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 is 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 the data variables must 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 population 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.

## 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.