# T-Test Essentials: Definition, Formula and Calculation

## One Sample T-Test Assumptions

This article describes the one sample t-test assumptions and provides examples of R code to check whether the assumptions are met before calculating the t-test.

Contents:

#### Related Book

Practical Statistics in R II - Comparing Groups: Numerical Variables

## Assumptions

The one-sample t-test assumes the following characteristics about the data:

• No significant outliers in the data
• Normality. the data should be approximately normally distributed

In this section, we’ll perform some preliminary tests to check whether these assumptions are met.

## Check one-sample t-test assumptions in R

### Prerequisites

Make sure you have installed the following R packages:

• ggpubr for creating easily publication ready plots
• rstatix provides pipe-friendly R functions for easy statistical analyses.
• datarium: contains required data sets for this chapter.

library(ggpubr)
library(rstatix)

### Demo data

Demo dataset: mice [in datarium package]. Contains the weight of 10 mice:

# Load and inspect the data
data(mice, package = "datarium")
head(mice, 3)
## # A tibble: 3 x 2
##   name  weight
##   <chr>  <dbl>
## 1 M_1     18.9
## 2 M_2     19.5
## 3 M_3     23.1

### Identify outliers

Outliers can be easily identified using boxplot methods, implemented in the R function identify_outliers() [rstatix package].

mice %>% identify_outliers(weight)
## [1] name       weight     is.outlier is.extreme
## <0 rows> (or 0-length row.names)

There were no extreme outliers.

Note that, in the situation where you have extreme outliers, this can be due to: 1) data entry errors, measurement errors or unusual values.

In this case, you could consider running the non parametric Wilcoxon test.

### Check normality assumption

The normality assumption can be checked by computing the Shapiro-Wilk test. If the data is normally distributed, the p-value should be greater than 0.05.

mice %>% shapiro_test(weight)
## # A tibble: 1 x 3
##   variable statistic     p
##   <chr>        <dbl> <dbl>
## 1 weight       0.923 0.382

From the output, the p-value is greater than the significance level 0.05 indicating that the distribution of the data are not significantly different from the normal distribution. In other words, we can assume the normality.

You can also create a QQ plot of the weight data. QQ plot draws the correlation between a given data and the normal distribution.

ggqqplot(mice, x = "weight")

All the points fall approximately along the (45-degree) reference line, for each group. So we can assume normality of the data.

Note that, if your sample size is greater than 50, the normal QQ plot is preferred because at larger sample sizes the Shapiro-Wilk test becomes very sensitive even to a minor deviation from normality.

If the data are not normally distributed, it’s recommended to use a non-parametric test such as the one-sample Wilcoxon signed-rank test. This test is similar to the one-sample t-test, but focuses on the median rather than the mean.

## Related article

T-test in R

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