Sampling Error Formula

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Sampling Error

What Is a Sampling Error?

A sampling error is a statistical error that occurs when an analyst does not select a sample that represents the entire population of data. As a result, the results found in the sample do not represent the results that would be obtained from the entire population.

Sampling is an analysis performed by selecting a number of observations from a larger population. The method of selection can produce both sampling errors and non-sampling errors.

Key Takeaways

  • A sampling error occurs when the sample used in the study is not representative of the whole population. 
  • Sampling is an analysis performed by selecting a number of observations from a larger population.
  • Even randomized samples will have some degree of sampling error because a sample is only an approximation of the population from which it is drawn.
  • The prevalence of sampling errors can be reduced by increasing the sample size.
  • In general, sampling errors can be placed into four categories: population-specific error, selection error, sample frame error, or non-response error.

Understanding Sampling Errors

A sampling error is a deviation in the sampled value versus the true population value. Sampling errors occur because the sample is not representative of the population or is biased in some way. Even randomized samples will have some degree of sampling error because a sample is only an approximation of the population from which it is drawn.

Calculating Sampling Error

The sampling error formula is used to calculate the overall sampling error in statistical analysis. The sampling error is calculated by dividing the standard deviation of the population by the square root of the size of the sample, and then multiplying the resultant with the Z-score value, which is based on the confidence interval.

Types of Sampling Errors

There are different categories of sampling errors.

Population-Specific Error

A population-specific error occurs when a researcher doesn’t understand who to survey.

Selection Error

Selection error occurs when the survey is self-selected, or when only those participants who are interested in the survey respond to the questions. Researchers can attempt to overcome selection error by finding ways to encourage participation.

Sample Frame Error

A sample frame error occurs when a sample is selected from the wrong population data.

Non-response Error

A non-response error occurs when a useful response is not obtained from the surveys because researchers were unable to contact potential respondents (or potential respondents refused to respond).

Eliminating Sampling Errors

The prevalence of sampling errors can be reduced by increasing the sample size. As the sample size increases, the sample gets closer to the actual population, which decreases the potential for deviations from the actual population. Consider that the average of a sample of 10 varies more than the average of a sample of 100. Steps can also be taken to ensure that the sample adequately represents the entire population.

Researchers might attempt to reduce sampling errors by replicating their study. This could be accomplished by taking the same measurements repeatedly, using more than one subject or multiple groups, or by undertaking multiple studies.

Random sampling is an additional way to minimize the occurrence of sampling errors. Random sampling establishes a systematic approach to selecting a sample. For example, rather than choosing participants to be interviewed haphazardly, a researcher might choose those whose names appear first, 10th, 20th, 30th, 40th, and so on, on the list.

Examples of Sampling Errors

Assume that XYZ Company provides a subscription-based service that allows consumers to pay a monthly fee to stream videos and other types of programming via an Internet connection.

The firm wants to survey homeowners who watch at least 10 hours of programming via the Internet per week and that pay for an existing video streaming service. XYZ wants to determine what percentage of the population is interested in a lower-priced subscription service. If XYZ does not think carefully about the sampling process, several types of sampling errors may occur.

A population specification error would occur if XYZ Company does not understand the specific types of consumers who should be included in the sample. For example, if XYZ creates a population of people between the ages of 15 and 25 years old, many of those consumers do not make the purchasing decision about a video streaming service because they may not work full-time. On the other hand, if XYZ put together a sample of working adults who make purchase decisions, the consumers in this group may not watch 10 hours of video programming each week.

Selection error also causes distortions in the results of a sample. A common example is a survey that only relies on a small portion of people who immediately respond. If XYZ makes an effort to follow up with consumers who don’t initially respond, the results of the survey may change. Furthermore, if XYZ excludes consumers who don’t respond right away, the sample results may not reflect the preferences of the entire population.

Sampling Error vs. Non-sampling Error

There are different types of errors that can occur when gathering statistical data. Sampling errors are the seemingly random differences between the characteristics of a sample population and those of the general population. Sampling errors arise because sample sizes are inevitably limited. (It is impossible to sample an entire population in a survey or a census.)

Company XYZ will also want to avoid non-sampling errors. Non-sampling errors are errors that result during data collection and cause the data to differ from the true values. Non-sampling errors are caused by human error, such as a mistake made in the survey process.

If one group of consumers only watches five hours of video programming a week and is included in the survey, that decision is a non-sampling error. Asking questions that are biased is another type of error.

What Is Sampling Error vs. Sampling Bias?

In statistics, sampling means selecting the group that you will actually collect data from in your research.

Sampling bias is the expectation, which is known in advance, that a sample won’t be representative of the true population. For instance, if the sample ends up having proportionally more women or young people than the overall population.

Sampling errors are statistical errors that arise when a sample does not represent the whole population once analyses have been undertaken.

Why Is Sampling Error Important?

Being aware of the presence of sampling errors is important because it can be an indicator of the level of confidence that can be placed in the results. Sampling error is also important in the context of a discussion about how much research results can vary.

How Do You Find the Sampling Error?

In survey research, sampling errors occur because all samples are representative samples: a smaller group that stands in for the whole of your research population. It’s impossible to survey the entire group of people you’d like to reach.

It’s not usually possible to quantify the degree of sampling error in a study since it’s impossible to collect the relevant data from the entire population you are studying. This is why researchers collect representative samples (and representative samples are the reason why there are sampling errors).

What Is Sampling Error vs. Standard Error?

Sampling error is derived from the standard error (SE) by multiplying it by a Z-score value to produce a confidence interval.

The standard error is computed by dividing the standard deviation by the square root of the sample size.

The Bottom Line

Sampling error occurs when a sample drawn from a population deviates somewhat from that true population. Large sampling errors can lead to incorrect estimates or inferences made about the population based on statistical analysis of that sample.

In general, sampling errors can be placed into four categories: population-specific error, selection error, sample frame error, or non-response error. A population-specific error occurs when the researcher does not understand who they should survey. A selection error occurs when respondents self-select their participation in the study. (This results in only those that are interested in responding, which skews the results.) A sample frame error occurs when the wrong sub-population is used to select a sample. Finally, a non-response error occurs when potential respondents are not successfully contacted or refuse to respond.

Formula to Calculate Sampling Error

Sampling Error Formula refers to the formula that is used in order to calculate statistical error that occurs in the situation where person conducting the test doesn’t select sample that represents the whole population under consideration and as per the formula Sampling Error is calculated by dividing the standard deviation of the population by the square root of the size of sample and then multiplying the resultant with the Z score value which is based on confidence interval.

Sampling Error = Z x (σ /√n)

Where,

  • Z is the Z score value based on the confidence interval
  • σ is the population standard deviation
  • n is the size of the sample

Step by Step Calculation of Sampling Error

The following are the steps to calculate sampling error.

  1. Gathered all set of data called the population. Compute the population means and population standard deviation.
  2. Now, one needs to determine the size of the sample, and further, the sample size has to be less than the population, and it should not be greater.
  3. Determine the confidence level, and accordingly, one can determine the value of Z score from its table.
  4. Now multiply Z score by the population standard deviation and divide the same by the square root of the sample size in order to arrive at a margin of error or sample size error.

Examples

Example #1

Suppose that the population standard deviation is 0.30, and the size of the sample is 100. What will the sampling error at 95% confidence level?

Solution

Here we have given the population standard deviation as well as the size of the sample. Therefore we can use the below formula to calculate the same.

Use the following data for the calculation.

  • Z Factor Value: 1.96
  • Population of Standard Deviation: 0.3
  • Sample Size: 100

Therefore, the calculation of the sampling error is as follows,

Sampling Error will be –

Example #2

Gautam is currently pursuing an accountancy course, and he has cleared his entrance exam. He has registered now for an intermediate level and will also be joining a senior accountant as an intern. He will be working in an audit of the manufacturing firms. 

One of the firms he was visiting for the first time was asked to check whether the bills for all the entries for purchases were reasonably available. The sample size he picked was 50, and the population standard deviation for the same was 0.50.

Based on available information, you are required to calculate sampling error at 95% and 99% confidence interval.

Solution

Here we are given the population standard deviation as well as the size of the sample; therefore, we can use the below formula to calculate the same.

Z score for 95% confidence level will be 1.96 (available from Z score table)

Use the following data for the calculation.

  • Z Factor Value: 1.96
  • Population of Standard Deviation: 0.50
  • Sample Size: 50

Therefore, the calculation is as follows,

Sampling Error will be –

Z score for 95% confidence level will be 2.58 (available from Z score table)

Use the following data for the calculation.

Therefore, the calculation is as follows,

Sampling Error will be –

As the confidence level increases, the sampling error also increases.

Example #3

In a school, the biometric session was organized so as to check the health of the students. The session was initiated with students of class X standard. In total, there are 30 students in the B division. Among them, 12 students were randomly selected to do a detailed checkup, and the rest was only a basic test was done. The report inferred that the average height of the students in B division is 154.

Solution

The population standard deviation was 9.39. Based on the above information, you are required to calculate the sampling error for 90% and 95% confidence interval.

Here we are given the population standard deviation as well as the size of the sample; therefore, we can use the below formula to compute the same.

Z score for 95% confidence level will be 1.96 (available from Z score table)

Use the following data for the calculation.

Therefore, the calculation of the sampling error is as follows,

Sampling Error will be –

Z score for 90% confidence level will be 1.645 (available from Z score table)

Use the following data for the calculation.

Therefore, the calculation is as follows,

Sampling Error will be –

s the confidence level decreases, the sampling error also decreases.

Relevance and Uses

This is very much vital to understand this concept as this shall depict how much one can expect that the survey results would, in fact, depict the actual view of the population overall. One needs to keep one thing in mind that a survey is performed using a smaller population called the sample size (also otherwise renowned as the respondents of the survey) to represent a bigger population.

It can be viewed as a way of calculating the effectiveness of the survey. When the sampling margin is higher, it shall represent that the survey consequences might stray from the actual total population representation. On the flip side, a sampling error or margin of error is smaller than that shall indicate that the consequences are now closer to the true representation of the population in total and which shall build a higher level of confidence about the survey that is under view.

How To Calculate Sampling Error in 6 Steps (With Examples)

Sampling error = confidence level × [standard deviation of population / (square root of sample size)]

The accuracy of a sample can affect the results of a study if a researcher selects a sample that doesn’t reflect the real composition of the population being studied. It’s important for samples to be accurate so they can represent statistics correctly. Calculating sampling error can help research professionals determine a sample’s efficacy by measuring how close it is to the targeted community. In this article, we discuss what sampling error is and how to calculate it in six steps.

What is a sampling error?

A sampling error is a calculation that measures statistical error when a tester uses a sample that doesn’t properly reflect the population under consideration. The results from studies with skewed samples may be incorrect. Sampling is an analysis that requires selecting several observations, usually from larger populations. For example, a researcher hypothesizes that people between the ages of 30 and 45 eat fruit at least once a week. They can select 100 people from their community in this age range to reflect the target population and observe their eating habits. Sampling error decreases as the sample size increases.

Here’s the formula for calculating sampling error:

Sampling error = confidence level × [standard deviation of population / (square root of sample size)]

Confidence levels are the percentage of samples researchers can expect to reflect the parameters of the entire population. The standard deviation of the population measures how scattered the researcher’s data is from the average value. The square root of the sample size is the value that, when multiplied by itself, equals the sample size value. Charts for confidence levels are available online to help researchers determine the level of confidence in their sample and find the corresponding number.

How to calculate sampling error

Here are six steps you can follow when calculating sampling error:

1. Record the sample size

This is the simplest number to find for the sampling error formula. Review your study to find the sample size. For example, if your sample included 60 people, use 60 in the formula.

2. Find the standard deviation of the population

Standard deviation measures the distance between each of your data points and the mean. Use the parameters of your population and the average value of your data to calculate the standard deviation. You can look up the formula for sample standard deviation if you’re unfamiliar with it.

3. Determine your confidence level

To determine the confidence level, use the confidence interval for your sample size. Confidence intervals specify the range of values in a sample that are likely to contain the accurate population mean. Take half of the confidence interval and multiply it by the square root of the sample size. Then divide this by the sample standard deviation to find your confidence level. Finally, use a confidence level table to find your equivalent score.

4. Calculate the square root of the sample size

Find the square root of your sample size. Your sample size may be a perfect square if its square root is a whole number. For example, four is a perfect square because it has a square root of two, while 10 isn’t a perfect square because it has a square root of 3.16.

5. Divide the standard deviation value by the square root value

Divide your population’s standard deviation by the sample size’s square root. It’s often easier to do this using a calculator rather than manually. Record the result to use in the next step.

6. Multiply the result by the confidence level

Finally, multiply the resulting number from the last step by the confidence level score you found earlier. Continue using a calculator to work with these complex decimals. The product represents the sampling error for your study.

Types of sampling errors

Here are some common sampling errors you might find in a study:

  • Population-specific error: A population-specific error can happen when a researcher doesn’t understand who to survey. You may avoid this mistake by understanding your research question before you select your sample or create a survey.
  • Selection error: This error occurs when respondents elect to participate in a study but then only those interested in the survey answer its questions. A researcher might overcome selection error by encouraging participation from the sample population.
  • Sample frame error: This refers to errors that happen when a researcher selects a sample from incorrect population data. Sample frame errors also occur when testers accidentally include respondents from outside the population of interest.
  • Nonresponse error: Nonresponse errors arise when researchers cannot contact potential respondents or participants don’t engage in the study.

Why is calculating sampling error important?

Sampling error is an important measurement to estimate the amount of uncertainty in a sample size. Statisticians usually use random samples to make assumptions about an entire population. It’s rare for the sample to be identical to the true population. Because it’s often difficult for an entire population to take part in a study, testers frequently manage some level of inaccuracy in their discoveries. By calculating sampling error, researchers can determine whether the sample size is unsatisfactory or unrepresentative of the larger population. They may then conduct another study or include the sampling error in their report.

How can you correct sampling errors?

Researchers can take several actions to reduce the amount of uncertainty in their studies. They may rely on probability sampling, where any individual in a population has a chance of participating in a study. This might make the sample size more representative of the entire population, and researchers can better generalize their results.

Here are some other tips for minimizing and controlling a sampling error:

  • Know your population to target the relevant sample.
  • Increase the sample size so the study gets closer to the actual population.
  • Perform an external record check to confirm the recorded data is consistent with written results.
  • Eliminate bias with random selections and confidential surveys.
  • Design samples carefully so they accurately reflect the target population.
  • Divide the sample into groups and test according to their composition in the actual population.
  • Train your team to carry out all procedures and activities in the study in the same way.

Example sampling error calculations

Here are some example calculations you can use as a guide when finding a sampling error:

Sampling with a higher confidence level

Mind Laboratories is a research company that wants to calculate the sampling error for one of its studies. Mind Laboratories’ study had a sample size of 100 people. The standard deviation of the population is 0.50. For a 99% confidence level, the score is 2.58. Mind Laboratories put these values into the formula for sampling error:

Sampling error = 2.58 x (.50 / √100)

Sampling error = 2.58 x (.50 / 10)

**Sampling error = 2.58 x 0.05**

**Sampling error = 0.12**

Sampling with a lower confidence level

Tech Flurry is a research organization that wants to find the sampling error for one of its studies, which has a lower confidence level. The study had a population of 324 and a standard deviation of 0.30. For an 80% confidence level, the score is 1.28. Tech Flurry uses these numbers in the formula for sampling error:

Sampling error = 1.28 x (.30 / √324)

**Sampling error = 1.28 x (.30 / 18)**

Sampling error = 1.28 x 0.016

*Sampling error = 0.02*

Sampling Error Definition

A Sampling Error can be defined as a Statistical Error that occurs when a sample that represents the entire population of data is not selected by an analyst and the results we find in the sample do not represent the actual results that can be obtained from the entire population.

What is Sampling and Sampling Error?

We can define Sampling as an analysis performed by selecting a number of observations generally from a larger population, and this selection produces both Sampling and Non-Sampling Errors.

Key Takeaways

  • Sampling Error can be defined as a Statistical Error that generally occurs when an analyst does not select a sample that represents the entire population of data and selects some part of the data.
  • The results found in the sample do not represent the results which can be obtained from the entire population.
  • This Error can be reduced by randomizing sample selection or by increasing the number of observations.

Sampling Error Meaning:

Let’s know the Sampling Error meaning. It can be defined as a deviation in sampled value versus the true population value due to the fact the sample selected is not representative of the population or biased in any way. Even the randomized samples will have some Sampling Error as it is only an approximation of the population from which it is drawn.

The Role of Sample Size

As has been illustrated above, the bigger is the sample size, the smaller will be the Sampling Error. The Sampling Error increases in proportion to the square root of the sample size that is denoted by n. For example, when we increase the sample size from 10 to 100, the Sampling Error halves, all else being equal.

Formula for Sampling Error

The Formula for Sampling Error refers to the formula that’s utilized in order to calculate statistical Error that happens within the situation where person conducting the test doesn’t select sample that represents the entire population into account and as per the formula Sampling Error is calculated by dividing the quality deviation of the population by the root of the dimensions of sample then multiplying the resultant with the Z score value which is predicated on confidence interval.

Where,

  • Z score value based on the confidence interval
  • σ denotes the population standard deviation
  • n denotes the sample size

Step by Step Calculation of Sampling and Sampling Error

Step 1) Gather all sets of knowledge called the population. Compute the population means and population variance .

Step 2) Now, one must determine the dimensions of the sample, and further the sample size has got to be but the population and it shouldn’t be greater.

Step 3) Now you need to determine the confidence level and accordingly one can determine the value of the Z score from its table.

Step 4) Now multiply Z score by the population variance and divide an equivalent by the root of the sample size so as to reach a margin of Error or sample size Error.

How can Sampling Error be Corrected?

Here are the steps for minimizing and controlling Sampling Error-

  • You can simply increase the sample size. A larger sample size generally leads to a more precise result because the study gets closer to the actual population size and the results obtained are more accurate.
  • Dividing the population into groups.
  • Important to know your population. 
  • Random selection results in the elimination of bias.
  • You can train your team. 
  • Performing an external record check.
  • Careful sample designs.
  • Take large enough samples.

Questions to be Solved (Sampling Error Example):

Sampling Error example 1) Suppose that the population standard deviation given is 0.40 and the size of the sample is equal to 2500 then find the Sampling Error at confidence level equal to 95%.

Solution) Let’s list down the data,

σ is equal to 0.40

Sample size (n) = 2500

The value of z at 95% of confidence level is equal to 1.96

Sampling Error example 2 ) Find the Sampling Error of the sample size equal to 100 of the population with a standard deviation equal to 0.5 at 90% confidence level.

Answer)From the given data,

σ  is equal to 0.5

Sample size (n) = 100

The value of z at 90% of confidence level is equal to 1.645

Note: Z-value at 90% confidence level is equal to 1.64.

Common mistakes to avoid on Sampling Errors

Some common mistakes that should be avoided while solving Sampling Error problems are-

  • Sample Frame Error-Sample Frame Errors happen when the false subpopulation is used to determine a sample. This type of Error rises when there was a mistake in understanding the population and their choices before surveying them. This can take place often when the population’s choices are not studied before handling the surveys and has often resulted in big tragedies.
  • Selection Error-Selection Error happens when the respondent’s sect themselves as the participants in the survey. Selection Errors can be controlled by going to additional lengths to get participation. The survey process includes the processes of initiating pre-survey communication requesting cooperation, actual surveying, and post-survey follow-ups. If no reaction is received, a 2nd survey request follows, and maybe several discussions using different modes such as calls or person-to-person meetups.
  • Population Specification Error- Population Specification Error happens when the investigator does not comprehend who they are. There is no particular or precise population in mind to take the survey. These types of Errors occur because of multiple layers of decision-making in the first place. Here the population can be anyone as there are many generalized populations who participate in the activity. 
  • Non-Response- Non-response Errors happen when the respondents are dissimilar to those who do not respond. This may happen because the potential respondent was not contacted. This case can also happen if the respondent refuses to respond. The size of this non-response Error can be prevented through follow-up surveys using different modes. This Error can be minimized with a little extra attention towards managing the respondents.
  • Sampling Errors- Sampling Errors happen due to variations in the number of representatives of the sample that responds to the program. It can be controlled and reduced by using various methods. One of the ways is to carefully choose sample designs. Another solution is to take large samples in the first place. One more thing can be used and that is to use multiple contacts to assure the representative response.

FAQs on Sampling Error

1. What is the Sampling Error with Example? What are the Steps to Reduce Sampling Error?

Sampling Error meaning.Sampling Error can be defined as the difference between a population parameter and a sample statistic that is used to estimate it. Let’s take for example, the difference between a population mean and a sample mean is known to be the Sampling Error. Sampling Error  generally occurs because a portion, and not the entire population, is surveyed.

Here are the steps for minimizing Sampling Error

  • You can simply increase the sample size. A larger sample size generally leads to a more precise result because the study gets closer to the actual population size and the results obtained are more accurate.
  • Dividing the population into groups.
  • Important to know your population. 
  • Random selection results in the elimination of bias.
  • You can train your team. 
  • Performing an external record check.

2. Is Sampling Error Always Present?What are the Sources of Sampling Error?

Sampling Error is generally unavoidable.An estimate of Sampling Error  is the margin of Error that you commonly see with survey results. Because the Sampling Error is just an estimate, there is a small chance (typically five percent or less chance) that the margin of Error is actually larger than stated in the report.

Sampling Error generally occurs because survey information is observed from only a sample of the target population instead of from the entire population in general,increasing the size of the sample data basically leads to a decrease in the Sampling Error.

3. How to study the topic of Sampling Error?

Sampling Errors are a topic of Maths that is very high-scoring and easy if the student has understood the fundamentals. Sampling Error can be estimated in various ways, but the Error received is almost invariably an estimation of the actual Error rather than the real measure of the Error. To calculate the true population of a given situation, first, the student has to calculate the sample value. Mastering Sampling Error or any other topic of Mathematics requires a lot of practice. 

Students should start with solving NCERT exercises and then move onto reference books like RD Sharma and RS Aggarwal. Solutions to these books and NCERT exercises can be found on Vedantu’s website. One more important resource for studying Sampling Errors is solving previous year’s questions. This will help the student to pick up the question asking patterns and study accordingly. These papers with solutions can be found on Vedantu’s website. That is how a student can understand the fundamentals and get a good hold on the topic of Sampling Errors. 

Students can also refer to Vendatu’s official Youtube channel where they can watch FREE video lectures on Sampling Errors by the best teachers. Students can also find the list of important questions at Vedantu’s official website which is made for students to study when they lack time and have a desire to get good marks.

4. What are the different types of Sampling Errors?

There are different types of Sampling methods that are used by researchers in market research so that they do not need to research the entire population to collect workable data and insights. Sampling in market research is divided into 2 types, they are-

  • Probability Sampling- Probability Sampling is the type of Sampling procedure, where the researcher sets a selection of a few criteria and selects components of a population purely on a random basis. All the components have an equal opportunity to be a stake in the sample with probability Sampling.
  • Non-probability Sampling- Non-probability Sampling is a type of Sampling method where the researcher selects members for research at random. This Sampling method is not a specified or predefined choice process. This makes it hard for all elements of a population to have similar opportunities to be involved in a sample.

5. Why is studying Sampling Errors important?

Sampling is a method of selecting particular components or a subset of the population to make statistical generalizations from them and estimate aspects of the whole population. These results determine the decisions of the government or big business. Any Error in these surveys can waste a lot of money or change history. For Example- If a luxury car company launches its exclusive service in a country where the majority of the money lies only in the hands of a few people. they’ll be at a loss and have to shut down the company. That’s why Error Sampling is very important in day-to-day lives.

Solved Examples

Example 1: Suppose that the population standard deviation is 0.40 and the size of the sample is 2500 then find the sampling error at 95% confidence level.

Solution:
From the given data,
σ = 0.40
Sample size = n = 2500
Value of z at 95% of confidence level = 1.96
Sampling error = z × σ/√n
= 1.96 × 0.40/√(2500)
= 1.96 × 0.40/50
= 0.01568

Example 2: Find the sampling error of the sample size 100 of population with standard deviation 0.5 at 90% confidence level.

Solution:
From the given data,
From the given data,
σ = 0.5
Sample size = n = 100
Value of z at 90% of confidence level = 1.645
Sampling error = z × σ/√n
= 1.645 × 0.5/√(100)
= 1.645 × 0.5/10
= 0.08225

Note:
Z-value at 90% confidence level = 1.645
Z-value at 95% confidence level = 1.96
Z-value at 99% confidence level = 2.58

Example of Sampling Error Formula (With Excel Template)

Sampling Error Formula – Example #1

Let us take the example of a sample of 500 people from an entire population of 100 million who were surveyed whether or not they like Vanilla ice creams. 70% of the sample responded positively, saying that they like Vanilla ice creams. Calculate the sampling error for a 95% Confidence Level and 99% Confidence Level.

Solution:

Sampling Error is calculated using the formula given below

Sampling Error = Z * √(p * (1 – p) / n) * (1 – √(n /N))

z-score at 95%

  • Sampling Error = 1.96 * √[70% * (1 – 70%) / 500] * [1 – √(500 / 100000000)]
  • Sampling Error = 4.01%

Therefore, the sampling error for the sample at 95% confidence level is 4.01%.

z-score at 99%

  • Sampling Error = 2.58 * √[70% * (1 – 70%) / 500] * [1 – √(500 / 100000000)]
  • Sampling Error = 5.28%

Therefore, the sample’s sampling error at a 99% confidence level has gone up to 5.28%.

Therefore, it can be observed that the sampling error of any sample increases with the increase in confidence level.

Sampling Error Formula – Example #2

Now, again let us take the example of the above example and keep everything the same except the population size, which is to be assumed to be significantly lower in this case, say 2,000. Calculate the sampling error for a 95% confidence level and a 99% confidence level.

Solution:

Sampling Error is calculated using the formula given below

Sampling Error = Z * √(p * (1 – p) / n) * (1 – √(n /N))

z-score at 95%

  • Sampling Error = 1.96 * √[70% * (1 – 70%) / 500] * [1 – √(500 / 2000)]
  • Sampling Error = 2.01%

Therefore, the sampling error for the sample at 95% confidence level is 2.01%.

z-score at 99%

  • Sampling Error = 2.58 * √[70% * (1 – 70%) / 500] * [1 – √(500 / 2000)]
  • Sampling Error = 2.64%

Therefore, the sampling error for the sample at a 99% confidence level is 2.64%.

Therefore, it can be seen that the sampling error decreases with a decrease in population size. So, samples are a better representative of the smaller data population.

Explanation

The formula for Sampling Error analysis can be computed by using the following steps:

Step 1: Firstly, decide on the confidence level to be used for the estimation. Based on the selected confidence level, the z-score can be determined that is denoted by “Z”. For instance, the z-score for a 95% confidence level is 1.96.

Step 2: Next, determine the sample size for the estimation. It is the proportion of the population that is expected to represent the entire population, i.e. its sample characteristics will mostly be similar to that of the entire population. It is denoted by “n”.

Step 3: Next, determine the size of the entire population that is denoted by “N”.

Step 4: Next, determine the proportion of the people surveyed who are likely to respond either in a positive way or say “yes” as an answer to the survey question. It is expressed in percentage and denoted by “p”. So, (1 – p) denotes the percentage of the people with the alternate response.

Step 5: Final, the formula for sampling error can be derived based on the confidence level of the estimation (step 1), sample size (step 2), population size (step 3) and proportion of the population with a set response (step 4) as shown below.

Sampling Error = Z * √(p * (1 – p) / n) * (1 – √(n /N))

Sample Problems

Question 1. Find the sampling error at a 95% confidence level given the standard deviation of the population is 0.23 and the sample size is 2145.

Solution:

Given: Z = 95%, σ = 0.23 and n = 2145

Since, SE = Z x σ/√n

= 1.96 x (0.23/√2145)

= 1.96 x 0.00496608

SE = 0.009733

Question 2. Find the sampling error at a 90% confidence level given the standard deviation of the population is 0.2 and the sample size is 100.

Solution:

Given: Z = 92%, σ = 0.2 and n = 100

Since, SE = Z x σ/√n

= 1.645 x (0.2/√100)

= 1.645 x 0.02

SE = 0.0329

Question 3. Find the sampling error at a 99% confidence level given the standard deviation of the population is 0.2 and the sample size is 36.

Solution:

Given: Z = 99%, σ = 0.2 and n = 100

Since, SE = Z x σ/√n

= 2.58 x (0.2/√36)

= 2.58 x 0.0333

SE = 0.085914

Question 4. Find the sampling error at a 99% confidence level given the standard deviation of the population is 0.9 and the sample size is 49.

Solution:

Given: Z = 99%, σ = 0.9 and n = 49

Since, SE = Z x σ/√n

= 2.58 x (0.9/√49)

= 2.58 x 0.1285

SE = 0.33153

Question 5. Find the sampling error at a 95% confidence level given the standard deviation of the population is 0.3 and the sample size is 81.

Solution:

Given: Z = 95%, σ = 0.3 and n = 81

Since, SE = Z x σ/√n

= 1.96 x (0.3/√81)

= 1.96 x 0.03333

SE = 0.0653268

How to calculate sampling error for a sample proportion

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):

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:

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,

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.

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