# Relationship between degrees of freedom and shape distribution

### Degrees of freedom (statistics) - Wikipedia

Similarly, a t distribution having 15 degrees of freedom would be used with a sample of The t distribution can be used with any statistic having a bell-shaped . The following figures illustrate the relationship between the t-distribution with different degrees of freedom and the standard normal distribution. The first figure . We usually talk about degrees of freedom, which are often denoted by ν, The shape of the t‐distribution depends on ν.

It can found in the Stat Trek main menu under the Stat Tools tab.

- Degrees of freedom (statistics)

Or you can tap the button below. For example, suppose we were interested in the t statistic having a cumulative probability of 0.

### Chi Square Distribution

We would refer to the t statistic as t0. For example, with 2 degrees of freedom, t0.

Because the t distribution is symmetric about a mean of zero, the following is true. The CEO claims that an average Acme light bulb lasts days. A researcher randomly selects 15 bulbs for testing.

**An Introduction to the Chi-Square Distribution**

The sampled bulbs last an average of days, with a standard deviation of 50 days. If the CEO's claim were true, what is the probability that 15 randomly selected bulbs would have an average life of no more than days?

There are two ways to solve this problem, using the T Distribution Calculator.

Both approaches are presented below. Solution A is the traditional approach. It requires you to compute the t statistic, based on data presented in the problem description.

Then, you use the T Distribution Calculator to find the probability. Solution B is easier. You simply enter the problem data into the T Distribution Calculator. The calculator computes a t statistic "behind the scenes", and displays the probability. Both approaches come up with exactly the same answer. Solution A The first thing we need to do is compute the t statistic, based on the following equation: Now, we are ready to use the T Distribution Calculator.

## Degrees of Freedom Tutorial

Since we know the t statistic, we select "T score" from the Random Variable dropdown box. Then, we enter the following data: The t statistic is equal to - 0. The calculator displays the cumulative probability: Hence, if the true bulb life were days, there is a This time, we will work directly with the raw data from the problem. We will not compute the t statistic; the T Distribution Calculator will do that work for us.

Since we will work with the raw data, we select "Sample mean" from the Random Variable dropdown box. Assuming the CEO's claim is true, the population mean equals The sample mean equals The standard deviation of the sample is Hence, there is a Problem 2 Suppose scores on an IQ test are normally distributed, with a population mean of Suppose 20 people are randomly selected and tested.

## Student's t Distribution

The standard deviation in the sample group is What is the probability that the average test score in the sample group will be at most ? To solve this problem, we will work directly with the raw data from the problem.

Now, because we are interested in cell means differences in a specific way i. It is also important to realize that if all marginal means are fixed, the grand mean is fixed too. In other words, we do not have to worry about the grand mean anymore for calculating our df1 in a two-way ANOVA, because we are already worrying about the marginal means.

As a consequence, our df1 will not lose a degree of freedom because we do not want to get to a specific grand mean.

### interpretation - How to understand degrees of freedom? - Cross Validated

Our df1 will only lose degrees of freedom to get to the specific marginal means. Now, how many cell means are free to vary before we need to fill in the other cell means to get to the four marginal means in the 2 x 2 design? There goes one degree of freedom. We also know the marginal mean for brown eyed males and brown eyed females together.

And as we know the other two marginal means, we have no choice in what we put in the blue eyed females cell mean to get to the correct marginal means. So, we chose one cell mean, and the other three cell means had to be filled in as a consequence to get to the correct marginal means.

We only have one degree of freedom in df1 for a 2 x 2 design. The same number of groups and they might even contain the same observations, but we get a different number of degrees of freedom. So now you see that using the degrees of freedom, you can infer a lot about the design of the test. You could do the same mental exercise for a 2 x 3 design, but it is tedious for me to write up, so I am going to give you the general rule.

Every variable in your design has a certain number of levels. Variable 1 in the 2 x 3 design has 2 levels, variable 2 has 3 levels.

You get df1 when you multiply the levels of all variables with each other, but with each variable, subtract one level. By the way, most statistical programs give you this value for free.