This range of values where we think it is can get narrower (e.g. ![]() Basically, when we make a confidence interval we are sort of 1 saying that "We think the thing we are estimating is in this range of values". I am not going to type the confidence interval formula here you can watch my videos or check 1,000's of books and websites for that. If we assume degrees of freedom is 30 (n=31), then the 95% confidence interval is 100 ± 3.668. If we assume a degrees of freedom =1 (n=2), the 95% confidence interval is 100 ± 89.8464. For example, suppose we had a sample mean of 100 and sample standard deviation of 10. ![]() Going from df=1 to df=30 makes a huge difference in the width of a confidence interval. How much would it change? It depends on how wrong we want to make the degrees of freedom. What if we used the wrong degrees of freedom? The "statistics" would be right, but confidence intervals and inferences would be wrong. As your sample size increases, you get better information about both of them, and the degrees of freedom captures the precision of the estimation of the standard deviation (or variance). The t distribution was invented by William Gosset to take into account that to make a confidence interval of something like a sample mean, you usually know neither the mean nor the standard deviation. In this case it is a measure of how much information we have to estimate the variance of the parameter of interest. In the particular case of the t, it is much easier. If you use the wrong degrees of freedom, you are calculating the wrong area under the wrong curve to calculate things such as p values. In general, "degrees of freedom" is a term that is hard to grasp, and the best way to think about it is "It (or they, in the case of the F) is just a number that tells us the shape of a distribution".
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