Question

What is standard error and why is it use full

Answer 1

Standard error (SE) is a statistical tool that measures the variability of a sample and how well it represents a population. It's useful because it helps identify the accuracy and consistency of a sample, and how much variation there is likely to be between different samples of a population. 

Here are some ways standard error is used:

• Estimating population parameters: Standard error is used to estimate an interval around a population parameter when an effect size statistic is not available. 
• Calculating P values: Standard error is used to calculate P values in many circumstances. 
• Performing regression analyses and hypothesis testing: Standard error is used to perform regression analyses and hypothesis testing. 
• Constructing confidence intervals: Standard error is used to construct confidence intervals. 

The formula for standard error is SE = σ / √n, where σ is the standard deviation and n is the sample size. 

Answer 2

The standard error (SE) is a statistical measure that quantifies the amount of variation or dispersion of a sample statistic (such as the mean) relative to the true population parameter. It reflects how much a sample statistic is expected to vary due to random sampling error.

Formula for Standard Error

The standard error of the mean (SEM) is commonly calculated as:

SE\(= \frac{\sigma}{\sqrt{n}}\)

Where:

• \(\sigma\) is the standard deviation of the population.
• n is the sample size.

If the population standard deviation \((\sigma)\) is unknown, the sample standard deviation (s) is often used:

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

Why Is Standard Error Useful?

1. Estimates Precision of the Sample Statistic
   The SE provides insight into how precise the sample mean (or another statistic) is as an estimate of the population mean. A smaller SE indicates more reliable estimates.

2. Builds Confidence Intervals
   SE is used to calculate confidence intervals, which show the range within which the true population parameter is likely to fall. For example:

   Confidence Interval=\(\bar x\) ± z⋅SE

   Where \(\bar{x}\) is the sample mean and z is the critical value from the standard normal distribution.

3. Conducts Hypothesis Testing
   In statistical tests, the SE is used to determine the significance of sample results by comparing them to a null hypothesis.

4. Accounts for Sample Size
   Since SE decreases as the sample size increases (n), it highlights the effect of sample size on the reliability of an estimate. Larger samples yield smaller SEs and more accurate estimates.

5. Evaluates Sampling Variability
   It helps quantify how much variability is expected in repeated sampling from the population, providing a measure of the stability of the sample statistic.

Example

Suppose a study measures the average height of 100 individuals and finds a sample mean of 170 cm with a sample standard deviation of 10 cm. The SE of the mean would be:

SE \(= \frac{10}{\sqrt{100}} \)​=1cm.

This implies that the sample mean is expected to vary by about 1 cm around the true population mean due to random sampling variability.