Variance/SD of a sample ($s$) divides by n-1—this is the unbiased estimate of the population’s variance, since otherwise dividing by n underestimates the variance
- Often don’t know population statistics. Estimate pop mean with sample mean, and pop SD with sample SD $s=\sum{\frac{(x-\bar{x})^2}{n-1}}$.
Sampling distribution: distribution of a random sample statistic
- “Variance of the sampling distribution of the sample mean”
- Standard error: SD of the sampling distribution, $\sigma_{\bar{x}}=\sqrt{\frac{\sigma}{n}}$. But if you don’t know $\sigma$ of population, then estimate it with sample SD $s$.
https://stats.stackexchange.com/questions/116889/why-dont-we-use-the-unbiased-sample-variance-to-calculate-the-standard-error
When to use z (normal) vs t distribution/interval/test?
- Use z only when you know the true pop SD or when sample large enough, else use t (source)
- For t, use dof=n-1
- limit of t distribution as n approaches inf is normal
