How to Apply One Sample T-Test on All Columns of an R Data Frame

The one-sample t-test is a fundamental statistical test that helps determine whether the mean of a sample significantly differs from a specified value (often called the "population mean" or "null value"). In R, we often have multiple columns of data in a data frame, and we might want to perform a one-sample t-test for each column simultaneously. This article provides a complete guide on how to apply a one-sample t-test to all columns of a data frame in the R Programming Language.

Introduction to One-Sample t-Test

A one-sample t-test is used to test whether the mean of a single sample is significantly different from a known or hypothesized population mean. It is particularly useful when you have one group of data and want to compare it against a reference value. For example, suppose we have a data frame containing scores of students in different subjects, and we want to test if the mean score in each subject differs from a specified value (e.g., 75).

When to Use a One-Sample t-Test

Use a one-sample t-test when:

• You have one sample of data and want to compare its mean to a known or hypothesized population mean.
• Your data is approximately normally distributed (especially important when sample size is small).

To conduct a one-sample t-test in R, you can use the t.test() function. Here’s a simple example using a single column of data:

# Sample data
sample_data <- c(78, 75, 82, 70, 88, 92, 85, 80)

# Conduct one-sample t-test
t.test(sample_data, mu = 75)  # Test if mean differs from 75

Output:

	One Sample t-test

data:  sample_data
t = 2.4876, df = 7, p-value = 0.04174
alternative hypothesis: true mean is not equal to 75
95 percent confidence interval:
 75.30896 87.19104
sample estimates:
mean of x 
    81.25 

If the p-value is less than 0.05, you reject the null hypothesis and conclude that the sample mean differs significantly from 75.

Applying One-Sample t-Test to All Columns in an R Data Frame

When dealing with a data frame with multiple columns, you may want to apply the one-sample t-test to all columns at once. We’ll use functions such as sapply() or lapply() to iterate over each column. Let's assume we want to test whether the mean of each column differs from a population mean of 75. We'll use the sapply() function to iterate over each column and apply t.test().

# Sample data frame with 3 columns
data <- data.frame(
  Math = c(78, 75, 82, 70, 88, 92, 85, 80, 76, 81),
  Science = c(85, 88, 80, 78, 92, 86, 89, 91, 87, 84),
  English = c(72, 74, 76, 71, 75, 79, 73, 77, 80, 78)
)

# Display the data frame
print(data)

# Applying one-sample t-test to all columns
t_test_results <- sapply(data, function(column) {
  test_result <- t.test(column, mu = 75)
  return(c(
    Mean = mean(column),
    t_value = test_result$statistic,
    p_value = test_result$p.value,
    conf_low = test_result$conf.int[1],
    conf_high = test_result$conf.int[2]
  ))
})

# Display the results in a readable format
t_test_results <- as.data.frame(t(t_test_results))
print(t_test_results)

Output:

     Math Science English
1    78      85      72
2    75      88      74
3    82      80      76
4    70      78      71
5    88      92      75
6    92      86      79
7    85      89      73
8    80      91      77
9    76      87      80
10   81      84      78


      Mean t_value.t      p_value conf_low conf_high
Math    80.7  2.780947 2.136765e-02 76.06334  85.33666
Science 86.0  7.778175 2.769079e-05 82.80083  89.19917
English 75.5  0.522233 6.141173e-01 73.33415  77.66585

If the p-value for a column is less than 0.05, it indicates that the mean of that column is significantly different from 75.

Conclusion

Applying a one-sample t-test to all columns of a data frame in R is straightforward with the use of functions like sapply() and t.test(). This method allows for efficient hypothesis testing across multiple columns, making it easier to identify which variables significantly differ from a specified value. Visualization using ggplot2 can further enhance the interpretation of your results. By mastering these techniques, you can confidently perform one-sample t-tests on multiple columns, enabling more robust statistical analysis of your data.