Standard Errors for Common Statistics: Parameters, Estimators, and Formulas

Standard Errors in Inferential Statistics

Standard error (SE) quantifies the variability of a sample statistic as an estimate of a population parameter. Understanding the relationship between population parameters, their sample estimators, and the associated standard errors is fundamental in inferential statistics, especially for constructing confidence intervals and conducting hypothesis tests.

Key Parameters, Statistics, and Their Standard Errors

The following table summarizes common population parameters, their corresponding sample statistics, and the formulas for their standard errors:

Parameter (Population value)	Statistic (Sample value)	Standard Error
Population Mean ($\mu$)	Sample Mean ($\overline{x}$)	$SE(\overline{x}) = \frac{s}{\sqrt{n}}$
Population Proportion ($p$)	Sample Proportion ($\hat{p}$)	$SE(\hat{p}) = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
Difference of Population Means ($\mu_1 - \mu_2$)	Difference of Sample Means ($\overline{x}_1 - \overline{x}_2$)	$SE(\overline{x}_1 - \overline{x}_2) = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$
Mean of Differences (Paired Data) ($\mu_d$)	Sample Mean of Differences ($\overline{x}_d$)	$SE(\overline{x}_d) = \frac{s_d}{\sqrt{n_d}}$
Population Slope (Regression) ($\beta_1$)	Sample Slope ($b_1$)	$SE(b_1) = \sqrt{\frac{s^2}{SS_{XX}}}$

Definitions and Applications

• Population Parameter: A fixed value describing a characteristic of the entire population (e.g., $\mu$, $p$).

• Sample Statistic: A value computed from sample data, used to estimate the population parameter (e.g., $\overline{x}$, $\hat{p}$).

• Standard Error (SE): The estimated standard deviation of a sample statistic's sampling distribution. It measures the precision of the statistic as an estimator of the parameter.

Examples

• Estimating a Mean: If a sample of $n=25$ students has a sample mean test score $\overline{x}=80$ and sample standard deviation $s=10$, the standard error is $SE(\overline{x}) = \frac{10}{\sqrt{25}} = 2$.

• Estimating a Proportion: In a survey of $n=100$ people, if $\hat{p}=0.6$ favor a policy, $SE(\hat{p}) = \sqrt{\frac{0.6 \times 0.4}{100}} = 0.049$.

• Comparing Two Means: For two groups with $n_1=30$, $s_1=5$ and $n_2=40$, $s_2=6$, $SE(\overline{x}_1 - \overline{x}_2) = \sqrt{\frac{25}{30} + \frac{36}{40}} \approx 1.36$.

Additional info:

• Standard errors are crucial for constructing confidence intervals and conducting hypothesis tests.

• For regression, $SS_{XX}$ is the sum of squares of the independent variable deviations from their mean.