In this post, we will discuss sampling error and the size of sampling error. Sampling error is the difference between a sample statistic (such as a sample mean) and the true population parameter (the actual population mean). Sampling error arises because a sample is being studied instead of the entire population.
Table of Contents
The word “error” in sampling error may be misleading for someone. It does not mean that you made a mistake in your research process. Sampling error is a statistical concept that exists even when your sampling is perfectly random and your execution is flawless.
Cause of Sampling Error
Sampling error is caused by random chance. When someone randomly selects a subset of a population, that specific subset will never have the exact same characteristics as the entire population. This chance variation is sampling error. For example
Suppose you have a large bowl of soup (consider it the population), and you taste a single spoonful (it is a sample). The flavour of that spoonful will probably be very close to the whole bowl, but it might be a tiny bit saltier or have one more piece of vegetable than the average spoonful. This small natural difference is the “Sampling Error”. It is not a mistake that you made; it is an inevitable result of sampling.
How is it measured?
Let $\hat{\theta}$ be a sample statistic and let $\theta$ be its true population parameter, then sampling error is
$$Sampling\,\, Error = \hat{\theta} – \theta$$
For example, $\overline{x}$ be the sample mean and $\mu$ is the true population parameter then
$$Sampling\,\, Error = \overline{x} = \mu$$
The most common way to quantify Sampling Error is the computation of standard error (SE). The computation of the standard error of the mean (SEM) estimates how much the sample average is likely to vary from the true population mean. A smaller standard error means less variability and more precision in the estimate.
The standard error formula is
$$SE = \frac{s}{\sqrt{n}}$$
where $s$ is the sample standard deviation and $n$ is the sample size.
Factors Affecting the Size of Sampling Error
Two main factors control the size of sampling error:
- Sample Size (
n): This is the most important factor.- Larger Sample Size → Smaller Sampling Error. As the sample size increases, the sample becomes a better and better representation of the population. That is, the sampling error shrinks.
- This is why national polls survey thousands of people, not just a few dozen.
- Population Variability (Standard Deviation
s):- More Variable Population → Larger Sampling Error. If the individuals in the population are very diverse (e.g., “ages of all people in a country”), any given sample might be less representative. If the population is very homogeneous (e.g., “diameters of ball bearings from the same machine”), a small sample will be very accurate.
This relationship is captured in the formula for the Standard Error above.
Sampling Error vs. Sampling Bias
This is a crucial distinction.
| Feature | Sampling Error | Sampling Bias (a non-sampling error) |
|---|---|---|
| Cause | Random chance | Flawed sampling method |
| Nature | Unavoidable and measurable | Avoidable and problematic |
| Effect | Causes imprecision (scatter) | Causes inaccuracy (shift) |
| Solution | Increase sample size | Fix the sampling 333method |
- Sampling Error: Firing a rifle multiple times at a target. The shots will cluster tightly (small error) or be spread out (large error) around the bullseye.
- Sampling Bias: The rifle’s scope is miscalibrated. All your shots are consistently off-target in one direction, missing the true bullseye.
Sampling Error: Real World Example
Suppose you want to know the average height of all 10000 students at the university (the population). The average height is 5’8″ (the parameter is known to you). You take a random sample of 100 students and calculate their average height. It comes out to 5’7.5″. You take another random sample of 100 different students, the average for this sample is 5’8.5″.
The difference between your first sample’s results (5’7.5′) and the true value (5’8″) is -0.5inches. This is the sampling error for that first sample. The difference for the second sample is +0.5 inches. This is the sampling error for the second sample.
This variation is natural and expected. Similarly, if the sample size is increased to 500 students, the sample averages (e.g., 5’7.9″, 5’8.1″) would likely be much closer to the true 5’8″, meaning that the sampling error would be smaller.
Sampling Error: Summary
- What it is: Natural variation between a sample and the population.
- What it’s not: A mistake or bias in the research design.
- Why it matters: It tells us the precision of our sample-based estimates.
- How to reduce it: Increase the sample size.
- How to measure it: Calculate the Standard Error (SE).
FAQs about Sampling Error and Size of Sampling Error
- What is sampling error?
- What is meant by the size of sampling error?
- How can sampling error be reduced?
- Give some real-world examples related to sampling error.
- How is sampling error computed?
- Describe the causes of sampling error.
- What is the difference between error, sampling error, and sampling bias
