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Stratified sampling technique.


Stratified sampling is a method of sampling from a population. In statistical surveys, when subpopulation within an overall population vary, it is advantageous to sample each subpopulation (stratum) independently. Stratification is the process of dividing members of the population into homogeneous subgroups before sampling. The strata should be mutually exclusive: every element in the population must be assigned to only one stratum. The strata should also be collectively exhaustive: no population element can be excluded. Then simple random sampling or systematic sampling is applied within each stratum. This often improves the Representative of the sample by reducing sampling error. It can produce a weighted mean that has less variability than the arithmetic mean of a simple random sample of the population. – PowerPoint PPT presentation

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Title: Stratified sampling technique.

  • By -Shashank kshandakar
    M.V.Sc(1st year)
    of lesit

  • Definitions and concepts in sampling
  • Principles of sampling
  • Advantages of sampling
  • Types of sampling
  • Stratified random sampling its properties

Important definitions
  • Population population is an aggregates of
    object under study.
  • Sample- A finite sub-set of statistical
    individual in a population is called a sample.
  • Sampling Unit Sets of units considered for
    selection in some stage of sampling.
  • Sampling scheme-Method of selecting sampling
    units from population

Important definitions.
  • Census-The process of complete enumeration of
    all the elements or units in the population
  • Sample size-The number of elements or units in a
    sample is known as sample size.
  • Accuracy-The amount of deviation of the
    estimate from the true value.
  • Precision-The deviation of the estimate from the
    average value. It is reciprocal of the variance

Sampling Frame
  • Sampling Frame- A complete list, map or other
    acceptable materials which serve as a guide to
    the population to be covered is known as frame.
  • A sampling frame which has the property that we
    can identify every single element and include any
    in our sample
  • The sampling frame must be representative of the
  • A sampling frame must be up-to-date.

Important definitions
  • Sampling fraction- Ratio of sample size(n) to
    the population size(n) i.e. (n/N)
  • Finite population correction-1- (Sampling
  • Sample space-A space consisting of all possible
    sample is called sample space
  • Relation between cost (per unit of sample
    selection) precision

Important definitions
  • Sampling distribution-The aggregates of the
    various value of the statistic under
    consideration so obtained(one from each sample)
    may be grouped into a frequency distribution
    which is known as sampling distribution of the
  • Standard error-The standard deviation of the
    sampling distribution of a statistic is known as
    standard error.

Important definitions
  • Parameter-A descriptive measure computed from
    the data of a population is called as parameter.
  • Statistic-A descriptive measure computed from a
    sample data is called as statistic.
  • Estimator-An estimator is a rule, function or
    formula of variates for estimating the population
  • Estimates -A particular value of an estimator
    from a fixed set of value of a random sample is
    known as estimate.

It is very easy and convenient to draw the
sample from homogenous population
The population having significant variations
(Heterogeneous), observation of multiple
individual needed to find all possible
characteristics that may exist
When sampling is not necessary?
  • When population is very small.
  • When we have extensive resources.
  • When we dont expect a very high response.

  • Sampling is the process of selecting units or
    elements from the study population in such a way
    that the units or elements selected represent the
    whole population

Principles of Sampling Theory
  • Principle of Statistical Regularities.
  • Principle of Inertia of large number.
  • Principle of Validity.
  • Principle of Optimization.

Principal Steps in a Sample Survey
  • Statement of objectives.
  • Definition of population to be studied.
  • Determination of sampling frame and sampling
  • Selection of proper sampling design.
  • Organization of field work.
  • Summary and analysis of data.

The Sampling Process
Plan procedure for selecting sampling units
Determine if a probability or non-probability
sampling method will be chosen
Determine sample size
Select actual sampling units
Select a Sampling Frame
Define the Target population
Conduct fieldwork
Advantage of sampling
  • 1. Reduced cost of survey.
  • 2. Greater speed of getting results.
  • 3. Greater accuracy of results.
  • 4. Greater scope.(When its impossible to study
    the whole population)
  • 5. Adaptability.

Type of sampling method
  • Probability sampling
  • Non-probability sampling.

Probability sampling
  • The term probability sampling is used when the
    selection of the sample is purely based
    on chance. 
  • Every unit of the population has known nonzero
    probability of being selected for the sample. The
    probability of selection may be equal or unequal
    but it should be non-zero and should be
    known. The probability sampling is also called
    the random sampling

Types Probability sampling
  • Probability sampling includes
  • Simple Random Sampling,
  • Systematic Sampling,
  • Stratified Random Sampling,
  • Cluster Sampling
  • Multistage Sampling.
  • Multiphase sampling

Simple random sampling
  • Applicable when population is small, homogeneous
    readily available but there is no guarantee
    that all segment of population will be
    represented in SRS
  • All unit within the frame have an equal
  • It provides for greatest number of possible
  • Simplest common method of sampling
  • Every sample are drawn unit by unit with equal
    probability of selection on the basis of sample
    drawn SRS may be divide into two group

Simple random sampling with replacement
  • Simple random sampling with replacement
  • If a unit is selected noted then
    returned back to the population before the next
    selection is made this procedure is repeated n
    times is known as Simple random sampling with
  • Number of sample required Z2a pq
  • Var(yn)

Simple random sampling without replacement
  • If a unit is selected noted not returned back
    to the population (selected unit is not available
    for further selection) this procedure is known as
    Simple random sampling with replacement
  • Number of sample requiredn0/1( n0-1)/N
  • Var(yn)

Advantage of SRS
  • Minimal knowledge of population needed
  • Economical required less time.
  • Easy to analyze data
  • Disadvantage of SRS
  • High cost low frequency of use
  • Requires sampling frame
  • Does not use researchers expertise
  • Larger risk of random error than stratified

  • Determine the size of the target population.
  • Decide on the margin of error. As much as
    possible the margin of error should not be higher
    than 5.
  • Use the formula
  • n N
  • 1 Ne2 (Pagoso , et
  • n sample size
  • N the size of the population
  • e the margin of error
  • Compute the sample proportion by dividing the
    result in number 3 by the population.

  • Population is 5,346
  • Margin of error is 3
  • Using the formula
  • n ___5,346_
  • 1 5346(.03)2
  • n 920
  • Sample proportion () 920 / 5346
  • 17

Procedure of selection of a random sample
  • Lottery method
  • Use of random number table
  • Tippets random number table
  • Fisher Yates random number table
  • Kendall smith random number table
  • A million random digit number table.

Test for randomness of selected random number
  • Frequency test
  • Serial test
  • Gap test
  • Poker test.

Stratified random sampling
  • Most commonly used sampling technique
  • Stratum ( - Clear division into which
    some thing is separated
  • The population (N) unit is divided into K group
    or sub-population called strata
  • The sample is drawn by first dividing the
    population into homogeneous sub-populations
    (strata) and then drawing samples from each of
    the sub-population (stratum).
  • It ensures proportionate representation of
    character under study when drawing a sample from
    a heterogeneous population and hence increases
    the quantity of information for a given cost.

How to Stratify and How Many Strata?
  • Strata should be homogeneous to increase
  • Usually 5-10 Strata are taken. If too many strata
    then the sample size within strata is too low.

Principle of stratification
  • Strata should be non-overlapping should
    together comprise the whole population.
  • Strata are homogenous themselves with respect to
    characters under study.
  • Population are divided into strata on the basis
    of sex , age, purpose ( milch, drought, dual )
    breed, land holding capacity of livestock owners,
    or any auxiliary information.

Auxiliary information
  • Past data or some other information related
    to the character on which we divide the
    population into strata such that
  • Within strata -Variable are homogenous.

    Between strata -Variable
    are heterogeneous.

Sampling technique Use of auxiliary information
Stratified sampling technique Construction of strata
PPS Sample selection, to get more efficient estimator of the population parameter
Ratio regression estimation For estimation purpose.
Advantage of stratification
  • Administrative convenience
  • Improving sampling design
  • Stratification makes it possible to use different
    sampling design in different strata
  • Provide better cross section of population i.e.
    adequate representation from various group of the
  • Gain in precision in the estimation of population

Main problem in stratification
  • What should be the number of strata?
  • How to allocate n sample from different strata
  • How we distribute population into various strata
  • How to determine the boundary of the strata.

Allocation of sample size in different strata
  • Allocation of the sample to different strata is
    done by considering of 3 factors
  • Total no. of unit in the strata
  • Variability within the stratum
  • The cost of taking observation per sampling unit
    in the stratum
  • (N.B-a good allocation is one where maximum
    precision is obtained with minimum recourses)

Method of allocation
  • Equal allocation
  • Proportional allocation
  • Neyman allocation
  • Optimum allocation.

Equal allocation
  • Most convenience
  • Equal no. of sample are drawn from each stratum.
  • Not usually sensible unless all strata are equal
    size in terms of overall estimates precision.
    However, maybe good if you want to compare
    stratum means as your primary focus

Proportional allocation
  • Given by Bowley (1926)
  • Very common in practice
  • Sampling fraction of each stratum is same

Proportional allocation
Allocate proportional to the size of the
strata-very widely used
Neyman allocation
  • Also known as minimum variance allocation
  • 1st discovered by Tschuprow (1923), but
    rediscovered by J.Neyman (1934)
    (Assumption sampling cost per unit
    among different strata is same size of the
    sample is fixed)
  • Allocation of sample among different strata is
    based on joint consideration of stratum size
    stratum variation

Optimum allocation
  • Optimum allocation is mean to choose sample so as
  • Minimise the variance(maximise the precision) of
    the estimate for fixed sample size(n).
  • Minimise the variance(maximise the precision) of
    the estimate at fixed cost.
  • Minimise the total cost for fixed desired

Cluster Sample
  • A Cluster Sample is obtained by first grouping
    the elements of the population into clusters and
    then simple random sampling or other type of
    sampling is used to select the clusters.
  • This type of sampling is used when a sampling
    frame cannot be prepared for individual units in
    the population but can be prepared for some
    cluster of them or when substantial time or
    expense can be saved by collecting data from a
    modest number of clusters.

  • When drawing a cluster sample, the first task is
    to specify appropriate clusters. In doing so,
    consideration has to be made about the level of
    heterogeneity of elements within clusters.
  • If clusters are generally heterogeneous, then
    few large clusters may be selected to constitute
    the sample but if they are homogeneous, then many
    small-sized clusters should be used.

Difference between strata cluster
  • Although strata and clusters are both
    non-overlapping subsets of the population, they
    differ in several ways.
  • All strata are represented in the sample but
    only a subset of clusters are in the sample.
  • With stratified sampling, the best survey results
    occur when elements within strata are internally
    homogeneous. However, with cluster sampling, the
    best results occur when elements within clusters
    are internally heterogeneous.

Difference Between Cluster and Stratified Sampling
Population of L strata, stratum l contains nl
Population of C clusters
Take simple random sample in every stratum
Take srs of clusters, sample every unit in chosen
  • Sampling frame of entire population has to be
    prepared separately for each stratum
  • When examining multiple criteria, stratifying
    variables may be related to some, but not to
    others, further complicating the design, and
    potentially reducing the utility of the strata.
  • In some cases (such as designs with a large
    number of strata, or those with a specified
    minimum sample size per group), stratified
    sampling can potentially require a larger sample
    than would other methods

  • Stratification is sometimes introduced after the
    sampling phase in a process called
  • This approach is typically implemented due to a
    lack of prior knowledge of an appropriate
    stratifying variable or when the experimenter
    lacks the necessary information to create a
    stratifying variable during the sampling phase.
    Although the method is susceptible to the
    pitfalls of post hoc approaches, it can provide
    several benefits in the right situation.
    Implementation usually follows a simple random
    sample. In addition to allowing for
    stratification on an ancillary variable,
    poststratification can be used to implement
    weighting, which can improve the precision of a
    sample's estimates.

  • Choice-based sampling is one of the stratified
    sampling strategies. In this, data are stratified
    on the target and a sample is taken from each
    strata so that the rare target class will be more
    represented in the sample. The model is then
    built on this biased sample. The effects of the
    input variables on the target are often estimated
    with more precision with the choice-based sample
    even when a smaller overall sample size is taken,
    compared to a random sample. The results usually
    must be adjusted to correct for the oversampling

  • Cochran, W. G., 1977. Sampling Techniques, Third
    Edition. New York John Wiley SonsCochran, W.
    G., 1977. Sampling Techniques, Third Edition. New
    York John Wiley Sons
  • Des Raj and Chandhok, P. (1998). Sampling Survey
    Theory. Narosa Publishing House, New Delhi
  • Horvitz, D. G., and D.J. Thompson, 1952. A
    generalization of sampling without replacement
    from a finite universe. The Journal of the
    American Statistical Association 47663-685.
  • Murthy, M.N. (1977). Sampling Theory and Methods.
    Statistical Publishing Society, Calcutta.
  • Sukhatme, P.V., Sukhatme, B.V., Sukhatme, S and
    Ashok, C. (1984). Sampling Theory of Surveys with
    Applications. Indian Society of Agricultural
    Statistics, New Delhi

Thank you