Calculate the mean % error in five observations :
`80.0,80.5,81.0,81.5,82`

Options

A. `0.74%`
B. `1.74%`
C. `0.38%`
D. `1.38%`

Explanation

To calculate the mean percentage error in the given observations, we can follow these steps:

### Step 1: Calculate the Mean of the Observations
The observations provided are:
- 80.0
- 80.5
- 81.0
- 81.5
- 82.0

To find the mean, we sum all the observations and divide by the number of observations.

\[ \text{Mean} = \frac{(80.0 + 80.5 + 81.0 + 81.5 + 82.0)}{5} \]

Calculating the sum:
\[ 80.0 + 80.5 + 81.0 + 81.5 + 82.0 = 405.0 \]

Now, divide by 5:
\[ \text{Mean} = \frac{405.0}{5} = 81.0 \]

### Step 2: Calculate the Deviations from the Mean
Now, we will calculate the absolute deviations of each observation from the mean (81.0).

1. \( |80.0 - 81.0| = 1.0 \)
2. \( |80.5 - 81.0| = 0.5 \)
3. \( |81.0 - 81.0| = 0.0 \)
4. \( |81.5 - 81.0| = 0.5 \)
5. \( |82.0 - 81.0| = 1.0 \)

### Step 3: Sum of Absolute Deviations
Now, we sum the absolute deviations calculated in Step 2:

\[ \text{Total Deviation} = 1.0 + 0.5 + 0.0 + 0.5 + 1.0 = 3.0 \]

### Step 4: Calculate the Mean Absolute Deviation
To find the mean absolute deviation, we divide the total deviation by the number of observations:

\[ \text{Mean Absolute Deviation} = \frac{3.0}{5} = 0.6 \]

### Step 5: Calculate the Mean Percentage Error
Finally, to find the mean percentage error, we use the formula:

\[ \text{Mean Percentage Error} = \left( \frac{\text{Mean Absolute Deviation}}{\text{Mean}} \right) \times 100 \]

Substituting the values we calculated:

\[ \text{Mean Percentage Error} = \left( \frac{0.6}{81.0} \right) \times 100 \approx 0.7407\% \]

Rounding to two decimal places, we get:

\[ \text{Mean Percentage Error} \approx 0.74\% \]

### Final Answer
The mean percentage error in the five observations is approximately **0.74%**.

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