1.2.4. Sampling Error:
Benchmark Question
Imagine you are conducting a survey to determine the average height of students in a high school. You measure the heights of 30 students from one class and find an average height of 165 cm. If you measured the heights of all students in the school, would the average height be exactly the same as what you found in your sample? Why or why not?
In any sampling process, the results obtained from a sample might differ from the results that would be obtained if the entire population were surveyed. This difference is known as sampling error. Understanding and managing sampling error is crucial to ensuring that conclusions drawn from a sample are as accurate as possible.
Sampling Error is the difference between the result obtained from a sample and the result that would have been obtained if the entire population had been surveyed. It arises because the sample may not perfectly represent the population. The extent of sampling error depends on factors like:
- Less-Representative Sample: If the sample not perfectly reflect the characteristics of the whole population, the sampling error will be high.
- Small Sample size: The size of the sample directly impacts sampling error. Larger samples tend to provide more accurate estimates of the population parameters and reduce sampling error.
Definitions
Sampling Error (\(S_E\)) occurs due to the natural variability between a sample and the population from which it is drawn. It’s the difference between the sample statistic (like a sample mean) and the true population parameter (like the population mean).
The sample statistic often differs from the true population parameter. For example, the discrepancy between the sample mean and the population mean is referred to as the sampling error of the sample mean.
Example 1 Sampling Error
Consider a population with the following 6 observations:
[10, 12, 15, 18, 20, 22].
Suppose we want to illustrate sampling error by taking different samples from this data.
- [10, 12, 15], \( \bar{x} = \frac{10 + 12 + 15}{3} = 12.33 \)
- [12, 18, 22], \( \bar{x} = \frac{12 + 18 + 22}{3} = 17.33 \)
- [10, 15, 18, 20},\( \bar{x} = \frac{10 + 15 + 18 + 22}{4} = 16.25 \)
The sampling error can be observed by comparing these sample means to the population mean \( \mu \), which is;
\[\mu = \frac{10 + 12 + 15 + 18 + 20 + 22}{6} = 16.17 \]Sampling Error is the absolute difference between the sample mean and the population mean.
- For Sample 1: \(S_E = |\bar{x} - \mu\)| = 3.84
- For Sample 2: \(S_E = |\bar{x} - \mu\)| = 1.16
- For Sample 3: \(S_E = |\bar{x} - \mu\)| = 0.08
When selecting samples, sample size and representativeness play crucial roles in minimizing sampling error. For instance, Sample 1, which includes only small values (10, 12, 15), is not representative of the entire population and results in a higher sampling error of 3.84. Sample 2, which is more representative (10, 18, 22), reduces the sampling error to 1.16.
Finally, by increasing both the sample size and representativeness in Sample 3 (10, 15, 18, 22), the sampling error is further reduced to 0.08. This illustrates that as the sample becomes more representative and larger, the sampling error decreases, leading to more accurate estimations of the population parameter.
Activity 3: Sampling error
- Consider the above example 1, what will be the sampling error if the sample contain (10 & 12)?
- What happens to the sampling error if the sample size is equal to the population size?
Answers:
- If we select a sample containing the values (10, 12) from the population, we can calculate the sampling error for this sample. \[\bar{x} = \frac {10 + 12}{2} = 11\] Then, Sampling Error will be: \[S_E = |\bar{x} - \mu| = 5.17\] When the sample contains (10, 12), the sampling error is -5.17. This substantial error indicates that the sample is not representative of the population because it only includes smaller values from the dataset.
- When the sample size is equal to the population size, you are essentially conducting a census, where no sampling is required, and all population characteristics are fully captured. In this case, the sampling error is eliminated entirely because there is no discrepancy between the sample statistics and the population parameters. Therefore, the sampling error will be zero.
Reduction Techniques:
Sampling error arises because a sample may not perfectly reflect the characteristics of the entire population, particularly when the sample size is small. To minimize sampling error, consider the following techniques:
- Increase the Sample Size: Larger samples are generally more representative of the population and reduce the variability in estimates, leading to smaller sampling errors.
- Ensure Representative Sampling: Use appropriate sampling methods to ensure that the sample accurately reflects the diversity of the population. This might include techniques such as stratified sampling or systematic sampling to capture all relevant subgroups within the population.
How satisfied are you with this page?
Very Dissatisfied
Dissatisfied
Neutral
Satisfied
Very Satisfied
Comments
Post a Comment