 6.45 + 8.45 + 1.45 + 0.55 + 9.55 + 0.55 + 9.55 = 54.55 \]

### Step 4: Calculate the Mean Deviation
Finally, we calculate the mean deviation by dividing the total absolute deviations by the number of observations.

\[ \text{Mean Deviation} = \frac{\text{Total Absolute Deviations}}{n} = \frac{54.55}{11} \approx 4.95 \]

### Final Answer:
The mean deviation from the mean for the given data is approximately **4.95**.

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