The sum of squares of the deviation of the values of the variable is_____ when taken about their arithmetic mean

Explanation

To solve the question, we need to understand the concept of the sum of squares of deviations from the arithmetic mean.

### Step-by-Step Solution:

1. **Understanding Deviation**:
The deviation of a value \( x_i \) from the mean \( \bar{x} \) is given by \( x_i - \bar{x} \).

2. **Sum of Squares of Deviations**:
The sum of squares of deviations from the mean is calculated as:
\[ S = \sum (x_i - \bar{x})^2 \]
where \( S \) is the sum of squares of deviations, and \( x_i \) represents each value in the dataset.

3. **Properties of the Mean**:
One important property of the arithmetic mean is that it minimizes the sum of the squared deviations. This means that when we calculate the sum of squares of deviations about the mean, it will yield the smallest possible value compared to any other point.

4. **Conclusion**:
Therefore, the sum of squares of the deviations of the values of the variable when taken about their arithmetic mean is **minimum**.

### Final Answer:
The sum of squares of the deviation of the values of the variable is **minimum** when taken about their arithmetic mean.