Standard Error Calculator

What is Standard Error?

Standard Error (SE) is a statistical measure that quantifies the variability of sample means around the true population mean. It provides an estimate of how much sample means are expected to fluctuate from the population mean due to sampling variability.

Formula and Its Meaning

The formula for Standard Error is:

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

Where:

• \(s\) is the sample standard deviation
• \(n\) is the sample size

This formula shows that the Standard Error decreases as the sample size increases, indicating that larger samples provide more precise estimates of the population mean.

Calculation Steps

1. Calculate the mean of the sample data.
2. Calculate the standard deviation of the sample data.
3. Divide the standard deviation by the square root of the sample size.

Example Calculation

Let's calculate the Standard Error for the dataset: 5, 20, 40, 80, 100

1. Calculate the mean: \(\bar{x} = \frac{5 + 20 + 40 + 80 + 100}{5} = 49\)
2. Calculate the standard deviation:
   • Squared differences: \((5-49)^2, (20-49)^2, (40-49)^2, (80-49)^2, (100-49)^2\)
   • Variance: \(s^2 = \frac{1936 + 841 + 81 + 961 + 2601}{4} = 1605\)
   • Standard deviation: \(s = \sqrt{1605} \approx 40.06\)
3. Calculate the Standard Error: \(SE = \frac{40.06}{\sqrt{5}} \approx 17.91\)

Visual Representation

This scatter plot represents the example dataset. The red dashed line indicates the mean (49), and the green dashed lines show the standard error range (±17.91 from the mean).