Find the mean deviation from the mean for the following data :
6.5,5,5.25,5.5,4.75,4.5,6.25,7.75,8.5

Answer

To find the mean deviation from the mean for the given data set, we will follow these steps:

### Step 1: Calculate the Mean
First, we need to find the mean (average) of the data set. The data given is:
\[ 6.5, 5, 5.25, 5.5, 4.75, 4.5, 6.25, 7.75, 8.5 \]

**Calculation:**
1. Sum of the data:
\[ 6.5 + 5 + 5.25 + 5.5 + 4.75 + 4.5 + 6.25 + 7.75 + 8.5 = 54.5 \]
2. Count of observations (n):
\[ n = 9 \]
3. Mean (\( \bar{x} \)):
\[ \bar{x} = \frac{\text{Sum of data}}{n} = \frac{54.5}{9} = 6.0556 \quad (\text{approximately}) \]

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

**Deviations:**
- For \( 6.5 \): \( |6.5 - 6.0556| = 0.4444 \)
- For \( 5 \): \( |5 - 6.0556| = 1.0556 \)
- For \( 5.25 \): \( |5.25 - 6.0556| = 0.8056 \)
- For \( 5.5 \): \( |5.5 - 6.0556| = 0.5556 \)
- For \( 4.75 \): \( |4.75 - 6.0556| = 1.3056 \)
- For \( 4.5 \): \( |4.5 - 6.0556| = 1.5556 \)
- For \( 6.25 \): \( |6.25 - 6.0556| = 0.1944 \)
- For \( 7.75 \): \( |7.75 - 6.0556| = 1.6944 \)
- For \( 8.5 \): \( |8.5 - 6.0556| = 2.4444 \)

### Step 3: Calculate the Mean Deviation
Now, we will sum all the absolute deviations and divide by the number of observations to find the mean deviation.

**Sum of Deviations:**
\[ 0.4444 + 1.0556 + 0.8056 + 0.5556 + 1.3056 + 1.5556 + 0.1944 + 1.6944 + 2.4444 = 10.0552 \]

**Mean Deviation:**
\[ \text{Mean Deviation} = \frac{\text{Sum of Deviations}}{n} = \frac{10.0552}{9} \approx 1.1173 \]

### Final Answer
The mean deviation from the mean for the given data is approximately **1.12**.

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