Chi-squared distribution in EXCEL

Consider the Distribution of CHI-square. Using MS EXCEL functions ХИ2.DIS() construct graphs of the distribution function and probability density, explain the use of that distribution for purposes of mathematical statistics.

Distribution CHI-square (X2, ХИ2, English. Chisquared distribution) is used in different methods of mathematical statistics:

Definition: If x1, x2, ..., xn are independent random variables, distributed according to standard normal law N(0,1), the distribution of the random variable Y=x12+ x22 +...+ xn2 has distribution Xwith n degrees of freedom.

Distribution X2 depends on one parameter, called freedom (df, degrees of freedom). For example, when building confidence intervals for variance the number of degrees of freedom is equal to df=n-1, where n is the size of the fetch.

Density X2 expressed by the formula:br>

if y&gt,0

TIP: Read more about distribution Function and probability Density see article distribution Function and the probability density in MS EXCEL.

Graphic functions

Distribution X2 has an asymmetric shape, average value equal to n, variance equal to 2n.

the example file, on sheet Graph given graphics of the density function and probabilities cumulative distribution function.

note: To build a distribution function and probability density you can use a chart of type Graph or Point (with smooth lines and no dots). Learn more about charting, read the article Basic charts.

Useful ХИ2-distribution

Let x1, x2, ..., xn independent random variables, distributed according to normal with the same parameters ? and? and XCP is arithmetic average these values of x.
Then the random variable y is 

X2distribution with n-1 degree of freedom. Using the definition of sample variance the above expression can be rewritten in the following way:

Consequently, sampling distribution statistics y when sample from normal distribution, has X2distribution with n-1 degree of freedom.

This property we need when constructing the confidence interval in the estimate of the variance of the distribution. Because variance can only be a positive integer, and X2distribution is used to evaluate the y D. b. &gt,0, as specified in the determination.

ХИ2-distribution in MS EXCEL

In MS EXCEL, beginning with version 2010, X2distribution has a special ХИ2.DIS() English name – CHISQ.DIST(), which allows to calculate the probability density (see the formula above) and integrated distribution function (the probability that a random variable X with ХИ2-distribution, will take a value less than or equal to x, P{X &lt,= x}). 

note: as ХИ2-distribution is a special case of gamma distribution, the formula =GAMMA.DIS(x,n/2,2,TRUE) for positive integer n returns the same result as the formula =ХИ2.DIS(x,n, TRUE) or =1-ХИ2.DIS.PH(x,n). And the formula =GAMMA.DIS(x,n/2,2,FALSE) returns the same result as the formula =ХИ2.DIS(x,n, FALSE), i.e. probability density ХИ2 distribution.

 ХИ2.DIS.Pkh() returns the distribution function, specifically, right-sided probability, i.e. P{X > x}. It is obvious that true equality 
=ХИ2.DIS.PH(x,n)+ ХИ2.DIS(x,n,TRUE)=1
since the first term calculates the probability P{X &gt, x}, and the second is P{X &lt,= x}.

To MS EXCEL 2010 in EXCEL was just a function ХИ2РАСП(), which allows you to calculate the right probability, i.e. P{X > x}. Possible new features of MS EXCEL 2010 ХИ2.DIS()  ХИ2.DIS.Pkh() override the functionality of this feature. Function ХИ2РАСП() left in MS EXCEL 2010 for compatibility.

ХИ2.DIS() is the only function that returns a probability density ХИ2-distribution (the third argument must be equal to FALSE). The other functions return integral distribution function, i.e. the probability that a random variable will take the value from the specified range: P{X &lt,= x}.

the Above functions in MS EXCEL is given in the example file, on sheet Function.

Examples

Find the probability that a random variable X will take a value less than or equal to the specified x: P{X &lt,= x}. This can be done in several functions:

= ХИ2.DIS(x, n, TRUE) 
=1-ХИ2.DIS.PH(x, n)
=1-ХИ2РАСП(x, n)

 ХИ2.DIS.Pkh() returns the probability P{X &gt, x}, the so-called right-sided probability, and therefore to find P{X &lt,= x}, it is necessary to subtract the result from 1.

Find the probability that a random variable X will take a value greater than the specified x: P{X &gt, x}. This can be done in several functions:

= 1-ХИ2.DIS(x, n, TRUE) 
=ХИ2.DIS.PH(x, n)
=ХИ2РАСП(x, n)

Reverse function ХИ2-distribution

the Reverse function is used to calculate alpha-quantiles, i.e. to calculate the values x for a given probability alpha, x must satisfy the expression P{X &lt,= x}=alpha.

ХИ2.OBR() is used to calculate the confidence intervals for variance of normal distribution.

ХИ2.OBR.Pkh() is used to calculate the top quantile, i.e. if the argument of the function specified significance level of 0.05 for example, the function will return a value of the random variable x for which P{X > x}=0,05. As a comparison: function ХИ2.OBR() will return a value of the random variable x for which P{X&lt,=x}=0,05.

In MS EXCEL 2007 and earlier, instead of ХИ2.OBR.Pkh() used ХИ2ОБР().

the Above functions can be interchanged, because the following formulas return the same result:
=HEE.OBR(alpha,n)
=ХИ2.OBR.PCH(1-alpha,n)
=ХИ2ОБР(1 - alpha,n)

Some examples are given in the the example file, on sheet Function.

Functions MS EXCEL using ХИ2-distribution

In MS EXCEL there is another feature to use for calculations ХИ2-distribution is ХИ2.TEST(). An earlier version of her - ХИ2ТЕСТ(). For more information about using this function, see article about the test of independence Chi-square.

the following is the correspondence of Russian and English names of the functions:
ХИ2.DIS.Pkh() - eng. name CHISQ.DIST.RT, i.e., CHI-SQuared DISTribution Right Tail, the right-tailed Chi-square(d) distribution
ХИ2.OBR() - eng. name CHISQ.INV, i.e., CHI-SQuared distribution INVerse
ХИ2.Pkh.OBR() - eng. name CHISQ.INV.RT, i.e. CHI-SQuared distribution INVerse Right Tail
ХИ2РАСП() - eng. the name of the CHIDIST function is equivalent to a CHISQ.DIST.RT
ХИ2ОБР() - eng. name CHIINV, i.e. CHI-SQuared distribution INVerse

Estimation of distribution parameters

because usually ХИ2-distribution is used for the purposes of mathematical statistics (computation of confidence intervals test hypotheses, etc.) and almost never to build models of real values, then for this distribution the discussion of the estimates of the distribution parameters is not performed herein.

the Approach ХИ2 distribution normal distribution

the number of degrees of freedom n&gt,30 distribution X2 is well approximated by a normal distribution with average ?=n and variance ?=2*n (see the sample file sheet Approximation).

TIP: About other distributions of MS EXCEL, you can read the article Distribution of a random variable in MS EXCEL.

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