Let’s say the average height of men is 67.7 inches (172cm). If a man is 72 inches (183cm) tall, how tall will his full-grown son be?

Your first instinct might be: “Oh, that’s easy. The son will be as tall as his father.”

Although 72 inches (183cm) is a possible guesstimate for the son’s height, it is not very probable from a statistical point of view.

Correlation, regression, and standard deviation

Instead, we should compile datasets of the average height of sons whose fathers were 72 inches (183cm) tall.

We will then repeat the step above for fathers of heights 58 inches (147cm), 59 inches (150cm), 60 inches (152cm), 61 inches (155cm), and so on, until 78 inches (198cm).

Next, we can draw a graph of averages between the heights of fathers (x-axis) and the heights of sons (y-axis). The plotted points ‘x’ mark the average height of sons whose fathers are of a particular height.

The regression line is a line of best fit with equation y=a+bx. Since the regression line is a straight line with a gradient b, the slope b will be the regression coefficient. The correlation coefficient is 0.5.

Now we can see that the father’s height and the son’s height are positively correlated, but only by a correlation coefficient of 0.5. Note that the slope of standard deviation (SD) line is less steep than the slope of the regression line. For every 1 SD increase in the father’s height, the son’s height increases by merely 0.5 SD.

#demographics #predictions #biology #statistics #data-science #data visualization