One Sample T Test: Definition, Using & Example

What is a One Sample T Test?

Use a one sample t test to evaluate a population mean using a single sample. Usually, you conduct this hypothesis test to determine whether a population mean differs from a hypothesized value you specify. The hypothesized value can be theoretically important in the study area, a reference value, or a target.

For example, a beverage company claims its soda cans contain 12 ounces. A researcher randomly samples their cans and measures the amount of fluid in each one. A one-sample t-test can use the sample data to determine whether the entire population of soda cans differs from the hypothesized value of 12 ounces.

In this post, learn about the one-sample t-test, its hypotheses and assumptions, and how to interpret the results.

One Sample T Test Hypotheses

A one sample t test has the following hypotheses:

Null hypothesis (H₀): The population mean equals the hypothesized value (µ = H₀).
Alternative hypothesis (H_A): The population mean does not equal the hypothesized value (µ ≠ H₀).

If the p-value is less than your significance level (e.g., 0.05), you can reject the null hypothesis. The difference between the sample mean and the hypothesized value is statistically significant. Your sample provides strong enough evidence to conclude that the population mean does not equal the hypothesized value.

Learn how this analysis compares to the Z Test.

One Sample T Test Assumptions

For reliable one sample t test results, your data should satisfy the following assumptions:

Random Sample

Drawing a random sample from your target population helps ensure your data represent the population. Samples that don’t reflect that population tend to produce invalid results.

Continuous Data

One-sample t-tests require continuous data. These variables can take on any numeric value, and the scale can be split meaningfully into smaller increments. For example, temperature, height, weight, and volume are continuous data.

Read Comparing Hypothesis Tests for Continuous, Binary, and Count Data for more information..

Normally distributed data or your sample has more than 20 observations

This hypothesis test assumes your data follow the normal distribution. However, your data can be mildly skewed when the distribution is unimodal and your sample size is greater than 20 because of the central limit theorem.

Be sure to check for outliers because they can throw off the results.

Related posts: Central Limit Theorem, Skewed Distributions, and 5 Ways to Find Outliers.

Independent Observations

The one-sample t-test assumes that observations are independent of each other, meaning that the value of one observation does not influence or depend on another observation’s value. Violating this assumption can lead to inaccurate results because the test relies on the premise that each data point provides unique and separate information.

Example One Sample T Test

Let’s return to the 12-ounce soda can example and perform a one-sample t-test on the data. Imagine we randomly collected 30 cans of soda and measured their contents.

We want to determine whether the difference between the sample mean and the hypothesized value (12) is statistically significant. Download the CSV file that contains the example data: OneSampleTTest. Use my T Test Calculator to perform this analysis yourself. Just download and open the dataset in a spreadsheet. Then copy and paste the cells into my calculator!

Here is how a portion of the data appear in the worksheet.

The histogram shows the data are not skewed, and no outliers are present.

Interpreting the Results

Here’s how to read and report the results for a one sample t test.

The statistical output indicates that the sample mean (A) is 11.8013. Because the p-value (B) of 0.000 is less than our significance level of 0.05, the results are statistically significant. We reject the null hypothesis and conclude that the population mean does not equal 12 ounces. Specifically, it is less than that target value. The beverage company is underfilling the cans.

Learn more about Statistical Significance: Definition & Meaning.

The confidence interval (C) indicates the population mean for all cans is likely between 11.7358 and 11.8668 ounces. This range excludes our hypothesized value of 12 ounces, reaffirming the statistical significance. Learn more about confidence intervals.

To learn more about performing t-tests and how they work, read the following posts:

One Sample T Test: Definition, Using & Example

What is a One Sample T Test?

One Sample T Test Hypotheses