Identify the statistic you need the standard error for (mean, proportion, difference in means, regression coefficient, etc.)
Mean (sample mean):
If population standard deviation is known: (SE=sigma/sqrt{n})
If population standard deviation is unknown: (SE=s/sqrt{n})
Proportion (sample proportion):
(SE=sqrt{dfrac{hat p(1-hat p)}{n}})
(Alternative using hypothesized (p_0)): (SE=sqrt{dfrac{p_0(1-p_0)}{n}})
Difference in means (independent samples):
(SE=sqrt{dfrac{s_1^2}{n_1}+dfrac{s_2^2}{n_2}})
Difference in proportions (independent samples):
(SE=sqrt{dfrac{hat p_1(1-hat p_1)}{n_1}+dfrac{hat p_2(1-hat p_2)}{n_2}})
Paired difference (mean of paired differences):
Compute differences (d_i), then (SE=s_d/sqrt{n})
Regression coefficient ( hatbeta ):
(SE(hatbeta)=sqrt{text{Var}(hatbeta)}) using the model’s variance-covariance matrix (often reported directly by software)
Generic standard error from bootstrap (if applicable):
Resample many times, compute the statistic each time, then (SE=text{SD}(text{bootstrap estimates}))
Generic standard error from an estimated standard deviation of the estimator:
(SE=sqrt{widehat{text{Var}}(hattheta)})
Use the correct (n) (sample size for the estimator) and the correct variability term ((sigma), (s), (hat p(1-hat p)), or model-based variance)