Standard Error Calculator

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Standard Error

Definition

The standard error (SE) is a measure of the variability of a sample statistic (such as the mean) across multiple samples. It indicates how precisely the sample statistic estimates the population parameter.

Key Formulas

Standard Error of the Mean:

SE = \frac{s}{\sqrt{n}}

s = sample standard deviation
n = sample size

Examples

Standard Error Calculation:

For dataset: $[5, 7, 8, 9, 10]$

\begin{align*} \text{Mean} &= \frac{5 + 7 + 8 + 9 + 10}{5} = 7.8 \\ \text{Standard Deviation} &= \sqrt{\frac{(5 - 7.8)^2 + (7 - 7.8)^2 + (8 - 7.8)^2 + (9 - 7.8)^2 + (10 - 7.8)^2}{5 - 1}} \approx 1.92 \\ \text{Standard Error} &= \frac{1.92}{\sqrt{5}} \approx 0.86 \end{align*}

Interpretation Guide

Smaller SE indicates more precise estimate
SE decreases as sample size increases
Used in constructing confidence intervals

Limitations & Considerations

Assumes random sampling
Affected by outliers and non-normal distributions
May not be reliable for very small samples

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