Sample error formula

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.

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.

Sampling Error

=

Z

×

σ

n

where:

Z

=

Z

 score value based on the

 confidence interval (approx

=

1.96

)

σ

=

Population standard deviation

n

=

Size of the sample

begin{aligned}&text{Sampling Error}=Ztimesfrac{sigma}{sqrt{n}}\&textbf{where:}\&Z=Ztext{ score value based on the}\&qquad text{confidence interval (approx}=1.96)\&sigma=text{Population standard deviation}\&n=text{Size of the sample}end{aligned}

​Sampling Error=Z×n​σ​where:Z=Z score value based on the confidence interval (approx=1.96)σ=Population standard deviationn=Size of the sample​

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.

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.

• Let’s know what Sampling Error is. When the sample mean is used as a point estimate of the population mean,  then we can expect some Error can be expected owing to the fact that a subset, or sample of the population,is used to compute the point estimate. 

• The absolute value of the difference between the sample mean denoted as x̄, and the population mean is denoted by μ, written as |x̄ − μ|, is known as the Sampling Error.

• Probability statements about the Magnitude of the Sampling Error can be incorporated by Interval estimation.

• The Sampling distribution of x̄  basically provides the basis for such a statement.

In this article we are going to discuss what is Sampling Error, Sampling and Sampling Error, Sampling Error definition, Sampling Error formula and Sampling Error examples.

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.

Sampling Error = [Ztimes frac{sigma }{sqrt{n}}]

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

Formula of Sampling error = [Ztimes frac{sigma }{sqrt{n}}]

= [frac{0.40}{sqrt{2500}}times 1.96]

= [frac{0.40}{sqrt{50}}times 1.96=0.01568]

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

Formula of Sampling error = [Ztimes frac{sigma }{sqrt{n}}]

= [frac{0.5}{sqrt{100}}times 1.645]

= [frac{0.5}{sqrt{10}}times 1.645=0.08225].

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.