Find the sum of deviations of the following data measured from their actual mean.
7,8,10,12,18

Answer

To find the sum of deviations of the given data from their actual mean, we will follow these steps:

### Step 1: Calculate the Mean
To find the mean (average) of the data, we use the formula:

\[ \text{Mean} (\bar{x}) = \frac{\text{Sum of all observations}}{\text{Number of observations}} \]

The data given is: 7, 8, 10, 12, 18.

First, we calculate the sum of these observations:

\[ 7 + 8 + 10 + 12 + 18 = 55 \]

Now, we find the number of observations, which is 5.

Now, we can calculate the mean:

\[ \bar{x} = \frac{55}{5} = 11 \]

### Step 2: Calculate the Deviations from the Mean
Next, we will calculate the deviations of each observation from the mean. The deviation for each observation \(x_i\) is calculated as:

\[ \text{Deviation} = x_i - \bar{x} \]

Now we calculate the deviations for each observation:

1. For \(x_1 = 7\):
\[ 7 - 11 = -4 \]

2. For \(x_2 = 8\):
\[ 8 - 11 = -3 \]

3. For \(x_3 = 10\):
\[ 10 - 11 = -1 \]

4. For \(x_4 = 12\):
\[ 12 - 11 = 1 \]

5. For \(x_5 = 18\):
\[ 18 - 11 = 7 \]

### Step 3: Sum the Deviations
Now we will sum all the deviations calculated:

\[ \text{Sum of deviations} = (-4) + (-3) + (-1) + 1 + 7 \]

Calculating this step by step:

\[ -4 - 3 = -7 \]
\[ -7 - 1 = -8 \]
\[ -8 + 1 = -7 \]
\[ -7 + 7 = 0 \]

Thus, the sum of the deviations is:

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

### Final Answer
The sum of deviations of the data from their actual mean is **0**.

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