What is the Standard Error?
The standard error is an estimate of the standard deviation of the sampling distribution of a statistic (e.g., a sample mean or a sample proportion). In this lesson, we explain (1) how to measure the standard deviation and standard error of a sampling distribution and (2) why these measures are important.
How to Measure Variability
Statisticians use sample statistics to estimate population parameters. Naturally, the value of a statistic may vary from one sample to the next. The standard deviation of a sampling distribution and the standard error both measure the same thing. They quantify how much a sample statistic is expected to fluctuate from the true population parameter due to random sampling.
Standard Deviation of Sample Estimates
The table below shows formulas for computing the standard deviation of statistics from simple random samples . These formulas are valid when the population size is much larger (at least 20 times larger) than the sample size.
| Statistic | Standard Deviation |
|---|---|
| Sample mean, x | SD = σ / sqrt( n ) |
| Sample proportion, p | SD = sqrt [ P(1 - P) / n ] |
| Difference between means, x1 - x2 | SD = sqrt [ σ21 / n1 + σ22 / n2 ] |
| Difference between proportions, p1 - p2 | SD = sqrt [ P1(1-P1) / n1 + P2(1-P2) / n2 ] |
Don't worry about when and how to use these formulas. We'll cover those topics in future lessons. For now, here are the main things you need to know about the standard deviation (SD) of a sampling distribution:
- The standard deviation (SD) measures the variability of a sample statistic (e.g., a sample mean or a sample proportion).
- The standard deviation (SD) is computed from a population parameter (either the population standard deviation σ or the population proportion P) and sample size (n).
- The standard deviation (SD) of the sampling distribution is not the same as the standard deviation of the population distribution (σ), which measures the variability of individual data points in the population.
- The standard deviation (SD) of the sampling distribution is not the same as the standard deviation of the sample (s), which measures the variability of individual data points in the sample.
- The standard deviation (SD) decreases as the sample size (n) increases, reflecting greater precision with larger samples.
Standard Error of Sample Estimates
Sadly, the values of population parameters are often unknown, which often makes it impossible to compute the standard deviation of a statistic. When this occurs, use the standard error.
The table below shows formulas for computing the standard error of statistics from simple random samples. These formulas are valid when the population size is much larger (at least 20 times larger) than the sample size.
| Statistic | Standard Error |
|---|---|
| Sample mean, x | SE = s / sqrt( n ) |
| Sample proportion, p | SE = sqrt [ p(1 - p) / n ] |
| Difference between means, x1 - x2 | SE = sqrt [ s21 / n1 + s22 / n2 ] |
| Difference between proportions, p1 - p2 | SE = sqrt [ p1(1-p1) / n1 + p2(1-p2) / n2 ] |
Don't worry about when and how to use these formulas. We'll cover those topics in future lessons. For now, here are the main things you need to know about the standard error (SE) of a sampling distribution:
- Like the standard deviation (SD) of a sampling distribution, the standard error (SE) measures the variability of a sample statistic (e.g., a sample mean or a sample proportion).
- The standard error (SE) is a sample estimate of the standard deviation (SD).
- The standard error (SE) is computed from a sample statistics: either sample standard deviation (s) or the sample proprotion (p) and sample size (n).
- The standard deviation (SE) of the sampling distribution is not the same as the standard deviation of the population distribution (σ), which measures the variability of individual data points in the population.
- The standard deviation (SE) of the sampling distribution is not the same as the standard deviation of the sample (s), which measures the variability of individual data points in the sample.
- The standard error (SE) decreases as the sample size (n) increases, reflecting greater precision with larger samples.
Note: The equations for the standard error are identical to the equations for the standard deviation, except for one thing - the standard error equations use statistics where the standard deviation equations use parameters. Specifically, the standard error equations use p in place of P, and s in place of σ.
Why is the Standard Error Useful?
- Measures precision. The standard error measures how precise a sample statistic is as an estimate of the population parameter. A smaller standard error indicates greater precision.
- Confidence intervals. The standard error is used to construct confidence intervals. (We cover confidence intervals extensively in the next few lessons).
- Hypothesis testing. The standard error is used to determine whether a sample statistic is significantly different from a hypothesized value. (The final lessons in this tutorial cover hypothesis testing.)
- Real-world analyses. We seldom know values for population parameters required to calculate the standard deviation (SD) of a sampling distribution. But we often know values for sample statistics required to calculate the standard error. So, in the real world, the standard error (SE) is used more often than the standard deviation (SD).
In short, the standard error is critical for understanding how much a sample statistic is likely to vary due to random sampling, and it plays a key role in making inferences about population parameters, based on sample statistics.
Test Your Understanding
Problem 1
Which of the following statements is true.
I. The standard error of sample mean is computed from sample attributes .
II. The standard deviation of a sample mean is computed from sample attributes.
III. The standard error is a measure of central tendency.
(A) I only
(B) II only
(C) III only
(D) All of the above.
(E) None of the above.
Solution
The correct answer is (A). The standard error of a sample mean can be computed from a knowledge of sample attributes - sample size (n) and the sample standard deviation (s). The standard deviation of a sample mean cannot be computed solely from sample attributes; it requires a knowledge of the population standard deviation (σ), which is a population parameter - not a sample statistic. The standard error measures the variability of a sampling distribution, not the central tendency.