Question : What is Standard error of the mean?

Mejbah Ahammad -

The standard error of the mean (SEM) is a statistical term that measures the accuracy with which a sample represents a population. In other words, it estimates how much the sample mean is likely to deviate from the true population mean.

The formula to calculate the standard error of the mean is:

SEM = σ / √n

where:

  • SEM = standard error of the mean
  • σ = population standard deviation
  • n = sample size

To break it down step by step:

  1. Population standard deviation (σ): This is a measure of the amount of variation or dispersion in the population. It represents how spread out the values are from the mean.
  2. Sample size (n): This is the number of observations or data points in the sample.
  3. Square root of the sample size (√n): This is used to normalize the sample size, as larger samples tend to have smaller standard errors.

By dividing the population standard deviation by the square root of the sample size, we get an estimate of the standard error of the mean. This value represents the average distance between the sample mean and the true population mean.

For example, let's say we have a population with a standard deviation of 10 and we take a sample of 100 observations. The standard error of the mean would be:

SEM = 10 / √100 = 10 / 10 = 1

This means that we can expect the sample mean to be within 1 unit of the true population mean, on average.

Summary

  • Definition and Purpose: The Standard Error of the Mean (SEM) measures the precision of the sample mean in estimating the population mean, indicating the average distance the sample mean is expected to deviate from the actual population mean.
  • Calculation: SEM is calculated using the formula SEM = σ / √n
​​ where σ is the population standard deviation, and n is the sample size. This calculation shows that larger samples tend to produce a more accurate estimation of the population mean, as they have a smaller SEM.
  • Practical Example: For a population with a standard deviation of 10 and a sample size of 100, the SEM would be 1. This result suggests that the sample mean is likely to be within one unit of the true population mean, emphasizing the reliability of the sample mean as an estimator.