Systematic Random Sampling

What is Systematic Random Sampling?

Systematic Random Sampling is a type of probability sampling technique where researchers select every kth element from a list or population, starting from a randomly chosen point. It’s simple, structured, and widely used in statistics, social sciences, quality control, and market research.

1.0Systematic Random Sampling Definition

Systematic sampling refers to a method where the first unit is selected at random, and subsequent units are chosen at regular intervals throughout the population list. This method is efficient, especially when a complete list of the population is available.

2.0Systematic Random Sampling Formula

To determine the sampling interval k, use the formula:

$k=\frac{N}{n}$

Where:

• N = Total population size
• n = Desired sample size
• k = Sampling interval

Start by selecting a random number r between 1 and k, and then select every kth item thereafter:

r, r+k, r+2k, r+3k, \ldots

3.0Systematic Random Sampling Example

Example:
Suppose you want to select 10 students from a list of 100.

1. N = 100, n = 10 → k = 100/10 = 10
2. Randomly choose a number between 1 and 10, say 4.
3. Select the 4th, 14th, 24th, ..., 94th students.

This is a classic systematic random sampling example used in education, health surveys, and business analytics.

4.0Systematic Random Sampling Method (Step-by-Step)

1. List the population in a logical order.
2. Determine sample size n.
3. Calculate the interval k = N/n.
4. Select a random starting point r $\in [1,k]$.
5. Choose every kth member starting from r.

5.0What Is a Systematic Sample?

A systematic sample is the subset obtained using the systematic random sampling method. It ensures equal probability of selection but relies on the structure of the population list.

What is the systematic random sample?

A systematic random sample is one that includes elements selected at equal intervals from an ordered list, after a random starting point is chosen.

6.0Advantages of Systematic Random Sampling

• Easy to use and understand
• Requires only one random start
• Useful when population is ordered
• Time-saving and cost-effective
• Good alternative to simple random sampling

7.0Disadvantages of Systematic Random Sampling

• Can introduce bias if there's a hidden pattern in the population list
• Not ideal for small populations
• Randomness is lower than in simple random sampling
• Not suitable for unevenly distributed populations

8.0Example for Systematic Sampling in Real Life

• Manufacturing: Selecting every 5th item on a production line for quality check
• Health Research: Surveying every 10th patient visiting a clinic
• Education: Picking every 3rd student roll number for a feedback study

These real-life applications show the value and reliability of systematic sampling in various fields.

9.0Solved Examples on Systematic Random Sampling

Example 1: A company wants to select a sample of 10 employees from a list of 100 employees for a training program using systematic random sampling. What should be the sampling interval, and how is the sample selected?

Solution:

• Total population N = 100
• Required sample size n = 10

$\text{Sampling interval }(k)=\frac{N}{n}=\frac{100}{10}=10$

Suppose a random starting number is 4.

Then selected employees: 4, 14, 24, 34, ..., 94.

Answer: Sampling interval = 10; Selected employee positions: 4th, 14th, ..., 94th.

Example 2: From a population of 75 households, select a systematic sample of 5.

Solution:

• N = 75, n = 5

$k=\frac{75}{5}=15$

Let’s say the random start = 3

Then: 3, 18, 33, 48, 63

Answer: Sampled households are 3rd, 18th, 33rd, 48th, and 63rd.

Example 3: Population size is 95; sample size is 10. Perform systematic sampling.

Solution:

• N = 95, n = 10,

$k=\frac{95}{10}=9.5\approx 10$

Pick a random number between 1 and 10, say 7.

Sample: 7, 17, 27, ..., 97 → Stop at max index ≤ 95.

Answer: Use rounded interval (9 or 10); final sample = 7, 17, 27, 37, 47, 57, 67, 77, 87, 97 (if ≤ 95).

Example 4: A factory produces 200 bulbs daily. A quality manager wants to check every 20th bulb starting from the 5th. How many will be selected?

Solution:

• k = 20, starting at 5
• Selected: 5, 25, 45, ..., up to 200

Number of bulbs =

$\lfloor \frac{200-5}{20}\rfloor +1=10$

Answer: 10 bulbs are selected.

Example 5: In a housing survey of 250 residents, choose 25 participants.

Solution:

• $k=\frac{250}{25}=10$
• Let random start = 6
• Sample = 6, 16, 26, ..., 246

Answer: Interval = 10; Participants = every 10th person starting from 6.

10.0Practice Questions on Systematic Sampling

1. Select a systematic sample of 8 from a list of 80 books. What is the interval and sample if starting point is 5?
2. A researcher needs a sample of 12 from a population of 96 students. If the first selected student is number 4, list the rest.
3. A manufacturer produces 150 widgets daily. Every 15th widget is tested for quality. How many are tested per day? List first 5 selections if sampling starts at 3.
4. A marketing analyst selects every 7th shopper at a mall starting from the 2nd. If 70 shoppers are tracked, how many will be sampled?