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Chi-Squared Distribution

March 29, 2018

The Chi-Squared (χN) distribution (with N degrees of freedom) is the probability distribution of the sum of the squares of N random variables having a standard Normal distribution (μ = 0, σ = 1). Fig. 15 shows examples of the χN2 PDF  for different values of N. As N→∞ the χN2  distribution converges to the Normal distribution. For all of the χN2  distributions, the mean value is μ = N  and the variance is σ 2 = 2 N.

The formula for the PDF is

chisquared-equation16

where Γ is the Gamma function.

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Figure 14. χN2 Distribution for the Sum of N values of x2.

The squared values used to evaluate the mean-square are, therefore, χ12 distributed. This is illustrated in Fig. 15, using the car vibration signal shown in Fig. 2. The distribution is strongly biased toward small values as shown by the simplified formula for χ12 :

chisquared-equation17

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Figure 15. PDF of Squared Car Vibration Data from Fig. 2 Compared to χ12

 

Since the variance, σ2, is derived from the mean-square value (Eq. 3), a confidence interval for σ2 can be determined using the χN2 distribution. For a random variable x with a standard deviation of σx , the summation of N values-squared has a σx2χN2 distribution, so the variance has a  σx2χN2 / N  distribution. Since the χN2 distributions are not symmetrical, the confidence intervals for the variance estimate are not symmetrical. Fig. 16 illustrates this where the values of N / χN2 are plotted vs. N  for different confidence levels. The confidence interval for the variance is then stated as the interval:

chisquared-equation18

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Figure 16. Confidence Intervals Using the χ N2 distribution for Small Sample Sizes

 

For N > 100, the Normal distribution can be used.  The standard deviation of the variance estimate is √2/N σxand the (symmetrical) confidence intervals from Fig. 13 can be used.