Stratified Random

UNIT 3 STRATIFIED RANDOM Sampling

SAMPLING
Structure
3.1 Introduction
Objectives
3.2 Principles of Stratification
Notations and Terminology
3.3 Properties of Stratified Random Sampling
3.4 Mean and Variance for Proportions
3.5 Allocation of Sample Size
Equal Number of Units from Each Stratum
Proportional Allocation
Neyman’s Allocation
Optimum Allocation
3.6 Stratified Sampling versus Simple Random Sampling
Proportional Allocation Versus Simple Random Sampling
Neyman’s Allocation Versus Proportional Allocation
Neyman’s Allocation Versus Simple Random Sampling
Merits and Demerits of Stratified Random Sampling
3.7 Summary
3.8 Solutions/Answers

3.1 INTRODUCTION
When the units of the population are scattered and not completely
homogeneous in nature, then simple random sample does not give proper
representation of the population. So if the population is heterogeneous the
simple random sampling is not found suitable. In simple random sampling the
variance of the sample mean is proportional to the variability of the sampling
units in the population. So, in spite of increasing the sample size n or
sampling fraction n/N, the only other way of increasing the precision is to
device a sampling which will effectively reduce the variability of the sample
units, the population heterogeneity. One such method is stratified sampling
method.

In stratified sampling the whole population is to be divided in some

homogeneous groups or classes with respect to the characteristic under study
which are known as strata. That means, we have to do the stratification of the
population. Stratification means division into layers. The auxiliary
information related to the character under study may be used to divide the
population into various groups or strata in such a way that units within each
stratum are as homogeneous as possible and the strata are as widely different
as possible.

Thus, all strata would comprise the population. Then from each stratum
sample would be drawn and lastly all samples would be combined to get the
ultimate sample. For example, let us consider that population consists of N
units and these are distributed in a heterogeneous structure. Now first of all
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Statistical Techniques
we divide the population into ‘k’ non overlapping strata of sizes N1, N2, N3,
..., Nk such that each stratum becomes homogeneous. Evidently N = N1 + N2 +
N3 + ... + Nk. Then from first stratum a sample of size n1 would be drawn by
simple random sampling method. Similarly, from the second stratum a sample
of n2 units would be drawn and so on, up to kth stratum. Now all these k
samples would be combined to get the ultimate sample. So, the ultimate size
of sample would be n  n1  n 2  n 3  ...  n k . This method of sampling is
known as Stratified random sampling because here stratification is done first
to make population homogeneous and then samples are drawn randomly by
simple random sampling from each stratum.
The principles of stratification are explained in Section 3.2. The properties of
stratified random sampling are described in Section 3.3, whereas Section 3.4
provides the derivation of the mean and variance of proportions in stratified
random sampling. The allocation of sample size with the help of different
techniques is described in Section 3.5. The comparative study between
stratified random sampling and simple random sampling is given in Section
3.6.
Objectives
After studying this unit, you would be able to
 define the stratified random sampling;
 explain the principles of stratification;
 describe the properties of stratified random sampling;
 derive the mean and variance of proportions in stratified random
sampling;
 describe the allocation of sample size with the help of different
techniques; and
 calculate the estimate of population mean and variance of sample mean.

3.2 PRINCIPLES OF STRATIFICATION

The principles to be kept in mind while stratifying a population are given

below:

1. The strata should not be overlapping and should together comprise the
whole population.

2. The strata should be homogeneous within themselves and heterogeneous

between themselves with respect to characteristic under study.

3. If a investigator is facing difficulties in stratifying a population with

respect to the characteristic under study, then he/she has to consider the
administrative convenience as the basis for stratification.

4. If the limit of precision is given for certain sub-population then it should

be treated as stratum.

46
3.2.1 Notations and Terminology Stratified Random
Sampling
N = Population size
n = Sample size
k = Number of strata
Ni = Size of ith stratum
k
Then N   N i
i 1

ni = Size of sample drawn from i th stratum

k
Then n   n i
i1

X ij = Value of character under study for jth unit of ith stratum

Ni
1
X i  Population mean of i th stratum  X ij
Ni j1

1 k Ni
X  Population Mean    X ij
N i1 j1
k k
1
  i i 
N i1
N X 
i 1
Wi X i

Ni
where, Wi  is called the weight of i th stratum
N
S i2 = Population mean square of the i th stratum
Ni
1
 X  ,  j  1, 2, ... , N
2
 ij  Xi i & i = 1, 2, . .., k 
Ni 1 j1

x ij = Value of jthsample unit taken from ith stratum

ni
1
xi = x ij = Mean of sample units selected from ith stratum
ni j1

ni
1
s i2 
n i 1
 x
j1
ij  xi  ,
2
i  1, 2, ..., k 

= Sample mean square of the i th stratum

Let us consider the following sample means to estimate the populations mean
of ith stratum X which are:
1 k
xn   n i xi
n i 1
1 k k
and x st   Ni xi 
N i 1
W x
i1
i i

where, x st is the weighted mean of the strata sample means, weights being
equal to strata sizes. These two will be identical if n i  Ni
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Statistical Techniques
3.3 PROPERTIES OF STRATIFIED RANDOM
SAMPLING

Theorem 1: x st is an unbiased estimate of the population mean X i.e.

E x st   X
Proof: We have
1 k
x st   Ni xi
N i 1
Therefore,
1 k 
E x st   E   N i x i 
 N i1 
1 k
 N i E x i 
N i1
Since the sample units selected from each of stratum are simple random
sample, then we have
E x i   Xi
Therefore,
1 k
E x st    N i Xi
N i1
Ni
1 k 1
= 
N i1
Ni
Ni
X
j1
ij

1 k Ni
=  X ij  X
N i1 j1

Hence proved
Theorem 2: Prove that
k k
1 Si 1 1  2 2
Var x st    N N  n  n    n   Wi Si
N i 
i i i
N2 i1 i i1  i

Proof: We have
 k 
Var x st   Var   Wi x i 
 i 1 
k
 W i
2
Var x i 
i1

The covariance term vanish since the samples from different strata are
independent and the sample units in each stratum are the simple random
sample without replacement, we have

48
 1 1  2 Stratified Random
Var x     S Sampling
n N 
 
1 1  2
or Var ( x i ) =    Si
 n i Ni 

Therefore,
k
  2
Var x st    Wi2  1  1  Si
i 1 n N  i i 

1 k
Si2

N2
 N i N i  n i 
i 1 ni

From the above result the variance depends on Si2 the heterogeneity within the
strata. Thus, if Si2 are small i.e. strata are homogeneous then stratified
sampling schemes provides estimates with greater precision.

Theorem 3: If Si2 is not known then prove that estimate of the variance of the
sample mean of the stratified random sample is given by

k
1 1 
E Var x st       Wi2Si2
i 1  n i Ni 

Proof: In general Si2are not known. A simple random sample is drawn from
each stratum. If we assume a individual stratum as a population then the
sample, drawn from it, would be a simple random sample. If the sample is
drawn from ith stratum, the sample mean square si2would be an estimate of
population mean square Si2

 
i. e. E s i2  Si2 i  1, 2, ..., k … (1)
Accordingly an unbiased estimate of the variance is given by

k
1 1  2 2
Var x st       Wi s i
i 1  n i N i 

Therefore,
 n 1 1  
EVar x st    E     Wi2 s i2 
 i1  n i N i  

k
1 1 
     Wi2 E  s i2 
i 1  n i Ni 
Substituting from equation (1), we get

k
1 1 
EVar x st       Wi2 Si2
i1  n i Ni 

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Statistical Techniques
3.4 MEAN AND VARIANCE FOR PROPORTIONS
As in simple random sampling, we can divide a population into two classes
with respect to a attribute. Hence the units in the population are classified in
these two classes accordingly as it possesses or does not possess the given
attribute. After taking a sample of size n, we may be interested in estimating
the population proportion of the defined attribute.
If a unit possesses the attribute, it receives the code value 1 and if an unit does
not possesses the attribute, it receives the value 0. Let the number of units
belonging to A in the ith stratum of size Ni be Mi and if the sample of size ni
taken from ith stratum, the number of units belonging to A be mi. Denoting
the proportion of units belonging to A in the population, in the ithstratum and
sample from the ith stratum by , i and pi respectively, various formula for
mean and variance are as follows:

Mi m
πi and pi  i
Ni ni
k k
N
and π   i π i   Wi π i
i 1 N i 1
for i =1, 2,…, k
The estimated proportion pst under stratified sampling for the units belonging
to A is
k
p st   Wi p i
i 1

 k  k
Mean p st   E   Wi p i    Wi E p i 
 i1  i 1
since we draw SRS from each stratum so by Theorem 10 of Section 2.4 of
Unit 2 we have
E p i   π i

By putting this value in above formula, we have

k
Mean p st    Wi π i  π
i 1

Hence sample proportion under stratified sampling is unbiased estimate for

population proportion.
Now variance of pst is given by
k
 k  1
Var p st   Var   Wi p i   2
N
 N Vp 
2
i i
 i1  i 1

since we draw SRS from each stratum so by Theorem 11 of Section 2.4 of

Unit 2 we have

50
 N i  n i π i 1  πi  Stratified Random
Var pi   . Sampling
Ni  1 ni

By putting this value in above formula, we have

N i N i  n i  π i 1  π i 
k 2
1
Var p st   
N2 i 1 N i  1 n i
If Ni is large enough, consider 1/ Ni as negligible and Ni-1~Ni, formula for
Var (pst ) reduces to
1 k  1  i 
Var pst   2 
N i N i  n i  i
N i 1 ni

and if ni / Ni is negligible, therefore

k
i 1  i 
Var pst    Wi2
i 1 ni

An unbiased estimate of Var (pst ) is given by

k
1 Ni  n i
E Var p st    N n i pi qi
N2 i 1 i  1

where, qi = 1− p i

3.5 ALLOCATION OF SAMPLE SIZE

In stratified sampling, the allocation of the sample to different strata is done
by considering the following factors:
1. The total number of units in the stratum, i.e. stratum size;
2. The variability within the stratum; and
3. The cost in taking observations per sampling unit in the stratum.
A good allocation is one where maximum precision is obtained with minimum
resources. In other words, the criterion for allocation is to minimize the cost
for a given variation or minimize the variance for a fixed cost, thus making
the most effective use of the available resources.

Types of Allocation of Sample Size

It is evident from the formula for variance of xst that it depends on n i the
number of units selected at random from ith stratum. Hence, the problem
arises, what optimum value of n i ( i  1, 2,..., k ) can be chosen out of n, so that,
the variance is as small as possible. Four types of allocations are considered
here:
3.5.1 Equal Number of Units from Each Stratum
This is a situation of considerable practical interest for reasons of
administrative convenience. In this allocation method, the total sample size n
is divided equally among all the strata i.e. if the population is divided in k
strata then the size of sample for each stratum would be

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Statistical Techniques
n
ni  for all i =1, 2, …, k
k

3.5.2 Proportional Allocation

This allocation was originally proposed by Bowley in 1926. This procedure of
allocation is very common in practice because of its simplicity. As its name
indicates, proportional allocation means that we select a small sample from a
small stratum and a large sample from a large stratum. The sample size in
each stratum is fixed in such a way that for all the strata, the ratio ni Ni is
equal to n N i.e.
ni n

Ni N
n
or n i  N i
N
n i  n Wi
or n i  N i … (2)

In other words, the allocation of a sample of size n to different strata is to be

done in proportional to their sizes. We have variance of x st
k
 n  Wi2 2
Var x st    1  i  Si
i 1  Ni  ni

Thus, substituting the values of n i and ni Ni as n Wi and n N respectively in

variance formula then we get variance under proportional allocation

 n  k W S2
Var x st PROP  1    i i
 N  i 1 n
… (3)
1 1  k
     Wi Si2
 n N  i 1

In case, the sampling fraction (n/ N) is negligible

k
Wi Si2
Var x st  PROP  
i 1 n ... (4)

3.5.3 Neyman’s Allocation

This allocation of the total sample size n to the different stratum is called
minimum variance allocation and is due to Neyman (1934). This result was
first discovered by Tchuprow (1923) but remained unknown until it was
rediscovered independently by Neyman. This allocation of samples among
different strata is based on a joint consideration of the stratum size and the
stratum variance. In this allocation, it is assumed that the sampling cost per
unit among different strata is same and the size of the sample is fixed. Sample
sizes are allocated by

52
 WiS i N i Si Stratified Random
ni n k
n k
Sampling

WS
i 1
i i N
i 1
i Si … (5)

A formula for the minimum variance with fixed n is obtained by substituting

the value of ni in variance formula, then we get
2
 k  k
  Wi Si   Wi Si2
Var x st NEY   i 1   i 1
n N … (6)

3.5.4 Optimum Allocation

The variance of estimated mean depends on ni which can arbitrarily be fixed.
One more factor, which is none the less important, also influences the
variance of estimated mean. The allocation problem is two fold:
1. We attain maximum precision for the fixed cost of the survey; and
2. We attain the desired degree of precision for the minimum cost.
Thus, the allocation of the sample size in various strata, in accordance with
these two objectives, is known as optimum allocation.

In any stratum the cost of survey per sampling unit cannot be the same. That
is, in one stratum the cost of transportation may be different from the other.
Hence, it would not be wrong to allocate the cost of the survey in each stratum
differently.

Let ci be the cost per unit of survey in the ithstratum from which a sample of
size ni is stipulated. Also suppose c0 as the over head fixed cost of the survey.
In this way the total cost C of the survey comes out to be
k
C  c 0   ci n i
i 1
… (7)
c0 and ci are beyond our control. Hence we will determine the optimum value
of ni which minimizes the variance of stratified sample mean.
To determine the optimum value of ni, we consider the function

  Var x st   C
k k
1 1   
     Wi2 Si2  λ  c0   ci n i 
n
i 1  i N i   i 1  ... (8)
where  is constant and known as Lagrange’s multiplier.

Using the method of Lagrange’s multiplier we select n i and the constant 

to minimize  . Differentiating  with respect to n i , we have

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Statistical Techniques
Wi2S2i
  λ ci  0 … (9)
n i2
or
Wi Si
ni  … (10)
 ci

Since λ is an unknown quantity, it has to be determined interms of known

values. So, we take the sum over all in equation (10) and thus obtain
k
1 k Wi Si
 ni 
i 1 
 i1 ci

1 k Wi Si
or n


i1 ci

1 k Wi Si
  
n i1 ci

Substituting the value of  in equation (10), we get the value of n i

ni  n
W S c   n N S c 
i i i i i i
… (11)
k k

 W S c   N S c 
i 1
i i i
i 1
i i i

Thus, the relation (11) leads to the following important conclusions that we
have to take a larger sample in a given stratum if
1. The stratum size Ni is larger;
2. The stratum has larger variability (Si); and
3. The cost per unit is lower in the stratum.

3.6 STRATIFIED SAMPLING VERSUS SIMPLE

RANDOM SAMPLING
Now, we shall make a comparative study of simple random sampling without
replacement and stratified random sampling under different kinds of
allocations i.e. Proportional allocation and Neyman’s allocation.
3.6.1 Proportional Allocation versus Simple Random
Sampling
The variance of the estimate of stratified sample mean with proportional
allocation and variance of the sample mean of simple random sampling is
given respectively by
1 1  k
Var x st PROP      Wi Si2
 n N  i1 … (12)

54
 Ni Stratified Random
1
where, Si2 
N i  1 
Xij  Xi 2 Sampling
j1

1 1 
and Var x SRSWOR     S2
n N … (13)
k Ni
1
where, S2   Xij  X 2
N  1 i1 j1
In order to comparing (12) and (13) we shall first express S2 in terms of S2i
we have
1 k Ni
S2    X ij  X i  X i  X
2
 
N  1 i 1 j 1
k  Ni  k Ni
N  1S2    Xij  Xi  2    Xi  X  2
i 1  j1  i 1 j 1

k  Ni

 2 Xi  X  Xij  Xi  
i 1  j1 

k k
N  1S2   N i  1 Si2   N i Xi  X  2
i 1 i 1

The product term vanishes since

Ni

 X
j1
ij  Xi   0

being the sum of square of deviation from the stratum mean. If we assume that
Ni and consequently N are sufficiently large so that we can put Ni-1= Ni and
N-1 = N, then we get

k k
N S 2   N i Si2   N i X i  X 
2

i 1 i 1
k k
S2   Wi S2i   Wi Xi  X 
2
… (14)
i 1 i 1

Substituting in equation (13), we get

1 1  k 1 1  k
Var x       Wi Si2      Wi Xi  X 
2
SRSWOR
 n N  i 1  n N  i 1

1 1  k
Var x  SRSWOR
 Var x st PROP      Wi Xi  X   2

 n N  i1

Var x 
SRSWOR
 Var x st PROP … (15)

3.6.2 Neyman’s Allocation versus Proportional Allocation

Considering the variances of estimated sample mean in stratified random
sampling with proportional allocation and Neyman’s allocation respectively
we have
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Statistical Techniques k
1 1 
Var x st PROP     W S i
2
… (16)
n N i 1

and
2
1 k  1 k
Var x st NEY    Wi Si    Wi S2i
n  i1  N i1 … (17)
By subtracting equation (17) from equation (16) we get
1 1  k
Var x st PROP  Var x st NEY      Wi Si
2

 n N  i1
2
 1  k  1 k 
2
    Wi Si    Wi Si 
 n  i1  N i1 
2
1 k  k  
  Wi S2i    WiSi  
n  i 1  i 1  
k
1
  Wi (Si  S) 2
n i 1 … (18)
k
1 k
where, S   Wi Si   Ni Si is the weighted mean of the stratum sizes Ni
i 1 N i 1
Hence from equation (18) we can say

Var x st PROP  Var x st NEY

because, R.H.S. of equation (18) is non-negative.

3.6.3 Neyman’s Allocation versus Simple Random Sampling

From the relationship between the proportional allocation and simple random
sampling and the relation between proportional and Neyman allocation we
have
1 1  k
 
Var x SRSWOR  Var x st PROP      Wi Xi  X
2
 
 n N  i 1 … (19)
k
1
 W S  2
and Var x st PROP  Var x st NEY  i i S ... (20)
n i 1

By substituting the value of the variance under proportional allocation in

equation (19) from equation (20), we have
1 k
 
Var x SRSWOR  Var x st NEY   Wi Si  S
n i 1
2
 
1 1  k
     Wi X i  X   2
…(21)
 n N  i 1
That means
Var x SRSWOR  Var x st NEY
because both the terms in R.H.S. of equation (21) are positive. From the
results of the relations of variance of simple random sample mean and the
56
variance of stratified sample means with proportional and Neyman Stratified Random
Sampling
allocations, we can reach on the conclusion that
Var x SRSWOR  Var x st PROP  Var x st NEY
3.6.4 Merits and Demerits of Stratified Random Sampling
Merits

1. More Representative
Stratified random sampling ensures any desired representation in the
sample of the various strata in the population. It overruled the probability
of any essential group of the population being completely excluded in the
sample.

2. Greater Accuracy
Stratified random sampling provides estimate of parameters with
increased precision in comparison to simple random sampling. Stratified
random sampling also enables us to obtain the results of known precision
for each of the stratum.

3. Administrative Convenience
The stratified random samples would be more concentrated geographically
in comparison to simple random samples. Therefore, this method needs
less time and money involved in interviewing the supervision of the field
work can be done with greater case and convenience.
Demerits
However, stratified random sampling has some demerits too, which are:

1. May Contain Error due to Subjectiveness

In stratified random sampling the main objective is to stratify the
population in homogeneous strata. But stratification is a subjective issue
and so it may contain error.

2. Lower Efficiency
If the sizes of samples from different stratum are not properly determined
then stratified random sampling may yield a larger variance that means
lower efficiency.
Example 1: A sample of 60 persons is to be drawn from a population
consisting of 600 belonging to two villages A and B. The means and standard
deviations of their marks are given below:

Villages Strata Sizes Means Standard Deviation

Ni   
Xi  i 
Village A 400 60 20
Village B 200 120 80

Draw a sample using proportional allocation techniques.

57
Statistical Techniques
Solution: If we regard the villages A and B as representing two different
strata then the problem is to draw a stratified random sample of size 30 using
technique of proportional allocation. In proportional allocation, we have
n
ni  Ni
N
Therefore,
60
n1   400  40
600
60
n2   200  20
600
Thus, the required sample sizes for the villages A and B are 40 and 20
respectively.

E1) A sample of 100 employees is to be drawn from a population of collages

A and B. The population means and population mean squares of their
monthly wages are given below:
Village Ni Xi S i2

Collage A 300 25 25

Collage B 200 50 100

Draw the samples using proportional and Neyman allocation technique

and compare.

E2) Obtain the sample mean and estimate of the population mean for the
given information in Example 1 discussed above.

3.7 SUMMARY
In this unit, we have discussed:
1. The definition and procedure of stratified random sampling;
2. The principles of stratification;
3. The properties of stratified random sampling;
4. The mean and variance of proportions in stratified random sampling;
5. The allocation of sample size with the help of different techniques; and
6. Calculation of the estimate of population mean and variance of sample
mean.

3.8 SOLUTIONS /ANSWERS

E1) If we regard the collages A and B representing two different strata then
the problem is to draw as stratified sample of 100 employees using
technique of proportional allocation and Neyman’s allocation.
58
 In proportional allocation we have Stratified Random
Sampling
n 100
ni  Ni   Ni
N 500
100
n1   300  60
500
100
n2   200  40
500
In Neyman’s allocation, we have
N iSi
ni  n  k

NS
i 1
i i

NS
i 1
i i = 300  5 + 200  10

= 1500 + 2000 = 3500

NiSi
n i  100 
3500
300  5
n1  100   42.85  43
3500

200  5
n 2  100   57 .14  57
3500
Therefore, the samples regarding the colleges A and B for both
allocations are obtained as:

Proportional Neyman
Collage A 60 43
Collage B 40 57
Total 100 100

E2) We have the following data:

2 2
Village Ni Xi σi
S 2i 
N 2
σi N i Si2 Ni σ i2 Xi N i Xi
N 1

A 400 60 20 400.67 160267.11 160000 3600 3440000

B 200 120 80 6432.16 1286432.16 1280000 14400 880000

Totat 600 1446699.27 1440000 180000 4320000

1 k
X   Ni Xi
N i1

59
Statistical Techniques
1
  400  60  200 120
600

1 48000
  24000  24000   80
600 600
k
2

1 1 
N S i i
Var x PROP
   i 1

n N N

 600  60  1446699.27 1446699.27

  
 60  600  600 40000

= 36.1675

1 1 
Var x SRSWOR     S2
n N
where,
k ni
1
S2  
N  1 i 1 j1

Xij  X
2

1  k k 2
 
 N  1  i1
N i  2
i  
i 1
N i X i 
 X 


1 k 2
k

   N    N i X i2  NX 2 
N  1  i1 i i
i1 
1
 1440000  4320000  600  80  80 
599
1
 5760000  3840000   1920000
599 599
600  60 1920000
Var x SRSWOR  
600  60 599
Then the conclusion is
Var x st PROP  36.1675
Var x SRSWOR  48.08

Therefore, precision of x st can be obtained by

Var x SRS  Var x st PROP
Gain in precision = 100
Varx st PROP
48.04  36.1675
 100
36.1675
= 32.8 %

60