# How to Calculate Sampling Error in Statistics: A Step-by-Step Guide

Whether youre making a pool about the voting intention of a country or trying to infer the average height of the American population, you need to express the error of your estimations, as it will always exist.

This tool calculates the error of your sample given the sample size and proportion or standard deviation. If youre interested in the opposite problem: how probable is finding a range of sample means or proportions, look at the normal probability for sampling distributions and the sampling distribution of the sample proportion calculators.

Sampling error is an important concept in statistics that refers to the difference between a sample statistic and the true value for the overall population. Being able to accurately calculate sampling error is crucial for determining the reliability and precision of survey and experiment results.

In this comprehensive guide, we will walk through the basics of sampling error, the sampling error formula, examples, and steps to calculate it.

## What is Sampling Error?

In statistics, a sample is a subset of a population used to draw inferences about the whole population. Sampling error refers to the degree to which a sample statistic deviates from the true population parameter.

For example, say we have a population of 1 million people. We take a sample of 1,000 people and calculate the average height to be 5’8″. However, the true average height for the overall population is 5’9″. In this case, there is a sampling error of 1 inch between our sample statistic (5’8″) and the population parameter (5’9″).

All samples will have some degree of error since they are merely estimates of the true population value But we want to minimize the sampling error as much as possible to get accurate insights

Some key points about sampling error

• It occurs when the sample does not adequately represent the population
• It is the difference between a sample statistic and the true population parameter
• It decreases as the sample size increases
• It can be random or biased
• It can be quantified using statistical formulas

## Sampling Error Formula

The most common way to calculate sampling error is using the following statistical formula

``Sampling Error = z * (σ / √n)``

Where:

• z = z-score based on chosen confidence level
• σ = standard deviation of population
• n = sample size

Let’s break this down step-by-step:

1. Determine the z-score based on the desired confidence level. Common confidence levels are 90%, 95%, or 99%. Refer to a z-table to find the corresponding z-score. Higher confidence levels will have a higher z-score.

2. Calculate the population standard deviation (σ). This measures how much variability exists in the overall population.

3. Take the square root of the sample size (√n). The sample size is the number of observations in your sample.

4. Divide the population standard deviation by the square root of the sample size (σ / √n).

5. Multiply the result by the z-score. This gives you the sampling error!

The key things to note are:

• Higher confidence levels lead to higher sampling error
• Larger sample sizes decrease sampling error
• More variation in the population (larger standard deviation) increases error

So in practice, we want to maximize sample size and minimize population variance to reduce sampling error.

## Examples of Calculating Sampling Error

Let’s walk through some examples to illustrate how to calculate sampling error using real numbers:

Example 1

• Population = 20,000 people
• Sample size = 500 people
• Population standard deviation = 7 lbs
• Confidence level = 95%
• z-score for 95% confidence = 1.96

Plugging this into our formula:

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``Sampling Error = 1.96 * (7 / √500)             = 1.96 * (7 / 22.36)            = 1.96 * 0.313            = 0.61 lbs``

Therefore, at the 95% confidence level, our sampling error is 0.61 lbs.

Example 2

• Population = 1,500 households
• Sample size = 150 households
• Population standard deviation = \$8,200
• Confidence level = 99%
• z-score for 99% confidence = 2.58

Sampling Error = 2.58 * (\$8,200 / √150)
= 2.58 * (\$8,200 / 12.25)
= 2.58 * \$668
= \$1,722

Our sampling error at the 99% confidence level is \$1,722. The higher confidence increased the error.

Example 3

• Population = 5,000 patients
• Sample size = 100 patients
• Population standard deviation = 10 years
• Confidence level = 90%
• z-score for 90% confidence = 1.645

Sampling Error = 1.645 * (10 / √100)
= 1.645 * (10 / 10)
= 1.645 * 1
= 1.645 years

With a lower 90% confidence level, our sampling error decreased to 1.645 years.

## Steps to Calculate Sampling Error

Here is a summary of the key steps involved in calculating sampling error:

1. Identify the target population and sample size used in the study

2. Determine the confidence level (e.g. 90%, 95%, 99%)

3. Find the z-score that corresponds to the confidence level

4. Calculate the population standard deviation (σ)

5. Take the square root of the sample size (√n)

6. Divide the population standard deviation by the square root of the sample size (σ / √n)

7. Multiply the result by the z-score to get the sampling error

8. Report the sampling error along with the sample statistic to communicate the reliability. A higher error means lower reliability.

## Tips for Minimizing Sampling Error

Here are some tips to minimize sampling error in statistics:

• Increase the sample size – more observations reduce error
• Take random samples that accurately represent the population
• Repeat measurements and average the results
• Use stratified sampling techniques if needed
• Apply careful controls and consistency
• Report confidence intervals, not just point estimates
• Use less sensitive measurements and instruments

While some error is always present, following best practices can help keep sampling error to a minimum for reliable results.

## Common Questions About Sampling Error

What exactly does sampling error tell you?

Sampling error quantifies the amount that a sample statistic varies from the true population parameter. It is an estimate of the accuracy and reliability of the sample.

When would sampling error be zero?

The only time sampling error would be zero is if you somehow measured the entire population. Since this is often not feasible, some sampling error always exists. But larger, more representative samples have lower error.

Can sampling error be negative?

Yes, sampling error can be negative if the sample estimate is lower than the true population parameter. Error is absolute deviation from the true value in either direction.

Does a bigger sample size always decrease error?

In general, larger sample sizes result in lower sampling error. But once you hit a certain point, there are diminishing returns to increasing the sample size further. Beyond 30-50 observations, error reductions become small.

How is sampling error different from non-sampling error?

Sampling error stems from using a sample vs. the entire population. Non-sampling errors occur from issues in the research methodology or measurement process itself.

## Conclusion

Being able to accurately calculate and interpret sampling error is a crucial skill in statistics. The sampling error formula allows us to quantify the reliability of sample estimates compared to the true population. While some error always exists, following best practices like increasing sample size, taking random samples, and repeating measurements can minimize error. Sampling error is a valuable tool for understanding the precision and accuracy of survey and experimental results.

## What is sampling error? – definition

Theres not a universal consensus about the definition of sampling error. Some authors define it as the error caused by any source, i.e., sample variability, poor study design, or nonrepresentative sampling. Other authors make a differentiation between a random part of the error (sampling error) and a nonrandom part (known as “bias” or “nonsampling error”) and define the sampling error as something related only to the variability from sample to sample.

Here well take the last definition, which coincides with the Glossary of Statistical Terms published by professor P.B. Stark of the University of California.

The sampling error is also known as “sample error” or “sample margin of error.”

Considering this, we can mathematically define the sampling error as the difference between the population parameter and our statistic. The problem is we rarely know the population parameter value. Therefore, the actual sampling error is unknown, and we must estimate it.

Now that you know what a sampling error is, lets see how to calculate it.

## How to use this sample margin of error calculator

Suppose youre carrying out a study to determine the percentage (proportion) of citizens who intend to vote for the candidate called Don Quixote in the next elections. Youve taken a random sample of 500 citizens, and 400 of them affirm to have the intention to vote for Don Quijote in the next election. What is the margin of error of your pool? Follow these steps to know it:

• In the calculator, select “Sample proportion error” as the error to estimate.
• Input 30 as the sample size in the second box.
• Calculate the sample proportion p^=400/500=0.8hat p = 400/500 = 0.8p^=400/500=0.8, and input it in the sample proportion box.
• Select a confidence level. Well take 95%, as it is the most common.
• Thats it. The sampling error is ±0.0351 (3.51%).

We estimated an 80% vote intention with our sample, but the sampling error lets us indicate with a 95% confidence level that the actual population vote intention is within 76.49% and 83.51%.

You can check the results using the sampling error formula: ep=1.96×0.8(1−0.8)500=0.0351e_p = 1.96 times sqrt{frac{0.8(1-0.8)}{500}} = 0.0351ep​=1.96×5000.8(1−0.8)​​=0.0351

Now suppose youre investigating the caloric content of a new food product. Youre interested in the average energy content of a batch of that product. You take a sample of 30 and measure how many calories they contain. The mean caloric content of the sample is 600 kcal, with a standard deviation of 70 kcal. If you want to calculate the sampling error of your energy content estimation, these are the steps:

• Select “Sample mean error” as the error to estimate.
• Select “Sample standard deviation” as the info you know.
• Input 30 as the sample size.
• Input 70 as the sample standard deviation.
• Select a confidence level. Well take 95%, as it is the most common one.
• Thats it. The sampling error must be ±26.1384 kcal from the mean content.

What this result indicates: With a 95% confidence level, we can say that the mean caloric content of the studied batch lies between 626.1384 and 573.8616 kcal.

You can check the results using the sampling error formula, taking into account that, in this case, tα/2=2.0452t_{α/2} = 2.0452tα/2​=2.0452, therefore eXˉ=1.9670600=26.138e_{bar X} = 1.96 frac{70}{sqrt{600}} = 26.138eXˉ​=1.96600​70​=26.138.

What this result doesnt indicate: The result only gives information about the mean caloric content of a population of products. It provides the possible values of a sample mean and not the possible values for each individual unit of the product.

When we say the mean caloric content lies between 64.4 and 75.6 kcal, we state our statement as a confidence interval. Learn more about it in our confidence interval calculator.