Methods of Sampling (Random, Non Random)

ఒక statistical investigationలో populationలోని ప్రతి unit నుండి information collect చేయడం ఎల్లప్పుడూ practical కాదు. అందువల్ల populationలోని కొంత భాగాన్ని ఎంపిక చేసి దాని ఆధారంగా population గురించి conclusions తీసుకుంటాము. ఈ processను Sampling – నమూనా ఎంపిక అంటారు.

Basic Idea:

Population → Select a Part → Sample → Collect Data → Draw Conclusions

Population and Sample

Population – సమగ్ర సమూహం

Statistical investigationలో మనకు interest ఉన్న మొత్తం individuals, objects లేదా units సమూహాన్ని Population అంటారు.

Sample – నమూనా

Population నుండి investigation కోసం select చేసిన కొంత భాగాన్ని Sample అంటారు.

Example:

ఒక collegeలో మొత్తం students = 2,000

Study కోసం selected students = 200

Population (N) = 2,000
Sample (n) = 200

Meaning of Sampling

Populationలోని కొంతమంది unitsను select చేసి, selected units నుండి information collect చేసే processను Sampling అంటారు.

Sampling:

Population నుండి ఒక portionను select చేసి study చేయడం.

Representative Sample

Population యొక్క important characteristicsను సాధ్యమైనంత closely represent చేసే sampleను Representative Sample అంటారు.

Important:

Good Sample ≠ Merely Large Sample

Good Sample → Population characteristicsను properly represent చేయాలి.

Methods of Sampling

Sampling methodsను broadly రెండు groupsగా classify చేయవచ్చు:

SAMPLING METHODS

↓

1. Random Sampling

2. Non-Random Sampling

1. Random Sampling – యాదృచ్ఛిక నమూనా

Populationలోని ప్రతి memberకు sampleలో select అయ్యే chance ఉండే విధంగా sample select చేస్తే దానిని Random Sample అంటారు.

Random Sample:

Every member of the population is assigned some chance of being selected.

Exam Memory:

Random Sampling → Selection by Chance
Non-Random Sampling → Selection is not based purely on chance

Random Sampling Techniques

RANDOM SAMPLING

↓

Simple Random Sampling

Systematic Random Sampling

Stratified Random Sampling

Cluster Sampling

A. Simple Random Sampling

Population నుండి ఒక fixed size sampleను select చేసేటప్పుడు, అదే size ఉన్న ప్రతి possible sampleకు select అయ్యే same probability ఉంటే దానిని Simple Random Sampling అంటారు.

Example:

50 studentsలో 5 studentsను select చేయాలి.

50 students namesను slipsపై రాసి, వాటిని mix చేసి, randomగా 5 slips draw చేస్తే అది Simple Random Sample.

Common Methods:

Lottery Method
Random Numbers
Computer-generated Random Selection

B. Systematic Random Sampling

Population unitsను ఒక ordered listలో arrange చేసి, ఒక random starting pointను select చేసిన తరువాత every kth unitను sampleలో select చేసే methodను Systematic Random Sampling అంటారు.

Sampling Interval (k) = N ÷ n

ఇక్కడ:
N = Population Size
n = Required Sample Size
k = Sampling Interval

Find k = N ÷ n

↓

Choose Random Start between 1 and k

↓

Add k repeatedly

↓

Obtain Required Sample

Numerical Problem 1 – Systematic Sampling

Problem:

A population contains 2,000 persons. A systematic sample of 200 persons is required. Find the sampling interval.

k = N ÷ n

N = 2,000
n = 200

k = 2,000 ÷ 200 = 10

Answer: Sampling Interval = 10

అంటే ప్రతి 10th unitను select చేయాలి.

⚡ Shortcut:

Systematic Sampling questionలో Population + Sample Size ఇస్తే వెంటనే:

k = N/n

2000/200 → zeros cancel → 10

Numerical Problem 2 – Identify Selected Units

Problem:

N = 1,000, n = 100. Random starting number is 6. Find the first five selected units.

First find sampling interval:

k = 1000 ÷ 100 = 10

Starting number = 6

Selected units:

6, 16, 26, 36, 46, ...

Answer: 6, 16, 26, 36, 46

⚡ Shortcut:

Start = r
Interval = k

Sample units:
r, r+k, r+2k, r+3k, ...

Here:
6 → +10 → 16 → +10 → 26 → +10 → 36 → +10 → 46

Numerical Problem 3 – Textbook Pattern

Problem:

From 45,000 households, a sample of 150 households is required. Find the systematic sampling interval.

k = 45,000 ÷ 150 = 300

Answer: k = 300

One unit is randomly selected among the first 300 units and thereafter the corresponding unit from each successive interval of 300 is selected.

C. Stratified Random Sampling

Population heterogeneousగా ఉండి, అందులో distinct groups ఉన్నప్పుడు populationను homogeneous subgroups లేదా Strataగా divide చేసి, ప్రతి stratum నుండి random sample select చేసే methodను Stratified Random Sampling అంటారు.

Population

↓

Divide into Strata

↓

Take Random Sample from EACH Stratum

↓

Combine Selected Units

Example:

College Students
↓
UG Students | PG Students
↓
Random sample from UG + Random sample from PG

Exam Key:

Stratified Sampling → Sample is taken from EACH STRATUM.

Proportionate Stratified Sampling

Populationలో ప్రతి stratum ఎంత proportionలో ఉందో అదే proportionను sampleలో maintain చేయాలంటే:

Sample from a Stratum = (Stratum Size ÷ Population Size) × Total Sample Size

Numerical Problem 4 – Stratified Sampling

Problem:

A college has 1,000 students:

UG = 700 students
PG = 300 students

A proportionate stratified sample of 200 students is required. Find the number selected from each group.

UG:

(700 ÷ 1000) × 200 = 140

PG:

(300 ÷ 1000) × 200 = 60

Answer:

UG = 140 students
PG = 60 students

Total = 200 students

⚡ Shortcut – Percentage Method:

700 out of 1000 = 70%
300 out of 1000 = 30%

Sample = 200

70% of 200 = 140
30% of 200 = 60

When strata are given as easy percentages, percentage method is faster.

Numerical Problem 5 – Three Strata

Problem:

A population of 2,000 employees consists of:

Department A = 1,000
Department B = 600
Department C = 400

A proportionate sample of 200 is required. Find the sample from each department.

A = (1000/2000) × 200 = 100

B = (600/2000) × 200 = 60

C = (400/2000) × 200 = 40

Answer:

A = 100
B = 60
C = 40

⚡ Ratio Shortcut:

1000 : 600 : 400
= 5 : 3 : 2

Total ratio = 10
Sample = 200

One ratio part = 200/10 = 20

A = 5 × 20 = 100
B = 3 × 20 = 60
C = 2 × 20 = 40

D. Cluster Sampling

Populationను groups లేదా clustersగా divide చేసి, one or a few clustersను sampleగా select చేసే techniqueను Cluster Sampling అంటారు.

Cluster సాధారణంగా populationను represent చేసే geographical లేదా other natural grouping కావచ్చు.

Example:

ఒక districtలో 100 schools ఉన్నాయి.

Individual studentsను మొత్తం district నుండి select చేయకుండా, కొన్ని schoolsను clustersగా select చేసి study చేయడం → Cluster Sampling.

Stratified vs Cluster Sampling

Basis	Stratified Sampling	Cluster Sampling
Population Division	Divided into Strata	Divided into Clusters
Selection	Sample from each stratum	One or a few clusters selected
Key Word	EACH group	FEW groups
Example	Students selected from every academic category	Few schools selected from a district

⚡ Most Important Identification Shortcut:

Stratified = SOME from ALL groups

Cluster = ALL / units from SOME groups

Exam questionలో “from each group” కనిపిస్తే → Think Stratified
“few groups/clusters are selected” కనిపిస్తే → Think Cluster

2. Non-Random Sampling – యాదృచ్ఛికేతర నమూనా

Sample selection పూర్తిగా random chance ఆధారంగా జరగకుండా accessibility, investigator judgment లేదా predetermined quota వంటి factors ఆధారంగా జరిగితే దానిని Non-Random Sampling అంటారు.

NON-RANDOM SAMPLING

↓

Convenience Sampling

Judgment Sampling

Quota Sampling

A. Convenience Sampling

Populationలో investigatorకు అత్యంత easily accessibleగా ఉన్న membersను sampleగా select చేస్తే దానిని Convenience Sampling అంటారు.

Example:

ఒక professor మొత్తం university students GPA తెలుసుకోవడానికి తన statistics classలోని studentsనే sampleగా తీసుకోవడం → Convenience Sample.

⚡ Keyword Shortcut:

Convenience → Easy / Available / Nearby / Accessible

B. Judgment Sampling

Expert తన prior knowledge మరియు judgment ఆధారంగా populationలోని specific unitsను select చేస్తే దానిని Judgment Sampling అంటారు.

Example:

Agricultural expert తన experience ఆధారంగా కొన్ని representative farmsను study కోసం select చేయడం.

⚡ Keyword Shortcut:

Judgment → Expert Choice

C. Quota Sampling

Populationలోని different groupsకు predetermined quotas fix చేసి, ఆ quota ప్రకారం unitsను select చేసే non-random methodను Quota Sampling అంటారు.

Example:

Populationలో Men = 60%, Women = 40% ఉంటే 100-person surveyలో 60 men మరియు 40 women అనే quota fix చేయడం.

⚡ Keyword Shortcut:

Quota → Fixed Number / Fixed Proportion

Random vs Non-Random Sampling

Random Sampling	Non-Random Sampling
Selection involves chance	Selection is not purely based on chance
Designed to reduce selection bias	Greater possibility of selection bias
Simple, Systematic, Stratified, Cluster	Convenience, Judgment, Quota
Useful for statistical inference when properly designed	Generalisation may be more limited

Sampling Error

Sample survey నుండి obtained result మరియు entire populationను census చేసి ఉంటే obtained result మధ్య ఉండే differenceను Sampling Error అంటారు.

Sampling Error = Sample Statistic − Population Parameter

Mean విషయంలో:

Sampling Error = x̄ − μ

Important:

Sampling Error → Chance వల్ల వస్తుంది.

Sampling Error → Sample Surveyలో మాత్రమే ఉంటుంది.

Censusలో Sampling Error ఉండదు.

Numerical Problem 6 – Sampling Error

Problem:

Population values are:
70, 78, 80, 80, 95

A sample contains:
70, 80, 95

Find the sampling error of the mean.

Step 1: Population Mean

μ = (70 + 78 + 80 + 80 + 95) ÷ 5 = 403 ÷ 5 = 80.60

Step 2: Sample Mean

x̄ = (70 + 80 + 95) ÷ 3 = 245 ÷ 3 = 81.67

Step 3: Sampling Error

Sampling Error = 81.67 − 80.60 = +1.07

Answer: +1.07

Sample mean population mean కంటే 1.07 ఎక్కువ.

⚡ Shortcut:

Mean questionలో:

Sampling Error = x̄ − μ

Sample Mean minus Population Mean.

Positive → Sample overestimates population mean
Negative → Sample underestimates population mean

Non-Sampling Errors

Data collection, recording లేదా tabulationలో human mistakes వల్ల వచ్చే errorsను Non-Sampling Errors అంటారు.

Examples:

Selection Error
Non-response Error
Response Error
Voluntary Response Error
Recording / Coding Errors

Very Important Exam Difference:

Sampling Error → Sample Survey only

Non-Sampling Error → Sample Survey + Census రెండింటిలోనూ రావచ్చు

Numerical Problem 7 – Sampling vs Non-Sampling Error

Population mean = 80.60
Correct sample mean = 81.67

Suppose a recording mistake changes a value and the calculated sample mean becomes 82.33.

Find:
1. Total difference from population mean
2. Sampling error
3. Non-sampling component

Total Difference:

82.33 − 80.60 = 1.73

Sampling Error:

81.67 − 80.60 = 1.07

Non-Sampling Component:

1.73 − 1.07 = 0.66

Answer:

Total Difference = 1.73
Sampling Error = 1.07
Non-Sampling Error Component = 0.66

Numerical Shortcuts – Sampling

Question Type	Shortcut / Formula
Systematic Sampling Interval	k = N/n
Systematic Selected Units	r, r+k, r+2k, r+3k...
Proportionate Stratified Sample	nh = (Nh/N) × n
Easy Stratified Problem	Convert group size into % and apply same % to sample
Ratio-based Strata	Simplify ratio → divide sample by total ratio → multiply
Sampling Error of Mean	x̄ − μ
Stratified Identification	Some units from every group
Cluster Identification	Some groups/clusters selected
Convenience Identification	Easily available units
Judgment Identification	Expert chooses
Quota Identification	Fixed quota/proportion

Exam MCQs – Sampling Methods

1. In random sampling:

A) No member has a chance of selection
B) Members are selected only by investigator's judgment
C) Each population member has some chance of selection
D) Only convenient members are selected

Answer: C

2. In Simple Random Sampling:

A) Every same-size sample has the same chance of selection
B) Only experts select units
C) Only one stratum is studied
D) Convenient units are selected

Answer: A

3. Selecting every kth unit is known as:

A) Quota Sampling
B) Systematic Random Sampling
C) Judgment Sampling
D) Convenience Sampling

Answer: B

4. Sampling interval is:

A) n/N
B) N × n
C) N/n
D) N − n

Answer: C – N/n

5. A sample is selected from every stratum. This is:

A) Cluster Sampling
B) Stratified Sampling
C) Convenience Sampling
D) Judgment Sampling

Answer: B

6. A few geographical groups are selected from a population. This indicates:

A) Cluster Sampling
B) Stratified Sampling
C) Quota Sampling
D) Judgment Sampling

Answer: A

7. Selection of the most accessible members is:

A) Simple Random Sampling
B) Convenience Sampling
C) Stratified Sampling
D) Cluster Sampling

Answer: B

8. Selection based on expert knowledge is:

A) Judgment Sampling
B) Systematic Sampling
C) Cluster Sampling
D) Simple Random Sampling

Answer: A

9. Fixed representation of population groups is associated with:

A) Quota Sampling
B) Systematic Sampling
C) Cluster Sampling
D) Census

Answer: A

10. Sampling error occurs:

A) Only in Census
B) Only in Sample Survey
C) Neither in Census nor Sample Survey
D) Only during tabulation

Answer: B

Numerical MCQs

11. If N = 5,000 and n = 250, the systematic sampling interval is:

A) 10
B) 20
C) 25
D) 50

Shortcut: 5000 ÷ 250 = 20

Answer: B – 20

12. If k = 20 and the random start is 7, which is the correct sequence?

A) 7, 17, 27, 37
B) 7, 27, 47, 67
C) 20, 40, 60, 80
D) 7, 14, 21, 28

Shortcut: Keep adding k = 20.

Answer: B – 7, 27, 47, 67

13. A population contains 60% rural and 40% urban people. In a proportionate sample of 500, how many should be rural?

A) 200
B) 250
C) 300
D) 350

Shortcut: 60% of 500 = 3/5 × 500 = 300

Answer: C – 300

14. Population mean is 50 and sample mean is 52.5. Sampling error is:

A) −2.5
B) +2.5
C) 50
D) 102.5

Shortcut: x̄ − μ = 52.5 − 50 = +2.5

Answer: B – +2.5

Identify the Sampling Method – Fast Practice

Situation	Answer	Clue
Names drawn by lottery	Simple Random	Equal random chance
Every 20th customer	Systematic	Every kth unit
Sample taken from each income group	Stratified	Each stratum
Five villages selected for study	Cluster	Groups selected
Students available in one classroom selected	Convenience	Easily accessible
Experts choose representative firms	Judgment	Expert judgment
60 men and 40 women fixed in advance	Quota	Predetermined quota

One-Minute Master Table

Concept	Remember
Population	Entire group
Sample	Part of population
Random Sample	Selection involves chance
Simple Random	Every same-size sample has same chance
Systematic	Every kth unit
Sampling Interval	k = N/n
Stratified	Random sample from each stratum
Cluster	One or a few clusters selected
Convenience	Most accessible members
Judgment	Expert judgment
Quota	Fixed group representation
Sampling Error	Sample result − Population result
Sampling Error occurs in	Sample Survey only
Non-Sampling Error occurs in	Census + Sample Survey

చివరి నిమిషం పునశ్చరణ

Sampling
→ Populationలో కొంత భాగాన్ని study చేయడం

Random Sampling:
→ Selection involves chance

Random Techniques:
→ Simple Random
→ Systematic Random
→ Stratified Random
→ Cluster

Non-Random Techniques:
→ Convenience
→ Judgment
→ Quota

Simple Random
→ Every same-size sample has equal chance

Systematic
→ Every kth unit
→ k = N/n

Stratified
→ Sample from EACH stratum

Cluster
→ One or a few clusters selected

Convenience
→ Easily accessible units

Judgment
→ Expert choice

Quota
→ Fixed quota / proportion

Proportionate Stratified Formula:
n_h = (N_h/N) × n

Sampling Error:
x̄ − μ

Golden Shortcut:

Every kth → Systematic
Every group → Stratified
Few groups → Cluster
Easy availability → Convenience
Expert choice → Judgment
Fixed target → Quota

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