৪৯তম বিসিএস ⎯ পরিসংখ্যান [৯৮১]

SRS এ সমগ্র পপুলেশন থেকে স্যাম্পল নেই। কিন্তু, ক্লাস্টার টেকনিকে স্যাম্পলকে ভাগ করা হয় কিছু ক্লাস্টারে।

For example, to analyse total population, villages are sampled, then all units within selected clusters are studied. Here,  village are clusters.

এখানে প্রত্যেকটা গ্রুপের মধ্যে যে গ্রামবাসী থাকে, তারা ভিন্ন বৈশিষ্ট্য বিশিষ্ট, তাই internally heterogeneous. ( অভ্যন্তরীণভাবে ভিন্নধর্মী) 

ধরি পৃথিবীটা টোটাল পপুলেশন। এখানে প্রতিটা দেশ একেকটি স্ট্রাটা। এখানে দেশগুলো নিজেদের ভেতর একই (কারণ নাগরিকগণ একই বৈশিষ্ট্যের), কিন্তু একটা দেশের সাথে আরেকটা দেশের নাগরিকদের বৈশিষ্ট্য আলাদা। এটাই বলে- internal homogeneous ( ভেতরে একই) ও external heterogeneity ( বাইরে আলাদা)

If periodicity matches sampling interval, bias occurs; independence ensures unbiasedness.

অর্থাৎ সিসটেমেটিক স্যাম্পলিং এ যদি পিরিওডিসিটি থাকে এবং স্যাম্পলিং  ইন্টারভেলের সাথে সেটা ম্যাচ করে, তাহলে বায়াসডনেস হবে।

অর্থাৎ ধরি, একটা ক্লাসের ছাত্রদের পড়াশোনার মান যাচাই করব। তারা যদি এমনভাবে বসে থাকে যে, প্রতি পাঁচজন পর পর ভালো স্টুডেন্ট এবং আমরা যদি sample ও এভাবে প্রতি পাঁচজন পর পর একজনকে বাছাই করি- তখন সেটা স্যাম্পলিং এরর হবে। বায়াসড রেজাল্ট দিবে। 

k = N/n
   =120/10
   =12.

Source: Sampling, Cochran. 

Margin of error= Z×SE ≈1.96×SE  = 1.96 ×2 =3.92.

Why multiplying by 1.96?

Because at 95% confidence Z value becomes 1.96.

Cluster sampling reduces travel and cost by grouping geographically.

Neyman allocation uses nh ∝ Nh Sh, but with equal variance → n_h ∝ N_h.

Source: Sampling, Cochran. 

Simple random sampling is statistically straightforward and allows for easy application of inferential statistics.

However, it doesn't guarantee demographic representation, may be costly for large populations, and does not eliminate all types of bias (e.g., non-response bias).

Source: Business Statistics, MP Gupta, SP Gupta. 

A random number generator or a lottery method can be used to ensure that the selection is truly random and unbiased.

Other options describe different sampling techniques (like convenience or systematic sampling)

Source: Business Statistics, Md. Abdul Aziz 

One major limitation of simple random sampling is that it requires a full list of the population, which can be difficult or impractical for large or scattered populations.

Also, it doesn't guarantee subgroup representation (that's better handled by stratified sampling)

When performing simple random sampling without replacement, the number of possible samples of size 10 from a population of 100 is given by the combination formula:

ncr = n!/(r!×(n-r)!)
= 100!/(10!×90!)

One key limitation of sampling is the potential for sampling bias, which occurs if the sample is not representative of the population. This can lead to misleading conclusions. Proper sampling techniques are essential to reduce this risk.

Snowball sampling is useful for hidden or hard-to-reach populations (e.g., drug users, refugees).

Participants refer others with similar characteristics.

Quota sampling is a non-probability method where the sample reflects certain characteristics of the population in specified proportions (e.g., 50% male, 50% female).

Source: Business Statistics, MK Roy

Probability sampling techniques (simple random, stratified, systematic, cluster) give each unit a known probability of being chosen.

Others are non-probability methods.

Purposive (judgmental) sampling relies on the researcher’s judgment to select cases that best serve the research purpose.

This may create extreme subjectivity and thus extreme bias.

Cluster sampling randomly selects whole groups (clusters) such as villages, schools, or households, and then studies all (or some) members within those groups.

Convenience sampling selects units that are easily accessible to the researcher.

It is quick and cheap but prone to bias.

Here a researcher goes to a shopping mall and interviews whoever is available.

So  units  are easily accessible to the researcher

Systematic sampling selects every k-th element from an ordered list after a random start. Here, every 10th name is chosen.

Stratified sampling divides the population into homogeneous strata (subgroups) and then samples from each stratum, ensuring representation of all groups.

Source: Live mcq class lecture

A statistic is a numerical value that describes a characteristic of a sample.

In this case, the median income of surveyed employees (a sample) is a statistic.

A parameter is a numerical summary that describes a characteristic of an entire population.

Since the mean age of all citizens is being considered, it's a parameter.

A population is the whole group that we want to draw conclusions about.

It can be people, items, events, or observations depending on the context of the study.

A sample is a smaller group selected from a larger group (population) for the purpose of conducting research.

It should represent the population to make valid inferences.

Convenience sampling involves selecting subjects who are easily accessible (e.g., friends, nearby individuals).
This non-probability method often introduces selection bias and is not representative of the overall population.

Source: Business Statistics, MK Roy.

In cluster sampling, the population is divided into clusters (often based on geography or institutions), and entire clusters are randomly selected.

Either all individuals in selected clusters are studied, or a subsample is taken within those clusters.

Then that becomes multistage sampling. 

Multistage sampling is an extension of cluster sampling.

It becomes multistage when, after selecting clusters, the researcher does another round of sampling within those clusters (e.g., selecting households or individuals).

This method adds flexibility and is useful in large or geographically dispersed populations.

To ensure representation of various income groups, stratified sampling is ideal.

It divides the population into strata based on income levels and selects samples from each, maintaining proportionality and reducing bias.

Convenience sampling selects participants who are easily accessible, such as friends, colleagues, or people nearby.

This introduces strong selection bias, making the sample non-representative, and limiting the generalizability of results.

Source: Business Statistics (MP Gupta, SP Gupta.)  

Simple random sampling gives each member of the population an equal and independent chance of being selected, which helps minimize selection bias.

Cluster sampling often results in higher sampling error because individuals within clusters tend to be more similar to each other, reducing variability. In contrast, stratified sampling aims to increase precision by sampling across subgroups.

সহজ কথায়, clus sampling এ ক্লাস্টারগুলো বিভিন্ন ধরণের বৈশিষ্ট্য সম্পন্ন ইনডিভিজুয়াল নিয়ে গঠিত, যাদের মধ্যে অমিল থাকার কথা। আর stratified sampling এ স্ট্রাটা গুলো সমধর্মী বৈশিষ্ট্য সম্পন্ন ইনডিভিজুয়াল নিয়ে গঠিত যাদের মিল থাকার কথা। এর অন্যথা হলে সমস্যা সেটা। 

sampling error এর প্রাথমিক কারণ হচ্ছে - আমরা যে পপুলেশন না নিয়ে স্যাম্পল নিয়ে কাজ করছি- এটাই। স্যাম্পলিং ম্যাথড যতই ভালো আর ত্রুটিমুক্ত হোক, এরর আসবেই। 

Sampling error arises when a sample, rather than the entire population, is studied.

It reflects the natural variability between sample estimates and true population values, even if sampling is random and correctly conducted.

Non-sampling errors occur due to human or systematic mistakes, such as data entry errors, misinterpretation of responses, or faulty survey design.

C ছাড়া বাকিগুলো sampling error.

Larger sample sizes tend to reduce sampling error, as they provide a more accurate representation of the population.

However, increasing the sample doesn't eliminate non-sampling errors like measurement or recording mistakes.

Unlike sampling error, non-sampling error can happen even when the entire population (census) is surveyed.

This includes errors like misreporting, misunderstanding questions, interviewer bias, and data processing mistakes.

Sample size is heavily influenced by the confidence level (e.g., 95%) and the margin of error (e.g., ±5%).

A higher confidence level or lower margin of error requires a larger sample size to maintain accuracy and reliability.

Source: Business Statistics (MK Roy) 

As the sample size increases, the sampling error typically decreases, leading to more precise estimates of population parameters.

However, it also increases cost and time, so there's a trade-off.

A smaller margin of error and a higher confidence level require a larger sample size to ensure the results are statistically reliable.

Scenario B is the most statistically demanding, thus needs the largest sample.

Snowball sampling is a non-probability technique often used to access hidden or hard-to-reach populations, where participants are recruited through referrals.

It lacks random selection, so not every individual has a known or equal chance of being included.

Among the options, only cluster sampling is a probability sampling method, where randomly selected clusters allow for generalization to the population with statistical confidence. 

Although the design starts with probability sampling (multi-stage: region → household), the actual inclusion of only available and willing participants introduces non-response bias, a non-probability element.

This makes it a mixed approach that affects representativeness.

In non-probability sampling, because participants are not chosen randomly, it's impossible to accurately calculate probabilities or measure sampling variability.

This limits the ability to compute confidence intervals, margins of error, or perform rigorous hypothesis testing.

In statistics, a large sample generally refers to a sample size of n ≥ 30. With such a size, the Central Limit Theorem applies, making the sample mean approximately normally distributed regardless of the population distribution.

Source: Business Statistics (Md. Abdul Aziz)

The sampling frame is the list, directory, or database of all population units from which a sample is actually selected. If the frame is incomplete, it leads to coverage error.

Common Examples of Sampling Frames:

-Electoral roll (list of registered voters).
-Customer database of a company.
-Telephone directory.
- School class attendance sheet.

An incomplete or inaccurate sampling frame causes under-coverage or over-coverage errors, which lead to biased results.

A correct sampling frame ensures that every unit of the population has a chance of being selected.

A large sample is needed when studying a large, heterogeneous population (like city income levels).

Small samples are more useful in in-depth studies, pilot studies, or controlled experiments, while large samples improve accuracy, generalizability, and normal approximation.

The Central Limit Theorem (CLT) states that for n ≥ 30, the sampling distribution of the mean tends to be normal, even if the population distribution is not.

This makes inferential statistics (Z-tests, confidence intervals, etc.) more accurate.

Source: Business Statistics (Md. Abdul Aziz)

Sample size depends on confidence level, variability, population size, and margin of error.

The “color coding of data sheets” is irrelevant to sample size determination.

Margin of error is inversely proportional to the square root of the sample size.

To cut the error in half (10% → 5%), the sample size must be approximately quadrupled. 

Source: Live MCQ class note

The main goal of determining sample size is to ensure that the findings are statistically valid, reliable, and generalizable to the population.

A sample that is too small may produce unreliable results, while an unnecessarily large sample wastes resources.