Sample & sample size: definition, types, examples, calculation, formula, tool

What is a sample?

A sample is a term from statistics and means that one selects a small quantity of data or elements from a larger quantity in order to make statements about the entire quantity that are derived from the smaller quantity. A sample is therefore used to draw conclusions about the so-called population without having to examine all elements. Applied to market research, this means that in surveys, for example, not all people, i.e. the basic population, are questioned in order to obtain certain findings, but only a sample. However, the population does not have to actually include all people, but only those who are relevant to a certain question. The basic population is also referred to synonymously as the population.

In order to be able to draw conclusions about the population from the results of a sample, the sample must be sufficiently large and representative. The size of the sample is calculated using various formulas, which can also be determined with the help of a sample calculator tool. “Representative” means that the sample reflects the characteristics of the population in the same proportion as they occur in the population.

If the sample is randomly selected and large enough, the results of the sample can be applied to the population with a certain probability. The probability with which the results can be transferred to the population depends on how representative and how large the sample is.

Is there a rule of thumb for the size of a sample?

There is no fixed rule of thumb for the size of a sample that applies to all cases. The size of a sample needed to be representative depends on several factors, including the size of the population, the desired precision of the results and the confidence level.

A frequently quoted rule of thumb is that one should always choose a sample large enough to be about 5% of the size of the population. However, this rule of thumb is not applicable in all cases and there is no guarantee that a sample that conforms to this rule of thumb is actually representative.

Factors that come into play when calculating the size of a sample

The size of a sample needed to be representative depends on several factors, including the size of the population, the desired confidence level and the margin of error.

Population

The size of the population indicates how many people in the population are eligible for the study. For example, the population of all homeowners in a country would include all people who own a house in that country. The size of the population would accordingly depend on how many homeowners there are in that country. For example, if 20 million people in the selected country own a house, then the population of all homeowners in that country would be 20 million people.

Confidence level

The confidence level (also called the confidence level or confidence grade) indicates how certain one can be that the results of the sample are transferable to the population. The higher the confidence level, the larger the sample usually has to be to reach this confidence level. The confidence level corresponds to a Z-score. This is a constant value that is needed for this equation. Here are the Z-scores for the most common confidence levels:

90% – Z-score = 1.645
95% – Z-score = 1.96
99% – Z-score = 2.576

Margin of error

The margin of error, also called confidence interval, indicates how far the result of a sample may deviate from the actual value in the population. It is given in percentage points and thus indicates how certain one can be that the result of the sample can actually be transferred to the population. The smaller the margin of error, the more accurate the result of the sample and the more certain one can be that it can also be transferred to the population.

Estimation or knowledge of the proportion or of certain characteristics

The estimate of proportion in the calculation of sample size is an estimate of the proportion of a particular characteristic in a total population. The estimate of proportion is often used in statistics to obtain estimates of the characteristics of populations by studying only a sample of the population. Proportion estimation can be used to obtain estimates of various properties of populations, such as the proportion of people in a population who have a particular disease or the proportion of people who have a particular political opinion. The accuracy of the proportion estimate depends on the size and representativeness of the sample. The larger and more representative the sample, the more accurate the estimate of the proportion will be.

Excursus: Lack of knowledge about the proportion or number of characteristics

There are cases where the characteristics or the number of persons for whom certain characteristics occur are not known. If you do not know the proportion or characteristics when calculating the sample, there are some possibilities how to proceed. One way is to use an estimate of the proportion based on previous studies or publications. You can also use an “average” proportion, for example 0.5, if you are not sure which proportion is most likely.

One option is to do what is called a “pilot study”. A pilot study is a small preliminary study that is used to determine the proportion that will be used for the main study. You can conduct a pilot study by taking a small sample from the population and determining the proportion of the observed trait. This proportion can then be used in the main study to calculate the sample size.

Another option is to use what is called “model-assisted estimation”. Model-assisted estimation uses statistical models to estimate the proportion based on known factors. The model-assisted estimation can then be used to calculate the sample size.

There are also other statistical methods that can be used to determine the proportion for sample size calculation where the proportion is not known. These include, for example, the bootstrapping method and the jackknife method. However, it is important to note that these methods are usually more complex and require more statistical knowledge and experience.

The question that everyone who conducts a survey or a study has to ask is about the size of the sample in order to get useful results. There are formulas that can be used to calculate the sample.

What is a sample size calculator?

A sample size calculator is a tool used to calculate the necessary sample size for a study. The sample size indicates how many people should be included in the study to provide reliable results.

The sample size calculator takes into account factors such as the size of the population, the confidence level, the desired margin of error and the proportion of characteristics to the population to calculate the necessary sample size. It is often used by people conducting studies or interpreting the results of studies to ensure that the study has enough data to produce reliable results and use resources effectively.

A sample size calculator can be useful in a variety of situations. Some examples where the sampling calculator can provide support are:

• When conducting a study and wanting to reliably calculate the sample size needed to provide reliable results.
• Understanding the impact of factors such as population size and desired precision on sample size.
• If you want to use resources effectively by minimising the necessary sample size to a level that is still representative
• Assessing the quality of studies and their results by considering sample size and confidence level.
• If you want to assess the validity of results of published studies by considering the sample size and confidence level.
• When one does not know the formula or is unsure how to apply it correctly. A sample size calculator tool can be used as an aid in this case to ensure that the sample size is calculated correctly.

mypinio sample size calculator: Which formula is used to calculate the sample in this tool?

There are different statistical methods that can be used to calculate the size of a sample that is needed to be able to conclude the population from the results of the sample. With the mypinio sample size calculator you can calculate the sample from a population. A sample size calculator is a tool that is used to calculate the size of a sample that is needed to determine a representative subset from the population.

In the mypinio sample size calculator the sample is calculated with the following two formulas:

“Margin of Error”

Formula 1: n = (z^2 * p * (1 – p)) / (e^2)

This formula is the so-called “margin of error” and was developed by the American statistician and mathematician William Gosset, who became known under the pseudonym “Student”. The formula is used to determine the accuracy of estimates in samples from a large population. Here n is the size of the sample, z is the “value of the standard error”, p is the estimated proportion of the population that has a certain property and e is the desired error (usually a value of 0.05 or 0.01 is used).

“Corrected sample size formula””

Formula 2: n’ = n / (1 + (n – 1) / N)

This formula is the so-called “corrected sample size” and was developed by the American statistician and mathematician George W. Snedecor. It is used to adjust the size of the sample in cases where the actual size of the population is known. Here n is the original sample size, n’ is the corrected sample size, and N is the size of the population. The formula is often used to improve the precision of estimates in surveys and other studies.

The variables in the formula for calculating a sample include the following parameters:

• n: Size of the sample. This is the result of the sample calculation
• Z: defined confidence interval as the standard normal distribution coefficient.
• N: the size of the population
• p̂: estimate of the proportion Probable or known number of items with the characteristic in the sample or total number of estimated individuals in the sample.
• e: the desired margin of error as a percentage.
• n: sample size
• n’: adjusted sample size

As you can see, the formula is not very easy to apply and requires mathematical knowledge. Therefore, there is the possibility of using a sample size calculator to calculate the sample. We have put this formula into our sample size calculator tool and the tool gives exact values for the calculation of your sample. In the following section you will find 2 examples for calculating the sample size.

Example of sample size calculation

Let’s say an electronics company wants to conduct a market research study on the usage habits of smart TVs among homeowners in a certain country. Now, there are about 20 million homeowners in that country, and obviously not all of them can be surveyed. Thus, the company chooses a representative size from all homeowners, i.e. a sample, in order to be able to draw conclusions about the population from this sample. You can also simulate the following calculations with the mypinio sample size calculator.

Example of a sample calculation:

• The population of all homeowners in this country is 20,000,000.
• The confidence level should be 95%
• The margin of error (confidence interval) may be 5%
• The estimate of the proportion is 50%

Accordingly, the calculation is as follows:

• N = 20000000 // The population of all homeowners in the country
• z = 1.96 // z-value for a confidence level of 95%
• p = 0.5 // estimate of the proportion of 50%
• e = 0.05 # margin of error (confidence interval).

Formula:
n = (z^2 * p * (1 – p)) / (e^2)

Formula with values inserted:
n = (1.69^2 * 0.5 * (1 – 0.5)) / (0.05^2) = 168.5

This is the value of the original sample size when the size of the population is infinite. In practice, however, the population will always be finite, so it is necessary to adjust the sample size to account for the actual size of the population. Adjust the sample size using the following formula:

Formula:
n’ = 168.5 / (1 + (168.5 – 1) / 20000000) = 385

Formula with inserted values:
n’ = n / (1 + (n – 1) / N)

The adjusted sample size of 385 is the actual sample size needed to achieve the desired precision and confidence in estimating the proportion in the population (in this case, the homeowners).

Thus, the company needs to survey at least 385 homeowners to infer the population.

Example 2 of a sample calculation:

However, if the company wants to have as low a standard deviation as possible and uses a confidence level of 99% and a margin of error of 1% when calculating the size of the sample, the company must survey 16628 homeowners.

The calculation then looks like this:

n = (2.576^2 * 0.5 * (1 – 0.5)) / (0.01^2) = 65401
n’ = 65401 / (1 + (65401 – 1) / 20000000) = 16628

Fortunately, you don’t need to know this formula by heart or apply it, because our sample calculator tool does the work for you. Simply use the following tool to calculate your sample.

Sample Size Calculator Tool

When is a sample considered “representative”?

Suppose a social science institute wants to find out what the attitude of the population is towards a certain political issue. The population consists of people of different ages, genders, levels of education and places of residence. To obtain a representative sample, the social scientist could, for example, make a random selection of people, ensuring that the sample reflects the same proportions as the population in terms of age, gender, education level and place of residence. The randomly selected people would then receive a survey asking them about their attitudes towards the political issue.

Since it might be difficult for the social science institute to get addresses of all people with the corresponding socio-demographic characteristics, so-called online panels are often commissioned. An online panel is a group of people who are willing to participate online in surveys, studies and other market research activities. Online panels are often used by companies, market research institutes and other organisations to collect information from a large number of people quickly and cheaply. The members of an online panel are selected according to certain characteristics such as age, gender, education level and place of residence to ensure that the panel is representative of the population.

→ Online panel

In a survey, a sample has a significant impact on the validity and reliability of the result. A good sample should be representative of the entire population being surveyed and be large enough to provide a reasonable estimate of the parameters of the population. If the sample is not representative or is too small, bias and inaccurate results may occur.

What are the different types of sampling methods?

Probability sampling

Probability sampling is a type of sampling in which each element in the population is assigned a specific, predetermined probability of being selected. This means that each element in the population is equally likely to be selected for the sample. Probability sampling is a good choice when you want to ensure that the sample is representative of the entire population and when you cannot examine all elements of the population for reasons of practicality. The results of analyses based on probability sampling can be generalised to the entire population, allowing statements to be made about the entire population.

Examples of probability samples

• Simple random sampling: In this method, each element of the population has the same probability of being selected for the sample. For example, you could take a simple random sample by randomly selecting every fifth person on a list of all the inhabitants of a city.
• Systematic sampling: In this method, the first element of the population is selected at random and then every nth element of the population is selected for the sample. For example, one could select every tenth person on a list of all residents in a city by randomly selecting the starting point and then selecting every tenth element on the list.
• Stratified sampling: In this method, the population is divided into different groups (strata) and a random sample is drawn from each group. For example, you could divide the population of a country into age strata and then draw a random sample from each age group.
• Cluster sampling: In this method, groups of elements in the population are randomly formed and then a random sample is drawn from each group. For example, you could form groups of patients in a hospital and then select one patient from each group for the sample.

Advantages of probability sampling

• Representativeness: Since each element in the population is equally likely to be selected for the sample, the sample is usually representative of the entire population.
• Generalisability: The results of analyses based on probability sampling can be generalised to the entire population. This means that statements can be made about the entire population.
• Comparability: Since all elements in the population are equally likely to be selected for the sample, the results of different probability samples can be compared with each other.
• Predictability: The size and composition of the sample can be calculated in advance, which improves the predictability of the results.
• Efficiency: In many cases, probability samples are more efficient than non-probability samples because they require a smaller sample to achieve the same precision.
• Independence: Since the sample is randomly selected from the population, the elements in the sample are independent of each other. This means that the behaviour of one element has no influence on the probability of another element being included in the sample.
• Enables the use of inferential statistics: Because the sample is representative of the population, inferential statistics can be used to make statements about the population.
• Reduces the effects of bias and prejudice: Because the sample is randomly selected, bias and prejudice are minimised when selecting items for the sample.
• Ease of documentation and review: Since the sample selection process is transparent and traceable, it is easier to document and review the process.

Disadvantages of probability sampling

• Time-consuming and costly: Probability sampling is usually more time-consuming and costly than non-probability sampling because it requires random selection of elements from the population.
• Difficulty in implementation: In some cases, it may be difficult to randomly select elements from the population, especially if the population is very large or difficult to access.
• Possible bias due to difficulties in selecting the items: In some cases, it may be difficult to select the items for the sample, especially if the population is very large or difficult to access.
• Possible bias due to errors in selecting the items: In some cases, there may be errors in selecting the items for the sample, e.g. if the items are not selected randomly or if an item is included in the sample more than once.
• Possible bias due to incompleteness of the sample: In some cases, the sample may be incomplete, e.g. if some elements of the sample are not accessible or if the sample is too small to be representative.

Non-probability sampling

In non-probability sampling, participants are selected based on convenience or accessibility rather than randomly. A non-probability sample is a type of sample in which each element in the population is not assigned a specific, predetermined probability of being selected. This means that some elements in the population are more likely and others less likely to be selected for the sample. A non-probability sample is usually easier and faster to collect than a probability sample because it does not rely on random selection of elements from the population. However, the results of analyses based on non-probability sampling cannot be generalised to the entire population, which means that one can only make limited statements about the population.

Examples of non-probability samples

• Random sampling: In this method, the sample is composed of elements that are easily accessible or available. For example, you could take a random sample by asking the first 10 people you meet on the street to participate in your survey.
• Quota sampling: In this method, the sample is made up of people who match certain predefined characteristics such as age, gender or level of education. For example, you could draw a quota sample by asking 100 people to participate in your survey, with a certain number of men and women and a certain number of people in each age group.
• Snowball sampling: In this method, the sample is made up of items recommended by other items in the sample. In snowball sampling, for example, you can ask one person to participate in your survey and then ask that person to recommend other people who meet certain criteria.
• Purposive sampling: In this method, the sample consists of people who are specifically selected because they have certain characteristics or knowledge about the topic of interest. For example, you could draw a purposive sample by selecting experts in a particular field to participate in your study.
• Network sampling: In this method, the sample is composed of elements who are connected through a social or professional network. For example, you could take a network sample by selecting a group of people connected through a LinkedIn group.
• Unrestricted sample: In this method, the sample is composed of elements selected without any predetermined criteria. For example, you could take an unrestricted sample by simply selecting the first 100 people you meet in a public place.

Advantages of non-probability sampling

• Quick and easy to conduct: non-probability samples are usually quicker and easier to collect than probability samples because they do not rely on randomly selecting elements from the population.
• Cost-effective: Non-probability samples are often less expensive than probability samples because they require less effort.
• Flexibility: Non-probability sampling allows for the selection of elements that are particularly relevant to the study, which increases flexibility.
• Suitability for special population groups: Non-probability sampling may be suitable in certain cases where it is difficult to take a representative sample from a specific population group.
• Possibility to answer specific questions: Non-probability sampling can be suitable in certain cases when the questions are very specific and it is not necessary to refer to the entire population.
• Suitability for small population sizes: Non-probability sampling can be suitable in certain cases when the population size is very small and it is difficult to take a representative sample.
• Suitability for rare population characteristics: Non-probability sampling can be suitable in certain cases when the characteristic you want to study is very rare in the population and it is difficult to take a representative sample.
• Possibility of gaining deep insights: Non-probability sampling can provide deep insights into certain population groups, as one can select elements that are particularly relevant to the study.
• Possibility to investigate qualitatively: Non-probability samples are particularly suitable for qualitatively oriented research where one wants to gain deep insights into the experience and understanding of the elements in the sample.

Disadvantages of non-probability samples

• Lack of representativeness: Since non-probability samples are not randomly selected from the population, they are usually not representative of the whole population. This means that the results cannot be generalised to the entire population.
• Lack of comparability: Since non-probability samples are not randomly selected from the population, the results are not always comparable with the results of other non-probability samples.
• Unpredictability: The size and composition of the non-probability sample cannot be predicted, which reduces the predictability of the results.
• Possible bias and prejudice: Since non-probability samples are not randomly selected from the population, there is a risk that bias and prejudice play a role in the selection of items for the sample.
• Possible bias in the results: Since non-probability samples are not representative of the whole population, there is a risk that the results are biased and do not reflect the actual conditions in the population.
• Difficulty in applying inferential statistics: Since the sample is not representative of the population, inferential statistics cannot always be used to make statements about the population.
• Difficulty in documentation and verification: As the sample selection process is not transparent and traceable, it can be difficult to document and verify the process.

Use the free sample size calculator tool from mypinio!

Is it too complicated for you to calculate the size of a sample using a formula? Then simply use the sample size calculator tool from mypinio. The sample size calculator already contains the required formula and you only have to adjust the confidence level, the characteristic proportion and the margin of error and enter the population. With mypinio’s sample size calculator you can quickly and effectively determine the sample size for your next survey or market research study and save yourself a lot of time and effort.

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