Why Squaring Deviations Prevents Cancellation in Variance Calculation

Conditions

• The data points are distributed around a central mean.
• The goal is to measure the spread or dispersion of the data.

Reasoning, step by step

1. Calculate the deviation for each data point: $x_i - \mu$.
2. Observe that some deviations are positive and some are negative.
3. Note that the sum of these raw deviations is zero: $\sum (x_i - \mu) = 0$.
4. Square each deviation: $(x_i - \mu)^2$, making all values positive or zero.
5. Sum the squared deviations to get a positive total that reflects the overall spread.
6. Divide by $n$ to find the average squared deviation (variance).

Example

In the video, the deviations are -3, 9, -15, -9, and 18. Their sum is 0. However, their squares are 9, 81, 225, 81, and 324, which sum to 720, a non-zero value indicating spread.

Common misconceptions

• Believing that the sum of deviations is a useful measure of spread.
• Thinking that squaring changes the order of magnitude unnecessarily without purpose.
• Confusing variance with the mean absolute deviation.

Watch the explanation

Connected concepts

Explore next

• What is the sum of the raw deviations from the mean?
• Why is the mean absolute deviation sometimes used instead of variance?
• How does squaring affect larger deviations compared to smaller ones?