T-tests are commonly used in statistics and econometrics to establish that the values of two outcomes or variables are different from one another.

The common assumptions made when doing a t-test include those regarding the scale of measurement, random sampling, normality of data distribution, adequacy of sample size and equality of variance in standard deviation.

### Key Takeaways

- A t-test a statistic method used to determine if there is a significant difference between the means of two groups based on a sample of data.
- The test relies on a set of assumptions for it to be interpreted properly and with validity.
- Among these assumptions, the data must be randomly sampled from the population of interest and that the data variables follow a normal distribution.

## The T-Test

The t-test was developed by a chemist working for the Guinness brewing company as a simple way to measure the consistent quality of stout. It was further developed and adapted, and now refers to any test of a statistical hypothesis in which the statistic being tested for is expected to correspond to a t-distribution if the null hypothesis is supported.

A t-test is an analysis of two populations means through the use of statistical examination; a t-test with two samples is commonly used with small sample sizes, testing the difference between the samples when the variances of two normal distributions are not known.

T-distribution is basically any continuous probability distribution that arises from an estimation of the mean of a normally distributed population using a small sample size and an unknown standard deviation for the population. The null hypothesis is the default assumption that no relationship exists between two different measured phenomena. (For related reading, see: *What does a strong null hypothesis mean?*)

## T-Test Assumptions

- The first assumption made regarding t-tests concerns the scale of measurement. The assumption for a t-test is that the scale of measurement applied to the data collected follows a continuous or ordinal scale, such as the scores for an IQ test.
- The second assumption made is that of a simple random sample, that the data is collected from a representative, randomly selected portion of the total population.
- The third assumption is the data, when plotted, results in a normal distribution, bell-shaped distribution curve. When a normal distribution is assumed, one can specify a level of probability (alpha level, level of significance,
*p*) as a criterion for acceptance. In most cases, a 5% value can be assumed. - The fourth assumption is a reasonably large sample size is used. A larger sample size means the distribution of results should approach a normal bell-shaped curve.
- The final assumption is homogeneity of variance. Homogeneous, or equal, variance exists when the standard deviations of samples are approximately equal.