Calculate the mean deviation (about mean) and standard deviation of the following data:
25,50,48,70,45,33,51,31,60

Answer

To solve the problem of calculating the mean deviation (about the mean) and standard deviation for the given data set: 25, 50, 48, 70, 45, 33, 51, 31, 60, we will follow these steps:

### Step 1: Calculate the Mean
First, we need to find the mean (average) of the data set.

\[ \text{Mean} (\bar{x}) = \frac{\sum x_i}{n} \]

Where \( n \) is the number of observations.

\[ \sum x_i = 25 + 50 + 48 + 70 + 45 + 33 + 51 + 31 + 60 = 413 \]

\[ n = 9 \]

\[ \bar{x} = \frac{413}{9} \approx 45.89 \]

### Step 2: Calculate the Mean Deviation
Next, we calculate the mean deviation about the mean. The mean deviation is given by:

\[ \text{Mean Deviation} = \frac{\sum |x_i - \bar{x}|}{n} \]

We will calculate \( |x_i - \bar{x}| \) for each observation:

- \( |25 - 45.89| = 20.89 \)
- \( |50 - 45.89| = 4.11 \)
- \( |48 - 45.89| = 2.11 \)
- \( |70 - 45.89| = 24.11 \)
- \( |45 - 45.89| = 0.89 \)
- \( |33 - 45.89| = 12.89 \)
- \( |51 - 45.89| = 5.11 \)
- \( |31 - 45.89| = 14.89 \)
- \( |60 - 45.89| = 14.11 \)

Now, we sum these absolute deviations:

\[ \sum |x_i - \bar{x}| = 20.89 + 4.11 + 2.11 + 24.11 + 0.89 + 12.89 + 5.11 + 14.89 + 14.11 = 99.11 \]

Now, we can calculate the mean deviation:

\[ \text{Mean Deviation} = \frac{99.11}{9} \approx 11.01 \]

### Step 3: Calculate the Standard Deviation
The standard deviation is calculated using the formula:

\[ \text{Standard Deviation} = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n}} \]

We first calculate \( (x_i - \bar{x})^2 \) for each observation:

- \( (25 - 45.89)^2 = 436.5921 \)
- \( (50 - 45.89)^2 = 16.9921 \)
- \( (48 - 45.89)^2 = 4.4521 \)
- \( (70 - 45.89)^2 = 580.5921 \)
- \( (45 - 45.89)^2 = 0.7921 \)
- \( (33 - 45.89)^2 = 168.7921 \)
- \( (51 - 45.89)^2 = 26.5921 \)
- \( (31 - 45.89)^2 = 219.5921 \)
- \( (60 - 45.89)^2 = 197.5921 \)

Now, we sum these squared deviations:

\[ \sum (x_i - \bar{x})^2 = 436.5921 + 16.9921 + 4.4521 + 580.5921 + 0.7921 + 168.7921 + 26.5921 + 219.5921 + 197.5921 = 1652.9 \]

Now we can calculate the standard deviation:

\[ \text{Standard Deviation} = \sqrt{\frac{1652.9}{9}} \approx \sqrt{183.65} \approx 13.55 \]

### Final Results
- Mean Deviation: \( \approx 11.01 \)
- Standard Deviation: \( \approx 13.55 \)

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