Can the Sum of Squares Be Negative?

No, the sum of squares cannot be negative.

The sum of squares (SS) is a statistical measure that represents the sum of the squared deviations from a mean or other central value. Since squaring any real number results in a non-negative value, and we are summing non-negative values, the result will always be non-negative. It can be zero, but never negative.

Explanation

The sum of squares is calculated as follows:

SS = Σ (x_i - x̄)²

Where:

• x_i represents each individual data point in the dataset.
• x̄ represents the mean (average) of the dataset.
• Σ represents the summation across all data points.

Because each term (x_i - x̄) is squared, the result is always a non-negative number. Adding together non-negative numbers can only result in a non-negative sum.

Example

Let's consider a simple dataset: 1, 2, 3

1. Calculate the mean: x̄ = (1 + 2 + 3) / 3 = 2
2. Calculate the deviations from the mean:
   • (1 - 2) = -1
   • (2 - 2) = 0
   • (3 - 2) = 1
3. Square the deviations:
   • (-1)² = 1
   • (0)² = 0
   • (1)² = 1
4. Sum the squared deviations: SS = 1 + 0 + 1 = 2

As you can see, the sum of squares in this example is 2, a positive value. It is impossible for this process to yield a negative result. If you calculate a negative SS, there has been a calculation error.