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What is a t-statistic?

What is a z-score?

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## T-Score vs. Z-Score: Overview

A **z-score** and a **t score **are both used in hypothesis testing. Few topics in elementary statistics cause more confusion to students than deciding when to use the z-score and when to use the t score. Generally, in elementary stats and AP stats, you’ll use a z-score in testing more often than a t score.

## T-score vs. z-score: When to use a t score

The general rule of thumb for *when* to use a t score is when your sample:

- Has a sample size below 30,
- Has an unknown population standard deviation.

You **must** know the standard deviation of the **population** *and *your sample size **should **be above 30 in order for you to be able to use the z-score. Otherwise, use the t-score.

The above chart is based on (from my experience), the “rule” you’re most likely to find in an elementary statistics class. That said, **this is one of those rules that isn’t set in stone, so you should always check with your textbook/teacher to make sure they aren’t suggesting something different.**

In real life though, it’s more common just to use the t-distribution as we usually don’t know sigma (SoSci, 1999).

“When a sample has more than 30 observations, the normal distribution

canbe used in place of the t distribution.” (Meier et.al, p. 191).

Note the use of the word *can *in the above quote; The use of the t-distribution is theoretically sound for all sample sizes, but you *can* choose to use the normal for sample above 30.

### T-Score vs. Z-Score: Z-score

Technically, z-scores are a conversion of individual scores into a standard form. The conversion allows you to more easily compare different data; it is based on your knowledge about the **population’s standard deviation and mean. **A z-score tells you how many standard deviations from the mean your result is. You can use your knowledge of normal distributions (like the 68 95 and 99.7 rule) or the z-table to determine what percentage of the population will fall below or above your result.

The z-score is calculated using the formula:

**z = (X-μ)/σ**

**Where**:

- σ is the population standard deviation and
- μ is the population mean.

The z-score formula doesn’t say anything about sample size; The rule of thumb applies that your sample size should be above 30 to use it.

### T-Score vs. Z-Score: T-score

Like z-scores, t-scores are also a conversion of individual scores into a standard form. However, t-scores are used **when you don’t know the population standard deviation**; You make an estimate by using your sample.

**T = (X – μ) / [ s/√(n) ].**

Where:

- s is the standard deviation of the sample.

If you have a larger sample (over 30), the t-distribution and z-distribution look pretty much the same. Therefore, you can use either. That said, if you know σ, it doesn’t make much sense to use a sample estimate instead of the “real thing”, so just substitute σ into the equation in place of s:

**T = (X – μ) / [ σ/√(n) ].**

This makes the equation identical to the one for the z-score; the only difference is you’re looking up the result in the T table, not the Z-table. For sample sizes over 30, you’ll get the same result.

## References

Meier et. al. (2014). Applied Statistics for Public and Nonprofit Administration. Cengage Learning.

SoSci. (1999). Article posted on Vermont Tech website. Retrieved 11/20/2016 from https://simon.cs.vt.edu/SoSci/converted/T-Dist/.

**Next**: When to use sigma/sqrt(n)

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*Statistical concepts explained visually* - Includes many concepts such as sample size, hypothesis tests, or logistic regression, explained by Stephanie Glen, founder of StatisticsHowTo.

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