The Chi-Square distribution is a continuous probability distribution that arises in the context of testing hypotheses about the variance of a normally distributed population. In R, the Chi-Square distribution can be calculated using the pchisq() function.

The pchisq() function takes two arguments:

1. q: the quantile of the Chi-Square distribution at which to evaluate the cumulative distribution function.
2. df: the degrees of freedom of the Chi-Square distribution.

For example, to calculate the probability that a Chi-Square random variable with 10 degrees of freedom is less than or equal to 5, we can use the following code:

 

 

pchisq(5, df = 10)

This will return a probability value, which represents the area under the Chi-Square distribution curve to the left of the quantile 5.

We can also use the qchisq() function to calculate the quantiles of the Chi-Square distribution.

For example, to find the 90th percentile of a Chi-Square distribution with 5 degrees of freedom, we can use the following code:

qchisq(0.9, df = 5)

This will return the quantile value, which represents the value below which 90% of the probability mass of the Chi-Square distribution lies.

Example – 2

Generating random numbers from a chi-square distribution

To generate random numbers from a chi-square distribution, you can use the rchisq function in R. The function takes two arguments: n, the number of random numbers to generate, and df, the degrees of freedom. The degrees of freedom parameter determines the shape of the distribution.

For example, to generate 10 random numbers from a chi-square distribution with 5 degrees of freedom, you can use the following code:

set.seed(123) # set seed for reproducibility
x <- rchisq(n = 10, df = 5)
print(x)

#[1] 1.086741 4.070908 4.251318 6.676105 4.760983 
# 7.017674 2.743997 6.278414 2.476452 2.873360

Calculating probabilities of the chi-square distribution

To calculate probabilities of the chi-square distribution, you can use the pchisq function in R. The function takes two arguments: q, the quantile or critical value, and df, the degrees of freedom. The function returns the cumulative probability of observing a chi-square value less than or equal to q.

For example, to calculate the probability of observing a chi-square value less than or equal to 10 with 5 degrees of freedom, you can use the following code:

p <- pchisq(q = 10, df = 5)
print(p)

#[1] 0.9375735

This means that the probability of observing a chi-square value less than or equal to 10 with 5 degrees of freedom is approximately 0.938.

Performing chi-square tests

# Create a contingency table
observed <- matrix(c(10, 20, 30, 40), nrow = 2, byrow = TRUE)
colnames(observed) <- c("Group 1", "Group 2")
rownames(observed) <- c("Male", "Female")

# Perform chi-square test
chisq.test(observed)

In this example, we create a contingency table with two groups (“Group 1” and “Group 2”) and two categories (“Male” and “Female”). The observed counts are 10 males in Group 1, 20 females in Group 1, 30 males in Group 2, and 40 females in Group 2.

The chisq.test function then performs a chi-square test on the contingency table and returns the test statistic, degrees of freedom, and p-value.

The output will look something like:

Pearson's Chi-squared test with Yates' continuity correction

data: observed
X-squared = 0.44643, df = 1, p-value = 0.504

 

Example – 3

# Generate 100 random numbers from a chi-square distribution 
# with 10 degrees of freedom

data <- rchisq(100, df = 10)

# Perform a chi-squared goodness-of-fit test
expected <- rep(1 / 10, 10)
test <-
chisq.test(table(cut(
data, breaks = seq(0, 30, length.out = 11))), p = expected)

# Print the test results
print(test)

Output

Chi-squared test for given probabilities

data: table(cut(data, breaks = seq(0, 30, length.out = 11)))
X-squared = 128.4, df = 9, p-value < 2.2e-16

In this example, we first use the rchisq() function to generate 100 random numbers from a chi-square distribution with 10 degrees of freedom. We then use the cut() function to group the data into 10 equally-spaced intervals between 0 and 30, and use table() to count the number of observations in each interval. We pass these counts to the chisq.test() function along with the expected probabilities for each interval, which are all equal to 1/10 in this case. The function then performs a chi-squared goodness-of-fit test and returns the test statistic, degrees of freedom, and p-value.