# Sampling Error Calculator

Created by Luis Hoyos
Reviewed by Steven Wooding
Based on research by
Stark PBGlossary of Statistical Terms;SticiGui (Statistics Tools for Internet and Classroom Instruction with a Graphical User Interface)
Last updated: Jun 05, 2023

Calculating the sampling error is an essential task in almost every statistical study.

Whether you're 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 you're 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.

• What is a sampling error?
• What is the standard error of the sample mean?
• How to calculate a sampling error.
• How to reduce sampling error.

## Population parameter vs. statistic

A parameter is a numeric characteristic of a population. Examples are:

• The mean height of American women over 20 years old.
• The proportion of citizens of a country who intend to vote for some candidate.
• The mean diastolic pressure of the citizens who don't exercise in a specific country.

On the other hand, a statistic is a point estimate or numeric characteristic of a sample. It is the data we usually can know.

These concepts are relevant to understanding the concept of sampling error.

## What is sampling error? – definition

There's 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 we'll 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, let's see how to calculate it.

## How to calculate the sampling error

The equation for the sampling error depends on whether we're trying to estimate a population proportion or a population mean. In any case, it consists of the product of a critical value related to the confidence level (z-score or t-statistic) and the standard error. The standard error estimates the standard deviation of the sampling distribution of the parameter under study. The most common example is calculating the standard error of the distribution of sample means, but we can apply it to any other parameter, i.e., variance, range, standard deviation, etc.

#### How to calculate sampling error for a sample proportion

If you're estimating a population proportion ($p$), the sampling error ($e_p$) of your sample proportion ($\hat p$) is:

$\small e_{\hat p} = z_{α/2}SE(\hat p) = z_{α/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}$

, where:

• $n$ – Sample size;
• $SE(\hat p)$ – Standard error of the sample proportion, which equals $\sqrt{\frac{\hat p(1-\hat p)}{n}}$.
• $z_{α/2}$ – Z-score, which depends on the required confidence level.

#### How to calculate the margin of error for a sample mean

If you don't know the population standard deviation ($σ$) (as usually happens in practice), you estimate the sample margin of error using the sample standard deviation ($s$):

$\small e_{\bar X} = t_{α/2}SE(\bar X) = t_{α/2} \frac{s}{\sqrt n}$

In the above equation:

• $SE(\bar X)$ – Standard error of the sample mean (its formula is $s/√n$), which is an estimate of the standard deviation of the sampling distribution of the means.
• $t_{α/2}$ – Equivalent value to $z_{α/2}$, with the difference that it considers the sample size.

If you do know ($σ$), you can calculate the sample margin of error of the sample mean ($e_{\bar X}$) with this formula:

$\small e_{\bar X} = z_{α/2}SE(\bar X) = z_{α/2} \frac{σ}{\sqrt n}$

💡 You can learn more about the $t_{α/2}$ value and its calculation in our critical value calculator.

## How to use this sample margin of error calculator

#### Example 1

Suppose you're 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. You've 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:

1. In the calculator, select "Sample proportion error" as the error to estimate.
2. Input 30 as the sample size in the second box.
3. Calculate the sample proportion $\hat p = 400/500 = 0.8$, and input it in the sample proportion box.
4. Select a confidence level. We'll take 95%, as it is the most common.
5. That's 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: $e_p = 1.96 \times \sqrt{\frac{0.8(1-0.8)}{500}} = 0.0351$

#### Example 2

Now suppose you're investigating the caloric content of a new food product. You're 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:

1. Select "Sample mean error" as the error to estimate.
2. Select "Sample standard deviation" as the info you know.
3. Input 30 as the sample size.
4. Input 70 as the sample standard deviation.
5. Select a confidence level. We'll take 95%, as it is the most common one.
6. That's 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, $t_{α/2} = 2.0452$, therefore $e_{\bar X} = 1.96 \frac{70}{\sqrt{600}} = 26.138$.

What this result doesn't 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.

## Is sampling error the same as a standard error?

No, sampling error is not the same as standard error, although they relate to each other.

• The standard error is the estimated standard deviation of a sampling distribution.
• The sampling error equals the standard error multiplied by a z-score or the t-statistic. It represents the error we incur when estimating a population parameter.
• Sampling error is the same as standard error only when the z-score or the t-statistic equal 1.

## Is standard error the same as the margin of error?

The standard error is not the same as the margin of error, but they relate to each other.

• The standard error is the estimated standard deviation of a statistic's sampling distribution.
• A margin of error is the standard error multiplied by a z-score or the t-statistic. It represents the error we incur when estimating a population parameter.
• The margin of error has a confidence interval associated with it.
• The standard error is the same as the margin of error only when the z-score or the t-statistic equal 1.

## FAQ

### How to reduce sampling error?

As long as we keep other factors constant, we can reduce sampling error by increasing the sample size. That occurs because sampling error is inversely proportional to the square root of the sample size.

### How to calculate a 95% confidence interval from standard error?

The way how you calculate a 95% confidence interval from standard error depends on the parameter to estimate and the available information:

• If you're estimating a population mean:
1. If you know the population standard deviation (σ), multiply the standard error (σ/√n) by the corresponding z-score, which is 1.96 for this confidence level.
2. If you only know the sample standard deviation (s), multiply the standard error (s/√n) by the corresponding t-value for 95%, which will depend on the sample size.
• If you're estimating a population proportion, multiply the standard error (√[p̂(1 - p̂)/n] by the corresponding z-score, zα/2 = 1.96.
Luis Hoyos
Error to estimate
Sample proportion error
Sample size (n)
Sample proportion (p̂)
Confidence level
%
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