One Sample T-Test

The one sample t-test is a statistical procedure used to determine whether a sample of observations could have been generated by a process with a specific mean. Suppose you are interested in determining whether an assembly line produces laptop computers that weigh five pounds. To test this hypothesis, you could collect a sample of laptop computers from the assembly line, measure their weights, and compare the sample with a value of five using a one-sample t-test.

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There are two kinds of hypotheses for a one sample t-test, the null hypothesis and the alternative hypothesis. The alternative hypothesis assumes that some difference exists between the true mean (\(\mu\)) and the comparison value (\(m_0\)), whereas the null hypothesis assumes that no difference exists. The purpose of the one sample t-test is to determine which of these hypotheses is more likely, given the sample data. The alternative hypothesis can assume one of three forms depending on the question being asked. If the goal is to measure any difference, regardless of direction, a two-tailed hypothesis is used. If the direction of the difference between the sample mean and the comparison value matters, either an upper-tailed or lower-tailed hypothesis is used. The null hypothesis remains the same for each type of one sample t-test. The hypotheses are formally defined below:

  • • The null hypothesis (\(H_0\)) assumes that the difference between the true mean (\(\mu\)) and the comparison value (\(m_0\)) is equal to zero.
  • • The two-tailed alternative hypothesis (\(H_1\)) assumes that the difference between the true mean (\(\mu\)) and the comparison value (\(m_0\)) is not equal to zero.
  • • The upper-tailed alternative hypothesis (\(H_1\)) assumes that the true mean (\(\mu\)) of the sample is greater than the comparison value (\(m_0\)).
  • • The lower-tailed alternative hypothesis (\(H_1\)) assumes that the true mean (\(\mu\)) of the sample is less than the comparison value (\(m_0\)).

The mathematical representations of the null and alternative hypotheses are defined below:

  • \(H_0:\ \mu\ =\ m_0\)
  • \(H_1:\ \mu\ \ne\ m_0\)    (two-tailed)
  • \(H_1:\ \mu\ >\ m_0\)    (upper-tailed)
  • \(H_1:\ \mu\ <\ m_0\)    (lower-tailed)

Note. It is important to remember that hypotheses are never about data, they are about the processes which produce the data. If you are interested in knowing whether the mean weight of a sample of laptops is equal to five pounds, the real question being asked is whether the process that produced those laptops has a mean of five.


As a parametric procedure (a procedure which estimates unknown parameters), the one sample t-test makes several assumptions. Although t-tests are quite robust, it is good practice to evaluate the degree of deviation from these assumptions in order to assess the quality of the results. The one sample t-test has four main assumptions:

  • • The dependent variable must be continuous (interval/ratio).
  • • The observations are independent of one another.
  • • The dependent variable should be approximately normally distributed.
  • • The dependent variable should not contain any outliers.

Level of Measurement

The one sample t-test requires the sample data to be numeric and continuous, as it is based on the normal distribution. Continuous data can take on any value within a range (income, height, weight, etc.). The opposite of continuous data is discrete data, which can only take on a few values (Low, Medium, High, etc.). Occasionally, discrete data can be used to approximate a continuous scale, such as with Likert-type scales.


Independence of observations is usually not testable, but can be reasonably assumed if the data collection process was random and impartial. In our example, we would want to select laptop computers at random, compared to using any systematic pattern. This ensures minimal risk of including some systematic bias in the data that would yield inaccurate results.


To test the assumption of normality, a variety of methods are available, but the simplest is to inspect the data visually using a histogram or a Q-Q scatterplot. Real-world data are almost never perfectly normal, so this assumption can be considered reasonably met if the shape looks approximately symmetric and bell-shaped. The data in the example figure below is approximately normally distributed.

histogram of a normally distributed variable

Figure 1. Histogram of an approximately normally distributed variable.


An outlier is a data value which is too extreme or not extreme enough to belong in the distribution of interest. Let's suppose in our example that the assembly machine ran out of a particular component, resulting in a laptop that was assembled at a much lower weight. This is a condition that is outside of our question of interest, and therefore we can remove that observation prior to conducting the analysis. However, just because a value is extreme does not make it an outlier. Let's suppose that our laptop assembly machine occasionally produces laptops which weigh significantly more or less than five pounds, our target value. In this case, these extreme values are absolutely essential to the question we are asking and should not be removed. Box-plots are useful for visualizing the variability in a sample, as well as locating any outliers. The boxplot on the left shows a sample with no outliers. The boxplot on the right shows a sample with one outlier.

boxplot of a normally distributed variable
boxplot of a cauchy distributed variable

Figure 2. Boxplots of a variable without outliers (left) and with an outlier (right).


The procedure for a one sample t-test can be summed up in four steps. The symbols to be used are defined below:

  • \(Y\ =\ \)Random sample
  • \(y_i\ =\ \)The \(i^{th}\) observation in \(Y\)
  • \(n\ =\ \)The sample size
  • \(m_0\ =\ \)The hypothesized value
  • \(\overline{y}\ =\ \)The sample mean
  • \(\hat{\sigma}\ =\ \)The sample standard deviation
  • \(T\ =\)The critical value of a t-distribution with (\(n\ -\ 1\)) degrees of freedom
  • \(t\ =\ \)The t-statistic (t-test statistic) for a one sample t-test
  • \(p\ =\ \)The \(p\)-value (probability value) for the t-statistic.

The four steps are listed below:

  • 1. Calculate the sample mean.
  • \(\overline{y}\ =\ \cfrac{y_1\ +\ y_2\ +\ \cdots\ +\ y_n}{n}\)
  • 2. Calculate the sample standard deviation.
  • \(\hat{\sigma}\ =\ \sqrt{\cfrac{(y_1\ -\ \overline{y})^2\ +\ (y_2\ -\ \overline{y})^2\ +\ \cdots\ +\ (y_n\ -\ \overline{y})^2}{n\ -\ 1}}\)
  • 3. Calculate the test statistic.
  • \(t\ =\ \cfrac{\overline{y}\ -\ m_0}{\hat{\sigma}/\sqrt{n}}\)
  • 4. Calculate the probability of observing the test statistic under the null hypothesis. This value is obtained by comparing t to a t-distribution with (\(n\ -\ 1\)) degrees of freedom. This can be done by looking up the value in a table, such as those found in many statistical textbooks, or with statistical software for more accurate results.
  • \(p\ =\ 2\ \cdot\ Pr(T\ >\ |t|)\)    (two-tailed)
  • \(p\ =\ Pr(T\ >\ t)\)    (upper-tailed)
  • \(p\ =\ Pr(T\ <\ t)\)    (lower-tailed)

Once the assumptions have been verified and the calculations are complete, all that remains is to determine whether the results provide sufficient evidence to favor the alternative hypothesis over the null hypothesis.


There are two types of significance to consider when interpreting the results of a one sample t-test, statistical significance and practical significance.

Statistical Significance

Statistical significance is determined by looking at the \(p\)-value. The \(p\)-value gives the probability of observing the test results under the null hypothesis. If the assumptions are true, smaller \(p\)-values indicate a result that is less likely to occur by chance. This also indicates decreased support for the null hypothesis, although this possibility can never be ruled out completely. The cutoff value at which statistical significance is claimed is decided on by the researcher but usually a value of .05 or less is chosen, ensuring approximately 95% confidence in the results.

Practical Significance

Practical significance depends on the subject matter. It is not uncommon, especially with large sample sizes, to observe a result that is statistically significant but not practically significant. In most cases, both types of significance are required in order to draw meaningful conclusions.