# Variability in the fitted values

For a dataset, I have fitted a model. The fitted or predicted values have less variability than the observed values. What does it imply?

• That your model explains a part of the observed variability. – Sergio Sep 12 at 14:09
• @Sergio Is it bad indication that the fitted values are not as spread as observed values? – user149054 Sep 12 at 14:34
• In the 1880's, Francis Galton discovered this was a universal phenomenon. It is part of his theory of "regression to the mean." He illustrated it with his celebrated quincunx. – whuber Sep 12 at 14:56
• Variability of observed values: $\sum(y_i-\bar{y})^2$. Variability of fitted values: $\sum(\hat{y}_i-\bar{y})^2$. Variability around your model: $\sum(y_i-\hat{y})^2$. Since $$\sum(y_i-\bar{y})^2=\sum(y_i-\hat{y})^2+\sum(\hat{y}_i-\bar{y})^2$$ the variability of fitted values can never be greater than the variability of observed values, and they can be equal only if you model explains absolutely nothing. – Sergio Sep 12 at 15:01

Variability of observed values: $$\sum(y_i-\bar{y})^2$$. Variability of fitted values: $$\sum(\hat{y}_i-\bar{y})^2$$. Variability around your model: $$\sum(y_i-\hat{y})^2$$. Since $$\sum(y_i-\bar{y})^2=\sum(y_i-\hat{y})^2+\sum(\hat{y}_i-\bar{y})^2$$ the variability of fitted values can never be greater than the variability of observed values, and they can be equal only if you model explains absolutely nothing