Find the mean of the following data :
30,32,24,34, 26,28, 30, 35, 33, 25
Show that the sum of the deviations of all the given observations from the mean is zero.

Answer

To find the mean of the given data and show that the sum of the deviations from the mean is zero, we will follow these steps:

### Step 1: Calculate the Mean
The formula for the mean (average) is given by:

\[ \text{Mean} = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]

**Calculating the sum of the observations:**

Given data: 30, 32, 24, 34, 26, 28, 30, 35, 33, 25

\[ \text{Sum} = 30 + 32 + 24 + 34 + 26 + 28 + 30 + 35 + 33 + 25 = 297 \]

**Counting the number of observations:**

There are 10 observations.

**Calculating the mean:**

\[ \text{Mean} = \frac{297}{10} = 29.7 \]

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

**Deviations:**
1. \(30 - 29.7 = 0.3\)
2. \(32 - 29.7 = 2.3\)
3. \(24 - 29.7 = -5.7\)
4. \(34 - 29.7 = 4.3\)
5. \(26 - 29.7 = -3.7\)
6. \(28 - 29.7 = -1.7\)
7. \(30 - 29.7 = 0.3\)
8. \(35 - 29.7 = 5.3\)
9. \(33 - 29.7 = 3.3\)
10. \(25 - 29.7 = -4.7\)

### Step 3: Sum of the Deviations
Now, we will sum all the deviations calculated above.

\[ \text{Sum of Deviations} = (0.3 + 2.3 - 5.7 + 4.3 - 3.7 - 1.7 + 0.3 + 5.3 + 3.3 - 4.7) \]

Calculating this step-by-step:

1. \(0.3 + 2.3 = 2.6\)
2. \(2.6 - 5.7 = -3.1\)
3. \(-3.1 + 4.3 = 1.2\)
4. \(1.2 - 3.7 = -2.5\)
5. \(-2.5 - 1.7 = -4.2\)
6. \(-4.2 + 0.3 = -3.9\)
7. \(-3.9 + 5.3 = 1.4\)
8. \(1.4 + 3.3 = 4.7\)
9. \(4.7 - 4.7 = 0\)

Thus, the sum of the deviations from the mean is:

\[ \text{Sum of Deviations} = 0 \]

### Conclusion
We have found that the mean of the given data is 29.7, and the sum of the deviations from the mean is 0, which confirms that the mean is correctly calculated.

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