## Systematic Sampling

**Definition:**

Systematic sampling is a probability sampling technique where a population is divided into equal-size intervals, and a starting point is randomly selected within the first interval. Then, every kth element in the population is selected for the sample, where k is the sampling interval calculated as the population size divided by the desired sample size.

### Types of Systematic Sampling

Types of Systematic Sampling are as follows:

**Systematic random sampling:**In this method, the first unit is selected at random from the population, and then subsequent units are selected at fixed intervals based on a predetermined pattern. For example, if we want to select a sample of 100 individuals from a population of 1000, we could randomly select one individual from the first 10, and then select every 10th individual after that.**Linear systematic sampling**: In this method, the population is arranged in a line or sequence, and units are selected at fixed intervals along that line. This is useful for populations that are arranged in a linear or sequential order, such as a list of customer names or a street of houses.**Circular systematic sampling:**In this method, the population is arranged in a circle or cycle, and units are selected at fixed intervals around the circle. This is useful for populations that have a circular or cyclical structure, such as days of the week or hours in a day.**Double sampling:**In this method, a small preliminary sample is taken from the population, and then a larger systematic sample is selected from the remaining population. This can be useful for reducing the cost of sampling or for ensuring that the sample is representative of specific subgroups within the population.**Stratified systematic sampling**: In this method, the population is first divided into strata or subgroups based on some characteristic (such as age or income), and then a systematic sample is selected from each stratum. This can help ensure that the sample is representative of the population as a whole, as well as of specific subgroups.**Cluster systematic sampling:**In this method, the population is divided into clusters or groups, and then a systematic sample of clusters is selected, with units within each selected cluster being sampled systematically. This can be useful when it is more practical to sample clusters than individuals (e.g., in large households or communities).**Random-start systematic sampling:**In this method, the starting point for selecting units is chosen randomly, rather than being based on a fixed pattern. This can help reduce bias in the sampling process and increase the representativeness of the sample.**Multi-stage systematic sampling**: In this method, a combination of different sampling methods (such as stratified and cluster sampling) is used to select units from the population. This can help ensure that the sample is representative of the population as a whole, while also allowing for efficient and cost-effective sampling.

### Systematic Sampling Method

Here are the basic steps for conducting a Systematic Sample Method:

**Define the population:**Determine the larger population you want to sample from.**Determine the sample size:**Decide on the number of units you want to include in your sample.**Calculate the sampling interval:**Divide the population size by the sample size to determine the sampling interval (k). For example, if you want to select a sample of 100 individuals from a population of 1000, the sampling interval would be 10 (i.e., 1000/100 = 10).**Choose a random starting point:**Choose a random number between 1 and k as your starting point.**Select the sample:**From your starting point, select every kth unit in the population until you have your desired sample size.

For example, let’s say you want to conduct a systematic sample of 100 households in a neighborhood of 1000 households. The sampling interval would be 10 (1000/100 = 10). You would choose a random number between 1 and 10 as your starting point (let’s say you choose 5). Then you would select every 10th household starting from the 5th household until you reach 100 households.

### Systematic Sampling Formula

The systematic sampling formula can be used to determine the sample size and the interval between selected elements.

The formula for systematic sampling is:

**n = N / (k)**

where:

- n is the sample size
- N is the population size
- k is the sampling interval

The sampling interval (k) is calculated by dividing the population size (N) by the desired sample size (n). For example, if we want to select a sample of 100 individuals from a population of 1000, the sampling interval would be calculated as follows:

k = N / n k = 1000 / 100 k = 10

In this example, we would select every 10th individual from the population to form the sample.

To use systematic sampling, we first select a random starting point from the population, which is any element between 1 and k. From that starting point, we select every kth element to form the sample. For example, if our starting point is 5, we would select elements 5, 15, 25, 35, and so on until we reach the desired sample size.

### Examples of Systematic Sampling

Here are some examples of systematic sampling:

**Quality control**: A production line manager wants to check the quality of products produced on a conveyor belt. The manager selects every tenth product that comes out of the belt to ensure that the quality meets the set standards.**Opinion survey**: A polling agency wants to conduct an opinion survey of the residents of a city. The agency selects every 100th person listed in the phonebook to participate in the survey.**Medical research**: A researcher wants to study the effects of a new medication on patients. The researcher selects every fifth patient that comes to a clinic to participate in the study.**Education research:**A researcher wants to study the performance of students in a school. The researcher selects every third student in a class to participate in the study.**Census survey**: A government agency wants to conduct a census survey of a population. The agency selects every 100th household in a region to participate in the survey.

**Systematic Sampling Example Situation**

Systematic Sampling Example Situation is as follows:

Let’s say we want to conduct a survey of 1000 households in a city to estimate the average monthly water usage. Instead of surveying all 1000 households, we can use systematic sampling to select a representative sample.

To use systematic sampling, we first need to determine the sampling interval (k). Let’s say we want to select a sample of 100 households. Using the formula for systematic sampling, we have:

k = N/n = 1000/100 = 10

This means that we need to select every 10th household from the population to form our sample. We can do this by choosing a random starting point between 1 and 10, and then selecting every 10th household thereafter.

For example, if we randomly select household number 3 as our starting point, we would select the following households for our sample: 3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 103, 113, 123, and so on, until we have selected 100 households in total.

By using systematic sampling, we can obtain a representative sample of the population with less effort and cost compared to surveying all households.

Once we have selected our sample of 100 households, we can survey them to obtain information about their monthly water usage. We can then use the data from the sample to estimate the average monthly water usage of the entire population.

To calculate the estimate of the population average, we can use the formula:

Population average estimate = (sum of sample observations) / sample size

For example, if our sample of 100 households had a total monthly water usage of 10,000 gallons, then our estimate of the population average monthly water usage would be:

Population average estimate = 10,000 / 100 = 100 gallons per household per month

### When to Use Systematic Sampling

Systematic Sampling is useful to use in the following situations:

**When****The population is large:**Systematic sampling is efficient and practical for large populations that are difficult or time-consuming to study in their entirety.**When****The sampling frame is well-ordered**: Systematic sampling requires a well-ordered sampling frame where there is no systematic pattern to the arrangement of items.**When Randomness is desired**: Systematic sampling ensures some level of randomness through the use of a randomly selected starting point, which is useful when a simple random sample is not feasible.**When Resources are limited:**Systematic sampling requires fewer resources than other sampling methods, making it a practical option for studies with limited resources.**When The research question is straightforward:**Systematic sampling is useful when the research question is straightforward, and the sample needs to be representative of the population.**When The sampling interval can be chosen appropriately:**The sampling interval needs to be chosen appropriately to avoid bias in the sample. This is possible when the population is homogenous and there are no subgroups that need to be analyzed separately.

### Applications of Systematic Sampling

Applications of Systematic Sampling are as follows:

**Agriculture:**Systematic sampling is used in agriculture to study the distribution of pests or diseases in crops. A systematic sampling of crops is taken to count the number of diseased or infected plants.**Market research:**Systematic sampling is used in market research to survey customers about their preferences and opinions on products. For example, a company may select every 10th customer who walks into a store to participate in a survey.**Environmental studies**: Systematic sampling is used in environmental studies to measure the quality of air, water, or soil in a particular area. Samples are taken at regular intervals to create a representative sample of the entire area.**Epidemiology:**Systematic sampling is used in epidemiology to study the prevalence of a disease in a population. Samples are taken systematically to estimate the number of people affected by the disease.**Education research:**Systematic sampling is used in education research to study the performance of students. Samples are taken systematically to estimate the average performance of students in a particular subject.**Quality control**: Systematic sampling is used in quality control to check the quality of products during production. Samples are taken systematically at regular intervals to ensure that the quality of products meets the set standards.

### Purpose of Systematic Sampling

The purpose of systematic sampling is to obtain a representative sample of a population by selecting every nth item from the population list. It is a convenient and straightforward method that can be used to save time and resources in large-scale studies. Systematic sampling has several purposes:

**Efficiency**: Systematic sampling is an efficient method of sampling because it requires fewer resources and less time than other sampling techniques. This is because the sampling intervals are predetermined, and the selection of the sample items is systematic.**Representation**: Systematic sampling ensures that the sample is representative of the population because it covers the entire population list. This reduces the chance of bias in the sample selection process.**Randomness**: Systematic sampling introduces an element of randomness into the sample selection process because the starting point of the sample selection is randomly determined.**Reliability**: Systematic sampling is a reliable method of sampling because it is easy to replicate. This means that the same sample can be obtained from the same population list, and the results will be consistent.**Cost-effective:**Systematic sampling is a cost-effective method of sampling because it requires fewer resources and less time than other sampling techniques. This makes it ideal for large-scale studies or when resources are limited.

### Characteristics of Systematic Sampling

The following are some of the characteristics of systematic sampling:

**Regular Sampling Intervals:**Systematic sampling involves selecting every kth item from the population, where k is a fixed interval. For example, if the population size is 1000 and the sample size is 100, then the sampling interval is 10 (i.e., 1000/100).**Random Starting Point**: The first item in the sample is selected randomly from the first k items in the population. This helps to avoid any bias that may be introduced by a specific starting point.**Easy to Implement**: Systematic sampling is a straightforward method to implement and is less time-consuming than some other sampling methods.**Representative Sample**: When the sampling interval is chosen appropriately, systematic sampling can provide a representative sample of the population.**Possibility of Bias**: Systematic sampling can introduce bias if there is a pattern in the population that is related to the sampling interval. For example, if a population is sorted by some characteristic and the sampling interval is chosen to match this sorting, then the resulting sample may not be representative of the population.**Sample Size**: The sample size in systematic sampling is determined by dividing the population size by the sampling interval. Therefore, the sample size is limited by the population size and the chosen sampling interval.

### Advantages of Systematic Sampling

There are several advantages of systematic sampling, including:

**Ease of Use**: Systematic sampling is easy to understand and implement, and it requires fewer resources than other sampling methods.**Representative Sample:**When the sampling interval is chosen appropriately, systematic sampling can provide a representative sample of the population. This means that the sample accurately reflects the characteristics of the population.**Efficient**: Systematic sampling is more efficient than some other sampling methods because it requires less time and effort to select a sample.**Reduction of Sampling Error**: Systematic sampling can help to reduce sampling error, which is the difference between the characteristics of the sample and the population. This is because systematic sampling ensures that each element in the population has an equal chance of being selected for the sample.**Suitable for Large Populations:**Systematic sampling is particularly useful for large populations, where other sampling methods may be impractical or too time-consuming.**Random Starting Point:**The use of a random starting point in systematic sampling reduces the possibility of bias in the sample.

### Disadvantages of Systematic Sampling

Disadvantages of Systematic Sampling are as follows:

**Possibility of Bias:**Systematic sampling can introduce bias if there is a pattern in the population that is related to the sampling interval. For example, if a population is sorted by some characteristic and the sampling interval is chosen to match this sorting, then the resulting sample may not be representative of the population.**Limited Sampling Frame:**The sampling frame (the list of items in the population) needs to be ordered, and the order should not have any systematic pattern. If the population is not ordered or the order has a pattern, then the sampling frame may not be representative of the population.**Lack of Flexibility:**Systematic sampling requires that the sampling interval is fixed before the sample is selected. This means that if there is a change in the population characteristics, the sampling interval may not be appropriate, and the sample may no longer be representative of the population.**Possible Overlap:**Systematic sampling can result in samples that overlap if the sampling interval is not chosen appropriately. This means that some elements of the population may be selected more than once, while others may not be selected at all.**Limited Randomness**: While systematic sampling uses a random starting point, it does not provide the same level of randomness as other sampling methods such as simple random sampling.