# Chi-Squared Distribution

March 29, 2018

Back to: Random Testing

Chi-squared () distribution is the distribution of a sum of squared random variables. Among other applications, it can be used to estimate the confidence interval for the variance of a random variable from a normal distribution.

A chi-squared distribution with N degrees of freedom () determines the probability of a normal distribution where the mean value () equals 0 and variance () equals 1.

Figure 3.15 provides examples of the probability density function (PDF) for different values of N. As N approaches infinity, the distribution converges with the normal distribution. For all the distributions, the mean value is and the variance is .

The formula for the PDF is as follows (Equation 16), where Γ is the Gamma function.

(1)

Equation 16

The squared values used to evaluate the mean-square are distributed. This is illustrated in Figure 3.15 using the car vibration signal shown in Figure 3.2. The distribution is strongly biased toward small values as shown by the simplified formula for :

(2)

Equation 17

As variance is derived from the mean-square value, the confidence interval for the variance can be determined using the distribution.

For a random variable x with a standard deviation of σ_{x}, the summation of N values-squared has a σ_{x}^{2}χ_{1}^{2} distribution. Therefore, the variance has a σ_{x}^{2}χ_{1}^{2}/N distribution.

As the distributions are not symmetrical, the estimated confidence intervals for the variance are not symmetrical. In Figure 3.16, the values of are plotted versus N for different confidence levels. The confidence interval for the variance is then stated as the interval:

(3)

For *N* > 100, normal distribution can be used. The standard deviation of the variance estimate is √2/*N* *σ*_{x}^{2 }and the (symmetrical) confidence intervals from Figure 3.13 can be used.