Sum of square of the deviations about mean is Minimum

The sum of the squares of deviations about the mean, also known as the sum of squared residuals or the sum of squared errors, is a measure of the variability or dispersion of a set of data points around their mean. It is often used in statistics to assess the goodness of fit of a regression model or to compare the variability of different datasets.

The sum of square of deviations about mean is minimum when the data points are located close to the mean. In other words, the closer the data points are to the mean, the smaller the sum of squares of deviations will be. This occurs because the deviations are squared before being summed, which means that larger deviations have a disproportionately larger impact on the sum.

Explanation:

When we calculate the sum of squares of deviations about the mean, we are essentially measuring how far each data point is from the mean, squaring those distances, and then summing them up. This gives us a measure of the overall variability or dispersion of the data.

When the data points are located close to the mean, the deviations will be small, and when we square these small deviations, we get even smaller values. As a result, the sum of squares of deviations will be relatively small.

On the other hand, when the data points are spread out from the mean, the deviations will be larger, and squaring these larger deviations will result in even larger values. Consequently, the sum of squares of deviations will be larger.

Therefore, the sum of squares of deviations about the mean is minimum when the data points are located close to the mean, indicating less variability or dispersion in the dataset.

Conclusion:

In conclusion, the sum of squares of deviations about the mean is minimum when the data points are located close to the mean. This measure of variability is used to assess the goodness of fit of regression models and compare the variability of different datasets. Understanding the concept of sum of squares of deviations can help in analyzing and interpreting statistical data.