## (Basic Statistics for Citizen Data Scientist)

# Chi-square Distribution

**Definition 1**: The **chi-square distribution** with *k* **degrees of freedom**, abbreviated χ^{2}(*k*), has probability density function

*k* does not have to be an integer and can be any positive real number.

Click here for more technical details about the chi-square distribution, including proofs of some of the propositions described below. Except for the proof of Corollary 2 knowledge of calculus will be required.

**Observation**: The chi-square distribution is the gamma distribution where *α = k*/2 and* β* = 2.

**Property 1**: The χ^{2}(*k*) distribution has mean* k* and variance 2*k*

**Observation**: The key statistical properties of the chi-square distribution are:

- Mean =
*k* - Median ≈
*k*(1–2/(9*k*))^3 - Mode = max (
*k*– 1, 0) - Range = [0.∞)
- Variance = 2
*k* - Skewness =
- Kurtosis = 12/
*k*

The following are the graphs of the pdf with degrees of freedom *df* = 5 and 10. As *df* grows larger the fat part of the curve shifts to the right and becomes more like the graph of a normal distribution.

**Figure 1 – Chart of chi-square distributions**

**Theorem 1**: Suppose *x* has standard normal distribution *N*(0, 1) and let *x _{1}, …, x_{k}*be

*k*independent sample values of

*x*, then the random variable

has a chi-square distribution χ^{2}(*k*).

**Corollary 1**:

- If
*x*has distribution*N*(0, 1) then*x*^{2}has distribution χ^{2}(1) - If
*x*~*N*(*μ, σ*) and*z*= (*x–μ*)/*σ*then over repeated samples*z*^{2}has distribution χ^{2}(1) - If
*x*are independent observations from a normal population with normal distribution_{1}, …, x_{k}*N*(*μ,σ*) and for each*i*,*z*= (*x–μ*)/*σ*, then the following random variable has a χ^{2}(*k*) distribution

**Property 2**: If *x* and y are independent and *x* has distribution χ^{2}(*m*) and y has distribution χ^{2}(*n*), then *x* + y has distribution χ^{2}(*m + n*)

**Theorem 2**: If *x* is drawn from a normally distributed population *N*(*μ,σ*) then for samples of size *n* the sample variance *s*^{2} has distribution

**Corollary 2**: *s*^{2} is an unbiased, consistent estimator of the population variance

**Corollary 3**: If *x* is drawn from a normally distributed population *N*(*μ, σ*), then for samples of size *n* the random variable has a χ^{2}(*n*–1), distribution

**Property 3**: The mean of the sample variance *s*^{2} is *σ*^{2} and the variance is

Proof: This can be seen from the proof of Corollary 2.

**Excel Functions**: Excel provides the following functions:

**CHIDIST**(*x, df*) = the probability that the chi-square distribution with *df* degrees of freedom is ≥ *x*; i.e. 1 – *F*(*x*) where *F* is the cumulative chi-square distribution function.

**CHIINV**(*α, df*) = the value *x* such that CHIDIST(*x, df*) = 1 – *α*; i.e. the value *x* such that the right tail of the chi-square distribution with area *α* occurs at *x*. This means that *F*(*x*) = 1 – *α*, where* F* is the cumulative chi-square distribution function.

With Excel 2010/2013 there are a number of new functions (**CHISQ.DIST, CHISQ.INV, CHISQ.DIST.RT **and **CHISQ.INV.RT**) that provide equivalent functionality to CHIDIST and CHIINV, but whose syntax is more consistent with other distribution functions. These functions are described in Built-in Statistical Functions.

In Excel 2010 CHISQ.DIST(*x, df*, TRUE) is the cumulative distribution function for the chi-square distribution with *df* degrees of freedom, i.e. 1 – CHIDIST(*x, df*), and CHISQ.DIST(*x*, *df*, FALSE) is the pdf for the chi-square distribution.

**Real Statistics Functions**: The Real Statistics Resource Pack provides the following functions.

**CHISQ_DIST**(*x, df, cum*) = GAMMA.DIST(*x, df*/2, 2, cum) = GAMMADIST(*x, df*/2, 2, *cum*)

**CHISQ_INV**(*p, df*) = GAMMA.INV(*p, df*/2, 2) = GAMMAINV(*p, df*/2, 2)

These functions provide better estimates of the chi-square distribution when *df* is not an integer. The first function is also useful in providing an estimate of the pdf for versions of Excel prior to Excel 2010, where CHISQ.DIST(*x, df*, FALSE) is not available.

The Real Statistics Resource also provides the following functions:

**CHISQ_DIST_RT**(*x, df*) = 1 – CHISQ_DIST(*x, df*, TRUE)

**CHISQ_INV_RT**(*p, df*) = 1 – CHISQ_INV(*p, df*)

**Example 1**: Suppose we take samples of size 10 from a population with normal distribution *N*(0, 2). Find the mean and variance of the sample distribution of *s*^{2}.

Statistics with R for Business Analysts – Normal Distribution

## Statistics for Beginners in Excel – Chi-square Distribution

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