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Why do we subtract 1 from the sample size to calculate degree of freedom?

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@B.Rabbit wrote:

Hello People,
I'm currently doing Inferential statistics course from Udacity. I'm stuck at t-distribution/degrees of freedom.
Suppose we are required to plot the t-distribution graph for a sample size 'n'. I don't quite understand why we fix the mean(sample mean) and draw samples to plot the t-distribution graph. Hence concluding that df(degrees of freedom) is n - 1. Instead we could draw random samples of the size n and compute the corresponding t-value and plot it in the graph without fixing the mean. Hence conclude df(degrees of freedom) as n. What is the flaw in my argument?
Regards

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