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**Difference between Z Distribution and T Distribution**

The **Z** distribution is a special case of the normal distribution with a mean of **0** and standard deviation of **1**. The **t-distribution** is similar to the **Z-distribution**, but is sensitive to sample size and is used for small or moderate samples when the population standard deviation is unknown. At large samples, the **z** and **t** **samples** are very similar.

The **t-statistic **is used to test hypotheses about an unknown population mean **u** when the value of "$ \sigma $" is unknown. The formula for the **t statistic** has the same structure as the **z-score formula**, except that the t statistic uses the estimated standard error in the denominator. The only difference between the **t formula** and the **z-score** formula is that the **z-score** uses the actual population variance, "$ \sigma^2 $" (or the standard deviation) and the **t formula** uses the corresponding sample variance (or standard deviation) when the population value is not known. Simply put, the basic difference between these two is that the t statistic uses sample variance **(s ^{2})** and the z-score uses the population variance "$ \left( \sigma^2 \right) $". To determine how well a

**t-statistic**approximates a

**z-score**, we must determine how well the sample variance approximates the population variance. Basically, for small samples, the t-statistic is used.

"$ \begin{align} Z = \dfrac{X - \mu}{\sigma} &&& t = \dfrac{\bar x - \mu_0}{\frac{s}{\sqrt{n}}}, \end{align}$"

The mean must be known prior to computing the sample variance. This places a restriction on sample variability such that only n-1 scores in a sample are free to vary. The value **n-1** is called the **degrees of freedom (df)** for the sample variance. Degrees of freedom describe the number of scores in a sample that are free to vary. Because the sample mean places a restriction on the value of one score in the sample, there are **n-1** degrees of freedom for the sample.

**The t Distribution and Shape of the t Distribution**

The** t-distribution** will approximate a normal distribution in the same way that a t statistic approximates a z-score. How well a t distribution approximates a normal distribution is determined by **degrees of freedom (df)**. The greater the **sample size (n)** is, the larger the **degrees of freedom (n-1)** are, and the better the **t-distribution** approximates the normal distribution. The exact shape of a t distribution changes with **df**. As **df** gets very large, the t distribution gets closer in shape to a normal **z-score distribution**. Distributions of t are bell-shaped and symmetrical and have a mean of zero. The **t-distribution** tends to be flatter and more spread out, whereas the normal **z-distribution** has more of a central peak.