Chi-square Distribution

Definition 1: The chi-square distribution with k degrees of freedom, abbreviated χ2(k), has probability density function

Chi square pdf

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 χ2(k) distribution has mean k and variance 2k

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

  • Mean = k
  • Median = k-2⁄3 for large k
  • Mode = k – 1 for k > 2
  • Range = [0.∞)
  • Variance = 2k
  • Skewness = \sqrt{8/k}
  • 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.

Chi-square distribution

Figure 1 – Chart of chi-square distributions

Theorem 1: Suppose x has standard normal distribution N(0, 1) and let x1, …, xk be k independent sample values of x, then the random variable

has a chi-square distribution χ2(k).

Corollary 1:

  1. If x has distribution N(0, 1) then x2 has distribution χ2(1)
  2. If x ~ N(μ, σ) and z = (x–μ)/σ then over repeated samples z2 has distribution χ2(1)
  3. If x1, …, xk are independent observations from a normal population with distribution N(μ,σ) and for each i, z = (x–μ)/σ , then the following random variable has a χ2(k) distribution


Property 2: If x and y are independent and x has distribution χ2(m) and y has distribution χ2(n), then x + y has distribution χ2(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 s2 has distribution


Corollary 2: s2 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 \frac{(n-1)s^2}{\sigma^2} has a χ2(n–1), distribution

Property 3: The mean of the sample variance s2 is σ2 and the variance is \frac{2\sigma^4}{n-1}

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 s2.

By Property 3


18 Responses to Chi-square Distribution

  1. Fred says:

    Hello Charles,
    I think there is a small typo in Theorem 1, the sum should be from i=1 to i=k I believe, not i=n.

    Many thanks,\


  2. Vendula says:

    Hello Charles, I would like to ask you for a help. I measured p-bodies in different cell lines and different times. I have groups for 0, 1, 2, 3 and more p-bodies. I have two replicates for each cell lines. May I use the chi-square test to compare, if there is any differece? And how handle the replicates, it is possible sum p-bodies for each replicate?
    thanks for you response.

    • Charles says:

      You haven’t provided enough information for me to give you a definitive answer, but it doesn’t sound like a fit for chi-square test of independence.

  3. shri says:

    Hi Charles,

    When I run a Chi-Sq Test in real statistics I get the following output:
    Chi Sq p-value X-Critical Sig Cramer V
    Pearson’s – 623.097 2.9E-122 26.296 Yes 0.345099

    Since X-Critical is less than Chi-Sq it gives the result that the variables are associated. In this case the p value is > 0.05 so i assumed its not significant. Do we not consider p-value ?

    Kind regards

    • Charles says:

      chisq-crit < chisq is equivalent to p-value < alpha. If the result is significant using the first inequality it will be significant using the second inequality and vice versa. Charles

  4. devi says:

    Four dice were thrown 112 times and the number of times 1 or 3 or 5 was thrown were as under
    Number of dice throwing 1 or 3 or 5 0 1 2 3 4
    Frequency 10 25 40 30 7

    Find the value of chi-square presuming that all dice were fair

  5. Jonathan Bechtel says:

    Hi Charles,

    This might be a silly question, but I want to be clear on something:

    Even though the chi sq distribution is X2(k), k would actually demarcate the x that’s in the PDF, correct?

  6. chandrakala says:

    how you reproduce this chi square graph? I mean what is the x and y-axis ?

  7. karan says:

    Hi sir,

    I have 200 measurements of a random variable for whom i have estimated mean and sigma. Now, i want to estimate the error bars on the standard deviation using chi-square function. I don’t know how to do that. Can you please help me on this.


  8. Colin says:

    Property 3: The mean of the sample variance s2 is σ2 and the variance is ?
    I cannot see the formula, it seems something wrong with the picture

    • Charles says:

      The variance is 2 times sigma raised to the 4th power divided by n-1.
      This formula is displayed using latex. I hope there isn’t a problem with latex displays. Can you see the formula in Corollary 3? It also uses latex.

      • Colin says:

        Thank you for your reply! I found the result in the proof section. Some picture in this website cannot display. It may attribute to the internet. But it doesn’t matter, your website is fabulous.

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