Statistics Definitions > T Statistic

## What is a T Statistic?

The T Statistic is used in a T test when you are deciding if you should support or reject the null hypothesis. It’s very similar to a Z-score and you use it in the same way: find a cut off point, find your t score, and compare the two. You use the t statistic when you have a small sample size, or if you don’t know the population standard deviation.

The T statistic doesn’t really tell you much on its own. It’s like the word “average” doesn’t mean anything on its own either, without some context. If I say “the average was 150,” it means nothing. If I say “the average weight of dogs seen in a veterinary office was 50lbs,” then the picture becomes clearer. In the same way, you need some more information along with your t statistic for it to make sense. You get this information by taking a sample and running a hypothesis test.

## What is the T Statistic used for?

When you run a hypothesis test, you **use the T statistic with a p value**. The p-value tells you what the odds are that your results could have happened by chance. Let’s say you and a group of friends score an average of 205 on a bowling game. You know the average bowler scores 79.7. Should you and your friends consider professional bowling? Or are those scores a fluke? Finding the t statistic and the probability value will give you a good idea. More technically, finding those values will give you evidence of a significant difference between your team’s mean and the population mean (i.e. everyone).

**The greater the T**, the more evidence you have that your team’s scores are significantly different from average. **A smaller T value **is evidence that your team’s score is *not* significantly different from average. It’s pretty obvious that your team’s score (205) is significantly different from 79.7, so you’d want to take a look at the probability value. If the p-value is larger than 5%, the odds are your team getting those scores are due to chance. Very small (under 5%), you’re onto something: think about going professional.

## T Score vs. Z-Score.

The Z-score allows you to decide if your sample is different from the population mean. In order to use z, you must know four things:

- The population mean.
- The population standard deviation.
- The sample mean.
- The sample size.

Usually in stats, you don’t know **anything **about a population, so instead of a Z score you use a T Test with a T Statistic. **The major difference between using a Z score and a T statistic is that you have to estimate the population standard deviation**. The T test is also used if you have a small sample size (less than 30).

## T Statistic Formula

The statistic can be found in so many different ways, **there is no single formula for it**. The formula depends on what type of test you are trying to do. For example:

- One sample t test. This is the most common type of t test you’ll come across in elementary statistics. You can test the mean of a single group against a known mean. For example, the average IQ is 100. You can test a class of children with a mean score of 90 to see if that’s significant, or if it just happened by chance.
- A paired t-test compares means from the same group at different times (say, one year apart). For example, you could try a new weight loss technique on a group of people and follow up a year later.

**It’s extremely unlikely you’ll want to calculate this statistic by hand**. The math is complicated, tedious, time-consuming and prone to errors. Instead, you’ll want to use software like SPSS for Statistics or Excel for Statistics.

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## References

Everitt, B. S.; Skrondal, A. (2010), The Cambridge Dictionary of Statistics, Cambridge University Press.

Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.

Levine, D. (2014). Even You Can Learn Statistics and Analytics: An Easy to Understand Guide to Statistics and Analytics 3rd Edition. Pearson FT Press