INTERACTIVE ANIMATION: THE CHI-SQUARE DISTRIBUTION

This interactive animation illustrates the Chi-square distribution.

Upper frame

In this frame is the standard normal distribution N(0, 1), together with a sample drawn from this distribution. The +1 and -1 marks are each one standard deviation (1) from the origin.

The vertical blue tick is the sample mean.

Lower frame

In this frame is the _n distribution curve for the value n as posted in the "Nb Points" display. Recall that this is the distribution curve of the Sum of the Squares of the abscissas of the sample points.

• Change the number of points and observe the changes of the shape fo the _n curve.

        * X² is a sum of squares and can therefore never be negative.

        * For n > 2, the curves always have the same general shape:

• They pass through the origin. This reflects the fact that it is very unlikely for X² to take very small values. This would mean that all the independent variables take simultaneously very small values, an unlikely circumstance.
• They tend toward 0 at infinity (like any probability distribution does). This reflects the fact that it is very unlikely for X² to take very large values (although there is no upper limit). This would mean that  all the independent variables take simultaneously very large values (positive and/or negative), also an unlikely circumstance.
• They have a unique maximum (mode). As n increases, this maximum gets lower and shifts to the right (while the area under the curve remains of course equal to 1). The position of this maximum is shown to be n - 2.

        * For n = 2, the curve steadily decreases from its largest value (0.5).

        * For n = 1, the vertical axis is an asymptote. The curve is not defined for X² = 0. Although ₁ can take arbitrary large values, the area under the curve is still 1.

₁  is the distribution of the square of a standard normal variable.

• Now click on the "Estimated mean" radio button. Two curves are now displayed:
• The black curve is still the _n of the Sum of the Squares of the abscissas of the sample points.
• The red curve is the _n-1 distribution of the Sum of the Squares of the distances of the sample points to the Estimated Mean. The maximum is higher and to the left of that of the _n curve. This is because the estimated variance is always less than the "true" variance (i.e. measured from the true mean as opposed to the estimated mean). This favors smaller values of the X² statistic, and hinders larger values of  X².

          Note that the number of degrees of freedom (upper right corner) is now down one unit.

• Click now on "Go", and observe the progressive build-up of the histogram of the values of X².