What is the t -distribution? Like the normal distribution, the t- distribution has a smooth shape. Like the normal distribution, the t- distribution is symmetric. If you think about folding it in half at the mean, each side will be the same.
Like a standard normal distribution or z-distribution , the t- distribution has a mean of zero. The normal distribution assumes that the population standard deviation is known.
The t- distribution does not make this assumption. The t- distribution is defined by the degrees of freedom. These are related to the sample size.
The t- distribution is most useful for small sample sizes, when the population standard deviation is not known, or both. As the sample size increases, the t- distribution becomes more similar to a normal distribution. All of the distributions have a smooth shape. All are symmetric. All have a mean of zero. There are actually many different t distributions.
The particular form of the t distribution is determined by its degrees of freedom. The degrees of freedom refers to the number of independent observations in a set of data. When estimating a mean score or a proportion from a single sample, the number of independent observations is equal to the sample size minus one. Hence, the distribution of the t statistic from samples of size 8 would be described by a t distribution having 8 - 1 or 7 degrees of freedom.
Similarly, a t distribution having 15 degrees of freedom would be used with a sample of size For other applications, the degrees of freedom may be calculated differently.
We will describe those computations as they come up. The t distribution can be used with any statistic having a bell-shaped distribution i. The sampling distribution of a statistic should be bell-shaped if any of the following conditions apply. The t distribution should not be used with small samples from populations that are not approximately normal.
When a sample of size n is drawn from a population having a normal or nearly normal distribution, the sample mean can be transformed into a t statistic, using the equation presented at the beginning of this lesson. We repeat that equation below:.
The t statistic produced by this transformation can be associated with a unique cumulative probability. This cumulative probability represents the likelihood of finding a sample mean less than or equal to x , given a random sample of size n.
The easiest way to find the probability associated with a particular t statistic is to use the T Distribution Calculator , a free tool provided by Stat Trek. Financial Ratios. Portfolio Management. Your Privacy Rights. To change or withdraw your consent choices for Investopedia.
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I Accept Show Purposes. Your Money. Personal Finance. Your Practice. Popular Courses. Trading Technical Analysis. What Is a T Distribution? Key Takeaways The T distribution is a continuous probability distribution of the z-score when the estimated standard deviation is used in the denominator rather than the true standard deviation.
Frequently asked questions about the t-distribution What is a t-distribution? The t -distribution is a way of describing a set of observations where most observations fall close to the mean , and the rest of the observations make up the tails on either side. The t -distribution forms a bell curve when plotted on a graph.
It can be described mathematically using the mean and the standard deviation. The t -distribution gives more probability to observations in the tails of the distribution than the standard normal distribution a. In this way, the t -distribution is more conservative than the standard normal distribution: to reach the same level of confidence or statistical significance , you will need to include a wider range of the data.
A t -score a. The t -score is the test statistic used in t -tests and regression tests. It can also be used to describe how far from the mean an observation is when the data follow a t -distribution. A test statistic is a number calculated by a statistical test. It describes how far your observed data is from the null hypothesis of no relationship between variables or no difference among sample groups.
The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis. Different test statistics are used in different statistical tests. A critical value is the value of the test statistic which defines the upper and lower bounds of a confidence interval , or which defines the threshold of statistical significance in a statistical test. It describes how far from the mean of the distribution you have to go to cover a certain amount of the total variation in the data i.
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