This document discusses concepts related to sampling and populations in biostatistics. It defines key terms like population, sample, sampling unit, target population, study population, and sample. It describes different types of sampling techniques including probability sampling (simple random sampling, systematic random sampling, stratified random sampling, cluster sampling, PPS sampling, multistage sampling, and multiphase sampling) and non-probability sampling (volunteer sampling, convenience sampling, quota sampling, snowball sampling). It also discusses concepts like sample statistic, population parameter, sampling error, and the importance of an adequate sample size.

1. Biostatistics
Dr Kanwal Preet Gill

2. CONCEPT OF SAMPLE AND
POPULATION
 Sampling :The process of drawing representative sample
from the population.
 Population: an aggregate of sampling units
 Sample: a small portion of the population which truly
represents the population with respect to study characteristics
of the population

3. CONCEPT OF SAMPLE AND
POPULATION
 Infinite number of samples can be drawn from a population.
Population
sample

5. Types of samples
 Probability/ random sampling
 Every unit has an equal probability of being selected
 Non-probability sample
 Assume that characteristic under study is evenly distributed
 Reliability can never be measured
 Representative sample:
 Sample that represents a population
 Sampling Unit:
 An element or group of elements of the population used for drawing a
sample

6. The target population, the study population and the sample
TARGET
POPULATION
STUDY
POPULATION
SAMPLE
POPULATION
INDIVIDUALS

7. Studying caries among 5-10 yr old children
 TARGET POPULATION
 All 5-10 year old children.
 STUDY POPULATION
 All 5-10 year old children coming to your OPD.
 SAMPLE
 10% of all 5-10 year old children coming to your OPD.

8. The target population, the study
population and the sample
 The study population should be adequately sized and
representative of the target population
 The sample should be adequately sized and representative of
the study population

9. Sample statistic
 The various summary values that describe a sample like
mean, standard deviation proportion etc are called sample
statistic.
 They are calculated from the individual sample
observations or measures and are often applied to calculate
the corresponding values for the population from which
the sample is purported to have been drawn.
 May or may not be valid estimators.

10. Population parameter
 Summary value that describes a population.
 Include constants like mean, variance, correlation coefficient.
 Denoted by Greek alphabets.

11. sampling error
Samples are never perfect replicas of their populations, so when a
conclusion is drawn about a population based on a sample, there will
always be what is known as sampling error.
Also called statistical error, sampling variability and is measured by
standard error.
Size of error can be reduced by increasing the sample size but cannot
be eliminated.

12. Sampling techniques
1. Probability sampling techniques
2. Non probability sampling techniques

13. Probability sampling techniques
Used for selecting samples from a population with each
unit of the study population having equal chance of either
being selected or being represented.

14. Probability sampling techniques
1. Simple random sampling
2. Systematic random sampling
3. Stratified random sampling
4. Cluster sampling
5. PPS sampling
6. Multistage sampling
7. Multiphase sampling

15. Simple random sampling
Every unit comprising of the study population is enumerated
and the requisite sample size as calculated is compiled by
random picking of numbers till the required number is achieved.
Done by use of random number tables /use of currency notes/
lottery draws

16. Simple Random sampling
Adv:
 Only complete sampling frame required
 No need for additional information
Disadv:
 Costly & not feasible for large population

17. Systematic Random Sampling
The entire study population is enumerated. Then, this number is
divided by the required sample size. This will be the sampling
interval.
From a random number, skip count with the sampling interval
till the sample size is achieved.

18. Stratified random sampling
When the study population is heterogeneous for some major
characteristics, then it is first divided into mutually exclusive
categories which are heterogeneous amongst one another but
homogeneous within each category for that characteristic.
These categories are called strata.

19. Stratified random sampling
From each stratum, sample selection is carried out on the lines
of simple random sampling.
Only so many units are selected per stratum such that total is
equal to the calculated sample size.

20. Cluster sampling
The cluster are chosen randomly. They are usually naturally
occurring or by geographic demarcations or are created by
using maps.
Each unit in the cluster is enumerated and a minimum of 7 units
per cluster are selected randomly.

21. Cluster sampling
The number of units can be increased if there is heterogeneity in
the cluster but it is always more desirable to increase the sample
size by increasing number of clusters.
Minimum recommended cluster number is 30 with 7 units each
taking sample size to 210.

22. Probability Proportional to size (PPS
sampling)
The sampling frame carries information about the varying size
of each sampling unit.
Define the number of clusters and divide the total study
population by number of clusters.
The value obtained is the sampling interval.

23. Probability Proportional to size (PPS
sampling)
Start from a random number between 1 and the value of SI and
skip count with SI in the cumulative frequency table of the
sampling units.
Select the cluster from sampling unit corresponding to the value
obtained by skip counting.
Larger sampling units have proportionately larger
representation.

24. Multistage Sampling
Selection of 1200 higher sec students in Amritsar
2 stage
Selection of 20 out of 100 higher
sec schools
Selection of 60 out of total higher
sec students from each school
3 stage
Selection of 20 out of 100 higher
sec schools
Selection of 2 out of total
sections in all schools
Selection of 30 students
from each section

25. Multiphase sampling
 Involves collection of basic information on a large sample size
 Successive collection of more specific information for sub-
samples out of earlier sample.
 Studied 100 students for BMI
 FBS for those who are obese

26. Non probability sampling techniques
A sample size is determined arbitrarily and selected on first
come first served basis.
Conclusions might or might not stand rigors of statistical
analysis.
Usually done as a prelude to formal data collection to see the
trend of observations or measures.

27. Non probability sampling techniques
1. Volunteer sampling
2. Convenience sampling
3. Quota sampling
4. Snowball sampling/ sisterhood sampling

28. Volunteer sampling
The participants choose to be a part of the study with or
without allurements with the pros and cons of the effects of the
study clearly stated beforehand.
Eg volunteers in a vaccine trial

29. Convenience sampling
Those subjects are chosen who are readily accessible and
available without paying any heed to the heterogeneity of the
subjects.
Eg exit poll opinions

30. Quota sampling
In this type of sampling, the focus is on getting specified
categories of people in the allocated number of sample size.
Eg sample of 100 newborn children with 50 males and 50
females.

31. Snowball sampling/ sisterhood sampling
starting from few initial subjects having outcome of interest, the
number is increased by asking the enrolled subjects about more
subjects who have the similar outcome.
Eg cases of breast cancer in a family.

32. Probability/ Chance
 Relative frequency or probable chances of occurrence with which a defined event is
expected to occur out of total possible occurrences.
 It is the measure of chance or uncertainty associated with a conclusion.
 Probability ranges from 0-1.
 It is the ratio of desired outcomes to total outcomes
 #desired/ #Total

33. Probability
 If I roll a number cube, there are six possibilities 1,2,3,4,5,6
 Each possibility has only one outcome, so each ahs a probability of 1/6
 1/6 + 1/6+1/6+1/6+1/6+1/6 = ?
 Let us toss a coin
 Probability of getting a tail =1/2
 Probability of getting a head =1/2
 Sum up of 2 probabilities =1/2+1/2 = ?

34. Probability and types of events
 Mutually exclusive events
 Independent Events
 Non-independent events

35. Mutually Exclusive Events & probability
 Probability of Mutually exclusive events
 Example : Probability of getting a head or a tail
 Probability of being a male child or a female child
 Probability of getting 2 or 3 when you throw a dice
 Probability of a child to be Rh +ve or Rh -ve

36. Mutually exclusive – Law of addition
 Let us we toss a coin - we get Head or Tail
 Probability of getting head (A)= ½
 Probability of getting Tail (B) = ½
 Total probability = P (A) + P (B)
= 1/2 + 1/2 = 1

37. Mutually exclusive events - example
Rh Factor Socio-economic Status Total
High Poor
+ve 90 80 170
-ve 10 20 30
100 100 200
What is the probability of being Rh +ve in high socio-economic status ?

38. Independent events & Probability
 When we throw a coin twice, what is the probability that we get Head and tail in 1st
and 2nd throw.
 What is the the probability that a newborn child will be boy and Rh +ve?

39. Independent Events – Law of Multiplication
 When we throw a coin twice, what is the probability the you will get Head and tail
in 1st and 2nd throw.
 Suppose we throw a coin
 Possibility of getting a head or a coin = ½
 Now we throw a coin 2nd time, it will be either a head or a tail
 What is the probability of getting head twice
 Probability of getting head 1st time = 1/2
 Probability of getting head 2nd time = 1/2
 Probability of getting head and Head = 1/2 X1/2 = 1/4

40. Example
 Two women are pregnant. Probability of newborn to be a girl is 0.488. Probability
that newborn is a boy is 0.512. what is the probability that both newborns are girls?

42. Following table shows countries by their region and by average business startup costs for a year.
Find the probability that the country is in the south Asia region, given that country's business
costs is high.
Region Very high High Low Total
East Asia & Pacific 3 13 8 24
Europe & Central Asia 0 8 17 25
Latin America &
Caribbean
4 23 5 32
Middle East & North
Africa
4 9 7 20
OECD 0 5 27 32
South Asia 0 7 1 8
Sub-Saharan Africa 21 22 4 47
Total 32 87 69 188

43. Conditional probability
 Probability that an event B will occur given that we already know the outcome of
an other event A.
 The prior occurrence of A causes the probability of B to change.
 P (A/B) = P(A) * P (B/A)

44.  In a locality of 1000 persons, 200 were attacked by cholera. In the population, 700 were
vaccinated against it. Among vaccinated, only 50 were attacked. Find out the probability that :
 A randomly selected person is attacked given that he is not vaccinated
 A randomly selected person is not attacked given that he is vaccinated
 Probability that randomly selected person is vaccinated = 700/1000
 Probability that randomly selected person is not vaccinated = 300/1000
 Probability that randomly selected person is attacked = 200/1000
 Probability that randomly selected person is not attacked = 800/1000
Attacked Not Attacked Total
Vaccinated 50 650 700
Not Vaccinated 150 150 300
Total 200 800 1000

45.  P attacked & not vaccinated
 = P (attacked) * P (not vaccinated in attacked)
 = 2/10 * 150/200 = 0.15 or 15%
 P not attacked and vaccinated
 = P (not attacked) * P (vaccine in not attacked)
 = 8/10 * 650/800 = 0.65 = 65%

46. A business owner noted the features of the 100 cars parked at the business. Here are
the results:
Given that a randomly selected car has a sunroof, find the probability the car has
doors.

47. Researchers surveyed one hundred students on which superpower they
would most like to have. The two-way table below displays data for the
sample of students who responded to the survey.
Superpower Male Female TOTAL
Fly 26 12 38
Invisibility 12 32 44
Other 10 8 18
TOTAL 48 52 100

48. In a wedding, 80 guests were asked whether they were a friend of the bride or of the
groom. Here are the results:
Given that a randomly selected guest is a friend of the groom, find the
are a friend of the bride.

49. Uses of probability
 1 Taking decision in case of uncertainty
 Probability of getting a boy / girl in 1st pregnancy
 Probability of getting a twin
 Probability of lung cancer among smokers
 2. deriving inferences about the population regarding parameter
 Percentage of people suffering from diabetes
 Efficacy of a treatment - comparison

50. Correlation & Regression
 Analysis based on multivariable distribution
 Correlation is described as the analysis which lets us know the association or the
absence of relationship between two variables.
 Regression analysis predicts the value of the dependent variable based on the
known value of independent variable.

51. Correlation
 It is when it is observed that a unit change in one variable is observed by an
equivalent change in the other variable.
 Correlation can be positive or negative
 The measures of correlations are:
 Karl Pearson’s correlation co-efficient
 Spearman’s Rank Test
 Scatter diagram

52. Regression
 In regression, there are two variable- one is independent (x) and the other is
dependent variable (y).
 A regression line of y on x is expressed as:
 Y = a + bx
 A- constant
 B-regression co-efficient.

53. Difference between correlation &
regression
 A statistical measure which determines the association of two quantities is known
as correlation, Regression describes how an independent variable is numerically
related to the dependent variable.
 Correlation is used to represent linear relationship, regression is used to fit the best
line.
 In correlation there is no difference between independent & dependent variable.
 Correlation tells us about strength of association while regression tells the impact
of unit change

54. testing of hypothesis
Variable Parametric Non-parametric
Comparison between 2
independent populations
Continuous
Discrete
Z test, t-test
Z test
Wilcoxon’s rank
Chi square
Comparison between 2 co-
related populations
Continuous
Discrete
Paired t-test
-
Wilcoxon’s signed rank
test
Mcnemar’s chi square
Comparison among several
independent populations
Continuous
Discrete
One-way ANOVA
-
Kruska-Wallis one way
ANOVA
Chi square
Comparison among several co-
related populations
Continuous
Discrete
Two-way ANOVA
-
Freidman’s two way
ANOVA
Mcnemar’s chi square

55. Importance of an adequate sized
sample

56. Too large a sample
 Waste of precious resources
 Overemphasis on trivial differences which have manifest as
significant but have no clinical/practical relevance.

57. Too small a sample size
Overlooking of significant differences between samples.

58. Formula for quantitative data
N= 4σ2
------------
L2
Where σ2 is the population variance i.e square of
population SD
L is the allowable error in specified Confidence limit.

59. `e.g mean SBP of employees in a computer firm is
120 mm Hg with SD of 10 mm of Hg. If the allowable
error is +-2mm of Hg at a risk of 5%, the size of
sample to verify the above mean and SD will be
N=4σ2
------------
L2
= 4*10*10
- ---------------------
2*2
= 100 employees

60. Formula for qualitative data
Conventionally, desired allowable error at 5% risk should not exceed the
true estimate of p by 10% or 20%.
N= 4pq
---------
L2
Where L is the allowable error i.e 10% or 20% of p
p is the positive attribute
q is 1-p

61. e.g Prevalence in the last year of worm infestation in children in a
slum is 40% . Calculate the size of sample required to know the
prevalence in the current year.
L is 10% of p
10% of 40 = 4
At 5% risk , n= 4pq/L2
= 4*40*60/ 4*4
= 600 children.