# Chi square distribution

## Shape of Distribution

### Basic Properties

- One parameter is required (Positive integer)
- Continuous distribution defined on semi-bounded range
- This distribution is asymmetric.

### Probability

- Probability density function
, where is gamma function.

- Cumulative distribution function
, where is incomplete gamma function.

- How to compute these on Excel.

1 2 3 4 5 6 A B Data Description 5 Value for which you want the distribution 9 Value of parameter N Formula Description (Result) =NTCHISQDIST(A2,A3,TRUE) Cumulative distribution function for the terms above =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
- How to compute this on Excel

1 2 3 4 A B Data Description 9 Value of parameter N Formula Description (Result) =NTCHISQMEAN(A2,A3) 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
Standard Deviation is a positive square root of Variance.

- How to compute this on Excel

1 2 3 4 A B Data Description 9 Value of parameter N Formula Description (Result) =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
- How to compute this on Excel

1 2 3 4 A B Data Description 9 Value of parameter Alpha Formula Description (Result) =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
- This distribution can be leptokurtic or platykurtic.
- How to compute this on Excel

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

## Random Numbers

- How to generate random numbers on Excel.

1 2 3 4

A B Data Description 9 Value of parameter N Formula Description (Result) =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