Download 05-SAMPLING TECHNIQUES

Quantitative Methods
Varsha Varde
Quantitative Methods
Sampling Techniques
Sampling
• The Process of Obtaining Information About a
Whole by Examining Only a Part
• Whole = Population
Part = Sample
• Everyday Life Concept
• Example: Physician makes diagnosis on the
basis of the findings of a small sample of blood
Auditors use sampling to draw
conclusions about large volumes of transactions
Market researchers use sample of
customers to determine market potential
Sample Inspection is done to accept
or reject a lot
?
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Why Sampling
• Population too large to be studied in full
• Sampling is Cheaper & Quicker as
compared to Census
• Necessary in destructive testing• Census not feasible-testing of medicines
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Purpose of Sampling
• To Estimate Value of a Population
Parameter (Mean, Variance, Proportions
etc.) on the basis of Value of the
Corresponding Sample Statistic.
• More Representative the Sample, More
Accurate is the Estimate.
• Bigger the Sample, Better the Estimate.
• Bigger the Sample, Greater is Cost & Time
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Terminology
 Population: Entire group of people, events, or objects of
interest in context of research
 Element: A single member of the population
 Population Frame: List of all elements in the population
from which a sample is drawn
 Example: List of all students in a college, list of all ent. events in
Mumbai in June 2010, list of all songs sung by Lata Mangeshkar
 Population Parameters: Proportion, Mean & Variance.
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Terminology
 Sample: A subset of population selected for data
collection in the research study
 Subject: A single member of the sample
 Sampling: Process of selecting sufficient number of
elements from the population
 Sampling saves time & cost of research
 Sample Statistics: Sample proportion, Sample mean
(central tendency) & sample variance (dispersion).
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Concept of Sampling Error
• Difference between the Actual Value of the
Characteristic of Population and the Value
Estimated from the Sample.
• The Art & Science of Sampling is to Apply
Appropriate Techniques to Minimize this
Risk, i.e. Minimizing Sampling Error.
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Statistical Assurance
Statistical Assurance About Minimum
Sampling Error (Risk) is Provided by Two
Parameters:
1. Precision: Quantum of Admissible Error
2. Reliability or Confidence Level: The
Probability that the Sample Estimate Will
Be In Fact Within the Stipulated Range
of Precision
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Concept of Precision
• Quantum of Admissible Error. Ideally Zero.
• Cannot be Zero Unless Sample is 100%.
• Precision Should be as small as possible
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Concept of Precision
• Precision (i.e. Error or Risk in Statistics)
Decreases as Sample Size Increases.
• But, Cost & Time of Estimation Increases
as Sample Size Increases.
• This is an Issue of Resource Allocation.
• Hence, You as Manager, Strike a Balance
and Decide Optimal Level of Precision.
• Note: Precision is Management Decision.
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Reliability or Confidence Level
• It is the Probability that the Sample Estimate
Will Be In Fact Within the Range of Precision
Set by You.
• This Prob Has to be Very High: 90%, 95%,99%.
• 100% Impossible Unless Sample is 100%.
• In Any Sampling Scenario, You Must First Set
Precision and Confidence Level.
• They will determine Required Sample Size
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Sample Size
• How Big Should Be My Sample?
• Sample Size Depends Upon the Sampling
Technique Selected for the Purpose.
• Therefore, First We Must Know About the
Various Sampling Techniques.
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Sampling Techniques
A Statistical /Probability Sample Should Be:
• Selected Objectively so that Inferences
Drawn from it are Reliable,
• Free from Personal Biases,
• Giving Equal or Known Chance of
Selection to Every Unit of the Population.
So, Sample Must Be Drawn Scientifically.
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Statistical Sampling Techniques
• Many Techniques Available.
• Selection of the Right One Depends Upon:
- Nature of the Population,
- Cost Budget,
- Time Constraint,
- Precision & Confidence Required
• Hence, Selection Falls in Your Domain.
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Simple Random Sampling
• Most Widely Used for Ease and Low Cost
• Equal Probability of Selection to All Units
in Population
• Random Number Tables (RNT) Available
• Internationally Tested for Randomness
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Random Number Table
73831
87202
27346
42658
51948
38726
67360
65465
98955
34797
90650
26365
82357
71994
04701
80200
80400
96962
16569
45336
73053
18361
04723
46016
19430
95116
35509
24437
46868
90743
67283
61384
99858
34025
51736
31247
45081
08987
38088
30791
01058
28686
61840
58777
51435
04458
64807
47978
05667
31668
06875
21213
04621
67232
92997
93798
62598
00033
19741
73503
70363
70553
20838
60008
28054
62915
64435
52836
87525
20419
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Simple Random Sampling Steps
1. Assign Sequential Numbers to All Units
2. Open Any Page of RNT. Start Anywhere
3. From This Starting Point Proceed
Vertically Downwards and Select As
Many Numbers As Required
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Random Number Table
73831
87202
27346
42658
51948
38726
67360
65465
98955
34797
90650
26365
82357
71994
04701
80200
80400
96962
16569
45336
73053
18361
04723
46016
19430
95116
35509
24437
46868
90743
67283
61384
99858
34025
51736
31247
45081
08987
38088
30791
01058
28686
61840
58777
51435
04458
64807
47978
05667
31668
06875
21213
04621
67232
92997
93798
62598
00033
19741
73503
70363
70553
20838
60008
28054
62915
64435
52836
87525
20419
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Example
• Quality Controller wishes to select a random
sample of 25 drums from the lot numbered
from 312 to 9233.
• Drums are already numbered
• Largest Number: 9233. Hence 4-digit Nos.
• Randomly select the starting point: 7383
• Hence, First Sample is Drum No. 7383
• Next No. is 6546. feasible. Accept it.
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• Next No. is 9895. Infeasible.
Discard it.
Random Sample of 25 Drums
73831
87202
27346
42658
51948
38726
67360
65465
98955
34797
90650
26365
82357
71994
04701
80200
80400
96962
16569
45336
73053
18361
04723
46016
19430
95116
35509
24437
46868
90743
67283
61384
99858
34025
51736
31247
45081
08987
38088
30791
01058
28686
61840
58777
51435
04458
64807
47978
05667
31668
06875
21213
04621
67232
92997
93798
62598
19741
70363
20838
28054
64435
87525
00033
73503
70553
60008
62915
52836
20419
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Systematic Sampling
•
•
•
•
•
Use When Pop is Already Arranged in an Order.
Example: Vouchers, Employee No., Batch No.
Variation of SRS. Faster. Speeds Up Sampling.
Does Not Use Random Number Tables.
Compute Skip Interval k = Ratio of Pop Size to
Sample Size.
• Randomly Select a Starting Number < k.
• Then Systematically Selects Every kth Number.
• Widely Used for Ease and Lower Cost.
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Example
• Internal Auditor wishes to select a sample
of 50 accounts receivable out of 520 such
accounts in a sales office.
• She opts for Systematic Sampling.
• Skip Interval k = 520 / 50 = 10.4
• Suppose Her Random no. below 10 is 7.
• Sample: Acct Nos. 7, 17, 27, 37,……, 497
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Stratified Sampling
• Example: 520 accounts receivable from 4
product divisions: Agro-Chemical (323),
Leather (54), Textile(22), Plastic (121).
• Sample of 50: 32, 5, 2 & 11 respectively
• Population Discernibly Heterogeneous
• Divide It into Several Parts (Called Strata)
• Each Stratum Homogeneous Within Itself
• Draw a Simple Random or Systematic
Sample from Each Stratum.
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Cluster Sampling
• Example: Hosiery Crates ( Each Crate Contains
Full Assortment of Sizes), Bldgs in Apt Complex
• Population Discernibly Heterogeneous
• Divide It into Several Clusters
• Each Cluster Heterogeneous Within Itself
• Draw SRS or Systematic Sample of Clusters
• Study Each Sampled Cluster Fully.
• Use When Population is Inherently Divided into
Heterogeneous Clusters.
• Convenient. Saves Cost & Time.
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Multi-Stage Sampling
• Samples are Drawn from Samples
• Example: Select 4 Out of 25 Working
Days, and Select Ten Sacks of Finished
Product from Each Selected Day’s Output
• This is 2-Stage Sampling.
• In Complex Situations, This Process Can
Go On for 3, 4 or Even More Stages.
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Determining Sample Size
Factors Influencing Sample Size:
• Precision (Your Decision)
• Confidence Level (Your Decision)
• Sampling Technique (Your Decision)
• Population Size (Known to You)
• Pop Parameter to be Estimated (KtY)
• Dispersion in Population (Known to You)
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Determining Sample Size
Effect of Factors Influencing Sample Size
• Lower Precision – Bigger Sample
• Higher Confidence – Bigger Sample
• Wider Dispersion in Pop – Bigger Sample
• Ironically, Population Size Affects Sample
Size Only Marginally
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Probability Sampling
 Example: A sample of 100 TVs to be
drawn from 10,000 TVs produced in June
2010
 Each TV has 100 ÷ 10,000 = 0.01 i.e. 1%
chance of being chosen
 Sampling Design tells researcher
precisely how to pick up 100 TVs
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1: Simple Random Sampling
Two lucky numbers to be drawn out of
100 tokens. Put all 100 tokens in a basket.
Stir well. Close eyes and pick up two
tokens
For larger population, assign serial
numbers to each element. Use a standard
table of random numbers. Select the
required number of elements one after
other
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But, enlisting large
p pulations is tedious. 31
A Case Study
HR Director of a software firm with 1926
engineers wants to find out desirability of
changing the current 10 – 6 working hours
to flexitime along with its benefits &
drawbacks perceived by the engineers
before the next board meeting
She would pick up a few engineers
randomly & ask them appropriate
questions.
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2: Systematic Sampling
A sample of 50 cars to be selected from
10,000 cars produced in 2009
10,000 ÷ 50 = 200. Select every 200th car
More precisely, select a random number
between 1 and 200, say 30. Select 30th car
Starting from 30th car, select every 200th
car: 30, 230, 430, 630, 830, 1030, 1230,
1430…
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A Case Study
Maruti Suzuki Ltd. wants to check
response of prospective buyers to the new
features introduced in its small car
segment
From the dealers alphabetical list, the
Company selects every 50th dealer &
sends a senior marketing manager to talk
to them.
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3: Stratified Random Sampling
If population contains identifiable
subgroups of elements, researcher must
provide proper representation to each
subgroup
Ex.: Population: All students of a college
Identifiable Subgroups: males / females;
arts/ science / commerce; brilliant /
average / poor
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Lata M. songs: ByVarsha
language,
solo / duet
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.3: Stratified Random Sampling
Process: Divide the population into
mutually exclusive identifiable subgroups
(strata)
Draw a simple random sample (or
systematic sample) from each stratum
Size of sample from each stratum directly
proportional to size of the stratum
Homogeneity within each stratum
Heterogeneity between strata.
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Study of Absenteeism (2%
sample)
Category (Stratum)
5 Strata
Managers
Total Number
7750
250
Sample Size
155
5
Junior Managers
500
10
Assistants
2000
40
Skilled Workers
4000
80
Unskilled Labour
1000
20
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.3: Stratified Random Sampling
Stratified random sampling involves
dividing population into strata
Hence, it needs higher time and cost
But, it provides desired precision with
smaller sample than sampling from nonstratified population
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4: Cluster Sampling
Used when population consists of several
groups of elements in such a manner that:
Groups are similar to each other and
Each group (CLUSTERS) is
heterogeneous
So, population has inter-group
homogeneity and intra-group
heterogeneity
Exactly opposite of stratified population
Process: Select aVarsha
few
clusters randomly. 39
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.4: Cluster Sampling Examples
 Complex of many identical buildings. We can select 5
out of 50 buildings
 A Mgmt Inst: 2000 students per year. 50 per batch. 40
batches run concurrently. Each has some active, some
ordinary & some passive students, and 75% boys, 25%
girls. Choose 4 batches and talk to all 200 students
without disturbing other 36 batches.
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.4: Cluster Sampling Examples
A truckload of mangoes in 4 dozen boxes.
Each box has upper layer of top quality
fruits. Quality & size drops layer by layer.
Thus, homogeneity between boxes &
heterogeneity within each box.
Draw a random or systematic sample of a
few boxes, open them and study them.
No need to open other boxes from the
truck.
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.4: Cluster Sampling
Convenient
Sample size smaller
Less time and cost
But, restrictive in application: You don’t
frequently get such populations.
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A Case Study
 Under a community health program for
tribals, it was necessary to discover their
current state of nutrition, health & beliefs
 Since adivasi padas are located at long
distances from each other in tribal areas,
a few adivasi padas were selected at
random and all residents from infants to
old ones were checked.
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