Chi Square Distribution

Shape of Distribution

Basic Properties

• One parameter $N$ is required (Positive integer)
• Continuous distribution defined on semi-bounded range $x \geq 0$
• This distribution is asymmetric.

Probability

• Probability density function

  $$f(x)=\frac{1}{2^{\frac{N}{2}}\Gamma\left(\frac{N}{2}\right)}\exp\left(-\frac{x}{2}\right)x^{\frac{N}{2}-1}$$

  , where $\Gamma(\cdot)$ is gamma function.

• Cumulative distribution function

  $$F(x)=\frac{\Gamma_{\frac{x}{2}}\left(\frac{N}{2}\right)}{\Gamma\left(\frac{N}{2}\right)}$$

  , where $\Gamma_{x}(\cdot)$ is incomplete gamma function.

• How to compute these on Excel.

	A	B
1	Data	Description
2	5	Value for which you want the distribution
3	9	Value of parameter N
4	Formula	Description (Result)
5	=NTCHISQDIST(A2,A3,TRUE)	Cumulative distribution function for the terms above
6	=NTCHISQDIST(A2,A3,FALSE)	Probability density function for the terms above

• Function reference : NTCHISQDIST

Characteristics

Mean -- Where is the "center" of the distribution? (Definition)

• Mean of the distribution is given as

  $$N$$

• How to compute this on Excel

	A	B
1	Data	Description
2	9	Value of parameter N
3	Formula	Description (Result)
4	=NTCHISQMEAN(A2)	Mean of the distribution for the terms above

• Function reference : NTCHISQMEAN

Standard Deviation -- How wide does the distribution spread? (Definition)

• Variance of the distribution is given as

  $$2N$$

  Standard Deviation is a positive square root of Variance.

• How to compute this on Excel

	A	B
1	Data	Description
2	9	Value of parameter N
3	Formula	Description (Result)
4	=NTCHISQSTDEV(A2)	Standard deviation of the distribution for the terms above

• Function reference : NTCHISQSTDEV

Skewness -- Which side is the distribution distorted into? (Definition)

• Skewness of the distribution is given as

  $$\sqrt{\frac{8}{N}}$$

• How to compute this on Excel

	A	B
1	Data	Description
2	9	Value of parameter Alpha
3	Formula	Description (Result)
4	=NTCHISQSKEW(A2)	Skewness of the distribution for the terms above

• Function reference : NTCHISQSKEW

Kurtosis -- Sharp or Dull, consequently Fat Tail or Thin Tail (Definition)

• Kurtosis of the distribution is given as

  $$\frac{12}{N}$$

• This distribution can be leptokurtic or platykurtic.

• How to compute this on Excel

	A	B
1	Data	Description
2	9	Value of parameter N
3	Formula	Description (Result)
4	=NTCHISQKURT(A2)	Kurtosis of the distribution for the terms above

• Function reference : NTCHISQKURT

Random Numbers

• How to generate random numbers on Excel.

	A	B
1	Data	Description
2	9	Value of parameter N
3	Formula	Description (Result)
4	=NTRANDCHISQ(100,A2,0)	100 chi square deviates based on Mersenne-Twister algorithm for which the parameters above

Note The formula in the example must be entered as an array formula. After copying the example to a blank worksheet, select the range A5:A104 starting with the formula cell. Press F2, and then press CTRL+SHIFT+ENTER.

• Function reference : NTRANDCHISQ

NtRand Functions

• If you already have parameters of the distribution
  • Generating random numbers based on Mersenne Twister algorithm: NTRANDCHISQ
  • Computing probability : NTCHISQDIST
  • Computing mean : NTCHISQMEAN
  • Computing standard deviation : NTCHISQSTDEV
  • Computing skewness : NTCHISQSKEW
  • Computing kurtosis : NTCHISQKURT
  • Computing moments above at once : NTCHISQMOM

Reference

• Wolfram Mathworld -- Chi Square Distribution
• Wikipedia -- Chi square distribution
• Statistics Online Computational Resource