chi-squared distribution

(noun)

A distribution with $k$ degrees of freedom is the distribution of a sum of the squares of $k$ independent standard normal random variables.

Related Terms

Examples of chi-squared distribution in the following topics:

  • Chi Square Distribution

    • Define the Chi Square distribution in terms of squared normal deviates
    • The Chi Square distribution is the distribution of the sum of squared standard normal deviates.
    • The mean of a Chi Square distribution is its degrees of freedom.
    • The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square.
    • Chi Square distributions with 2, 4, and 6 degrees of freedom

  • The chi-square distribution and finding areas

    • For instance, Figure 6.9(b) shows the upper tail of a chi-square distribution with 2 degrees of freedom.
    • Figure 6.9(d) shows a cutoff of 11.7 on a chi-square distribution with 7 degrees of freedom.
    • Figure 6.9(e) shows a cutoff of 10 on a chi-square distribution with 4 degrees of freedom.
    • Figure 6.9(f) shows a cutoff of 9.21 with a chi-square distribution with 3 df.
    • (a) Chi-square distribution with 3 degrees of freedom, area above 6.25 shaded.

  • Estimating a Population Variance

    • The chi-square distribution is used to construct confidence intervals for a population variance.
    • A chi-square distribution can be used to construct a confidence interval for this variance.
    • The chi-square distribution with a $k$ degree of freedom is the distribution of a sum of the squares of $k$ independent standard normal random variables.
    • In fact, the chi-square distribution enters all analyses of variance problems via its role in the $F$-distribution, which is the distribution of the ratio of two independent chi-squared random variables, each divided by their respective degrees of freedom.
    • The chi-square distribution is a family of curves, each determined by the degrees of freedom.

  • Finding a p-value for a chi-square distribution

    • This is where the chi-square distribution becomes useful.
    • How many degrees of freedom should be associated with the chi-square distribution used for ?
    • The chi-square distribution and p-value are shown in Figure 6.10.
    • Degrees of freedom: We only apply the chi-square technique when the table is associated with a chi-square distribution with 2 or more degrees of freedom.
    • The p-value for the juror hypothesis test is shaded in the chi-square distribution with df = 3.

  • Notation

    • where df = degrees of freedom depend on how chi-square is being used.
    • (If you want to practice calculating chi-square probabilities then use df = n−1.
    • For the χ2 distribution, the population mean is µ = df and the population standard deviation is $\sigma = \sqrt{2 \cdot df}$.
    • The random variable for a chi-square distribution with k degrees of freedom is the sum of k independent, squared standard normal variables.

  • Using simulation for goodness of fit tests

    • In short, we simulate a new sample based on the purported bin probabilities, then compute a chi-square test statistic $X^2_{sim}$.
    • We do this many times (e.g. 10,000 times), and then examine the distribution of these simulated chi-square test statistics.
    • Since the minimum bin count condition was satisfied, the chi-square distribution is an excellent approximation of the null distribution, meaning the results should be very similar.
    • Figure 6.21 shows the simulated null distribution using 100,000 simulated values with an overlaid curve of the chi-square distribution.
    • The precise null distribution for the juror example from Section 6.3 is shown as a histogram of simulated $X^2_{sim}$ statistics, and the theoretical chi-square distribution is also shown.

  • Randomization for two-way tables and chi-square

    • In short, we create a randomized contingency table, then compute a chi-square test statistic.
    • We repeat this many times using a computer, and then we examine the distribution of these simulated test statistics.
    • When the minimum threshold is met, the simulated null distribution will very closely resemble the chi-square distribution.
    • As before, we use the upper tail of the null distribution to calculate the p-value.

  • Student Learning Outcomes

  • Structure of the Chi-Squared Test

    • The chi-square test is used to determine if a distribution of observed frequencies differs from the theoretical expected frequencies.
    • The chi-square ($\chi^2$) test is a nonparametric statistical technique used to determine if a distribution of observed frequencies differs from the theoretical expected frequencies.
    • Second, we use the chi-square distribution.
    • We may observe data that give us a high test-statistic just by chance, but the chi-square distribution shows us how likely it is.
    • The approximation to the chi-squared distribution breaks down if expected frequencies are too low.

  • Chi-Square Probability Table

    • Areas in the chi-square table always refer to the right tail.