Sampling Error

When we use statistics to make predictions, we often face differences between what we expect and what happens. These differences are called errors. Two common types are:

• Random error: Happens by chance and tends to balance out with larger sample sizes.
• Sampling error: It arises naturally because we use a sample (not the entire population).

Sampling error arises naturally because we typically use a sample (a subset) to make inferences about a larger population, rather than measuring the entire population. Even with statistically sound sampling methods, the specific sample drawn may not perfectly mirror all characteristics of the entire population. This inherent discrepancy is the sampling error.

Sampling Error Formula

The size and shape of the sample are used to calculate the sampling error rate, which reflects the accuracy of the selection process. An important factor in identifying such an error is the selection basis, which is a type of systematic error caused by non-random sampling methods.

The formula to find the sampling error is given as follows:

Sampling Error (SE) = (1/√ N) 100

Where: N is the Sample Size

How to Reduce Sampling Error?

To reduce sampling error, two methods are:

• Increase Sample Size - A larger sample size brings you closer to the true population and reduces error. The sample should represent all groups (demographics) fairly. Sampling error decreases as sample size increases — they are inversely related.

• Stratification - If the population is similar (homogeneous), sampling is easy and often representative.
  But if the population has different groups (heterogeneous), it's harder. So, we divide the population into strata (groups with similar traits) and take samples from each group based on its size. This helps make the sample more accurate.

Other key Methods:

• Proportional Group Representation: - Use groups proportional to their existence in your overall target market.
  For example, if 40% of your target market consists of a certain demographic, ensure that you use 40% of this demographic in your survey study.

• Use Random Sampling: - In general, you need a more diverse, yet precise approach to recruiting participants for your survey.
  For example, you can draw a random sample of participants, but control who can take part in your survey based on demographic and psychographic information. You can also ask questions that participants must answer in a certain way to complete the survey.

Precautions Using Sampling Errors

Sample Size Too Small: When the sample size is too small, it may lead to errors.

Sampling Bias: It occurs when the members of the sample are unrepresentative of the population.

Sample Coverage Error: This could happen for a variety of reasons, including the sample being too small, the sample being unrepresentative of the population, or the sample being contaminated.

Sample Contamination: There may bea chance that where Sample may be diluted. This leads to less accuracy.

Sample Unrepresentativeness: This could happen for a variety of reasons, including the person being too busy to take the survey, the person refusing to take the survey, or the person being unable to take the survey for some reason.

Application Of Sampling Error in Computer Science

• Machine Learning – Helps explain why models give different results on different data samples.
• Databases – Used to guess answers faster using part of the data (sampling), with some error.
• Big Data – Makes huge data easy to handle by using samples, with error showing how close the results are.
• Simulations (Monte Carlo) – Uses samples to estimate answers; sampling error tells how accurate it is.
• Network Monitoring – Samples some data to save time; error shows how close the results are to real values.

Solved Question on Sampling Error

Example 1: A manufacturing company produces light bulbs. It is estimated that 2% of the light bulbs produced are defective. If a box contains 100 light bulbs, what is the probability that exactly 3 light bulbs in the box are defective?

Solution:

To find the probability of getting exactly 3 defective light bulbs out of 100, we can use the binomial probability formula:

P(X = 3) = (¹⁰⁰C₃) × (0.02)³ × (0.98)⁹⁷

≈ 0.1168 or 11.68%

Example 2: In a particular city, 25% of the residents have a certain disease. If 5 residents are selected at random, what is the probability that exactly 2 of them have the disease?

Solution:

Let X be the number of residents with the disease out of 5 selected.

P(X = 2) = (⁵C₂) × (0.25)² × (0.75)³

≈ 0.2734 or 27.34%

Example 3: A fair coin is tossed 10 times. What is the probability of getting exactly 6 heads?

Solution:

Tossing a fair coin 10 times is a binomial experiment with n = 10 and p = 0.5 (probability of getting a head).

P(X = 6) = (¹⁰C₆) × (0.5)⁶ × (0.5)⁴

≈ 0.2051 or 20.51%

Question 4: In a city, 40% of people prefer using public transport. If 6 people are selected randomly, reducing is the probability that exactly 4 of them prefer public transport?

Solution:

Let X be the number of defective screws. This is a binomial distribution with
n=8, p=0.10, and q=1−p = 0.90

P(X = 2) = \binom{8}{2} \cdot (0.10)^2 \cdot (0.90)^6P(X = 2) = 28 \cdot 0.01 \cdot 0.531441P(X = 2) \approx 0.1488

Unsolved Question on Sampling Error

Question 1: In a factory, 5% of items are defective. A quality inspector checks a random sample of 20 items. What is the probability that exactly 1 item is defective?

Question 2: A basketball player has a 70% chance of making a free throw. If she takes 8 free throws, what is the probability she makes exactly 6 of them?

Question 3: A student guesses on a multiple-choice quiz with 5 questions, each having 4 options (only one correct). What is the probability that the student gets exactly 2 questions correct?

Question 4:In a city, 40% of people prefer using public transport. If 6 people are selected randomly, what is the probability that exactly 4 of them prefer public transport?