Find the mean deviation using arithmetic mean for the following observations:
(a) 68,32,49,54,21,38,59,41,66,76
(b) 28,12,17,35,22,18,5,32

Explanation

To find the mean deviation using the arithmetic mean for the given observations, we will follow these steps for both parts of the question.

### Part (a): Observations: 68, 32, 49, 54, 21, 38, 59, 41, 66, 76

**Step 1: Calculate the Mean (x̄)**
The mean is calculated using the formula:
\[ x̄ = \frac{\sum x}{n} \]
Where \( \sum x \) is the sum of all observations and \( n \) is the number of observations.

- Sum of observations:
\[ 68 + 32 + 49 + 54 + 21 + 38 + 59 + 41 + 66 + 76 = 504 \]

- Number of observations (n):
\[ n = 10 \]

- Mean (x̄):
\[ x̄ = \frac{504}{10} = 50.4 \]

**Step 2: Calculate the Absolute Deviations (|x - x̄|)**
Now we will find the absolute deviations of each observation from the mean:
\[ |x - x̄| \]

- \( |68 - 50.4| = 17.6 \)
- \( |32 - 50.4| = 18.4 \)
- \( |49 - 50.4| = 1.4 \)
- \( |54 - 50.4| = 3.6 \)
- \( |21 - 50.4| = 29.4 \)
- \( |38 - 50.4| = 12.4 \)
- \( |59 - 50.4| = 8.6 \)
- \( |41 - 50.4| = 9.4 \)
- \( |66 - 50.4| = 15.6 \)
- \( |76 - 50.4| = 25.6 \)

**Step 3: Calculate the Sum of Absolute Deviations**
Now, we sum all the absolute deviations:
\[ \sum |x - x̄| = 17.6 + 18.4 + 1.4 + 3.6 + 29.4 + 12.4 + 8.6 + 9.4 + 15.6 + 25.6 = 142 \]

**Step 4: Calculate the Mean Deviation (MD)**
Finally, we calculate the mean deviation using the formula:
\[ MD = \frac{\sum |x - x̄|}{n} \]
\[ MD = \frac{142}{10} = 14.2 \]

### Part (b): Observations: 28, 12, 17, 35, 22, 18, 5, 32

**Step 1: Calculate the Mean (x̄)**
Using the same formula:
\[ x̄ = \frac{\sum x}{n} \]

- Sum of observations:
\[ 28 + 12 + 17 + 35 + 22 + 18 + 5 + 32 = 169 \]

- Number of observations (n):
\[ n = 8 \]

- Mean (x̄):
\[ x̄ = \frac{169}{8} = 21.125 \]

**Step 2: Calculate the Absolute Deviations (|x - x̄|)**
Now we will find the absolute deviations of each observation from the mean:
\[ |x - x̄| \]

- \( |28 - 21.125| = 6.875 \)
- \( |12 - 21.125| = 9.125 \)
- \( |17 - 21.125| = 4.125 \)
- \( |35 - 21.125| = 13.875 \)
- \( |22 - 21.125| = 0.875 \)
- \( |18 - 21.125| = 3.125 \)
- \( |5 - 21.125| = 16.125 \)
- \( |32 - 21.125| = 10.875 \)

**Step 3: Calculate the Sum of Absolute Deviations**
Now, we sum all the absolute deviations:
\[ \sum |x - x̄| = 6.875 + 9.125 + 4.125 + 13.875 + 0.875 + 3.125 + 16.125 + 10.875 = 65 \]

**Step 4: Calculate the Mean Deviation (MD)**
Finally, we calculate the mean deviation:
\[ MD = \frac{\sum |x - x̄|}{n} \]
\[ MD = \frac{65}{8} = 8.125 \]

### Final Answers:
- Mean Deviation for Part (a): **14.2**
- Mean Deviation for Part (b): **8.125**