One Sample t-test
Sometimes the objective is not to compare two groups, but to compare a sample mean against a known or expected value.
The One Sample t-test is used to determine whether the mean of a sample differs significantly from a specified value.
When to Use a One Sample t-test
This test is appropriate when:
- You have a single numerical (scale) variable
- You want to compare it to a known or hypothesized value
- The data is approximately normally distributed
Typical use cases include:
- Comparing average salary to an industry standard
- Checking if average exam score differs from a passing mark
- Comparing production output to a target value
Example Dataset
Assume a company claims that the average monthly sales per employee is ₹50,000.
You collect data from a sample of employees:
| Employee_ID | Monthly_Sales |
|---|---|
| 1501 | 47000 |
| 1502 | 52000 |
| 1503 | 48000 |
| 1504 | 51000 |
The question is: Is the true average sales value different from ₹50,000?
Formulating Hypotheses
Before running the test, we define hypotheses:
- Null hypothesis (H₀): μ = 50,000
- Alternative hypothesis (H₁): μ ≠ 50,000
The one-sample t-test evaluates whether the observed sample mean provides enough evidence to reject the null hypothesis.
Key Assumptions
The assumptions for a one-sample t-test include:
- Data is numeric
- Observations are independent
- Data is approximately normally distributed
Normality becomes especially important for small sample sizes.
Running One Sample t-test (Menu)
To run the test using SPSS menus:
- Go to Analyze → Compare Means → One-Sample T Test
- Move the variable into Test Variable(s)
- Enter the Test Value (e.g., 50000)
- Click OK
SPSS outputs descriptive statistics and the one-sample t-test results.
Using SPSS Syntax
T-TEST
/TESTVAL=50000
/VARIABLES=Monthly_Sales
/CRITERIA=CI(.95).
This syntax compares the sample mean against ₹50,000.
Interpreting the Output
Key values to examine:
- Mean Difference – difference from test value
- t-value – magnitude of difference
- Sig. (p-value) – statistical significance
Interpretation rule:
- p < 0.05 → sample mean significantly differs from test value
- p ≥ 0.05 → no significant difference
Always report both the mean and the p-value.
Common Mistakes
Typical errors include:
- Using the test for categorical data
- Ignoring normality assumptions
- Confusing one-sample with independent tests
Correct test selection ensures valid conclusions.
Quiz 1
What does a one sample t-test compare?
A sample mean against a known value.
Quiz 2
What type of variable is required?
Numerical (scale) variable.
Quiz 3
What does p < 0.05 indicate?
The sample mean significantly differs from the test value.
Quiz 4
Where do you enter the comparison value in SPSS?
In the Test Value field.
Quiz 5
Is normality important for small samples?
Yes.
Mini Practice
A factory claims that the average weight of packaged goods is 1 kg.
Collect a sample of weights and perform a one-sample t-test to verify this claim.
Use Analyze → Compare Means → One-Sample T Test, set Test Value = 1, and interpret the p-value.
What’s Next
In the next lesson, you will learn about Chi-Square Tests, used to analyze relationships between categorical variables.