One-Sample T-Test

                The One-sample t-test compares the mean of a normally distributed variable with a hypothesized value (true mean). The test uses the standard deviation of the sample s to estimate the population standard deviation . To run t-test for multiple variables at the same time, simply select the variable with hypothesized values instead of entering the hypothesized value.


The test assumes the population is normally distributed.

In case the population is not normally distributed, the test statistic has different and unknown distribution and, strictly speaking, the t-test cannot be used. However, according to the central limit theorem, if the sample size is large enough, we can use the t-test even if the normal distribution requirement is not met. Unfortunately there is no easy way to determine what N is large enough. In each particular case there is a separate limit that depends on how strongly the distribution differs from the normal distribution. Some sources quote N=30 as large enough, but even this sample size may turn out to be not large enough. Non-parametric tests may work as an alternative in this case.

How To

Run Statistics→Basic Statistics→One Sample T-Test...

Select one or more variables.

Select a variable with hypothesized values (if you have a distinct hypothesized value for each variable) or enter a hypothesized value as a number (if you have the same hypothesized value for each variable or you are running the t-test for one variable).

Null hypothesis is defined as H0: . Optionally, you can define the alternative hypothesis H1 in the Advanced Options:

o   H1: less than (lower-tailed) ,

o   H1: not equal (two-tailed),

o   H1: greater than (upper-tailed).

Default value is H1: Not equal (two-tailed).



                Report includes descriptive statistics summary and results of the one sample t-test for each variable.

Mean, Mean LCL, Mean UCL, Standard Error (of Mean), Sample Size – see the Descriptive Statistics procedure for more information. 

Hypothesized value (Test Constant, H0 Value) - the hypothesized value (fixed estimate).

Difference – difference between the sample mean and the hypothesized value.

Test statistic– the number   defined as               

where  is the standard deviation of the sample,   – sample mean,  – sample size.

Test statistic either exactly follows or closely approximates a t-distribution with N-1 degrees of freedom under the null hypothesis.

d.f. - appropriate degrees of freedom for each variable.

p-value (*-tailed) –p-value for the test, corresponding to the selected H1.

H1: Not equal (two-tailed): If the p-value is less than default  (default value 0.05), statistically, the sample mean is significantly different from the hypothesized value.