Revisiting Sampling Concepts

Population

• A population is all the possible members of a category

• Examples: • the heights of every male or every female• the temperature on every day since the beginning of time• Every person who ever has, and ever will, take a particular

drug

Sample

• A sample is some subset of a population

– Examples:• The heights of 10 students picked at random• The participants in a drug trial

• Researchers seek to select samples that accurately reflect the broader population from which they are drawn.

PopulationSample

Sample Statistics

PopulationParameters

Inference

Samples are drawn to infer something about population

Reasons to Sample

Ideally a decision maker would like to consider every item in the population but;

• To Contact the whole population would be time consuming e.g. Election polls

• The cost of such study might be too high

• In many cases whole population would be consumed if every part of it was considered

• The Sample results are adequate

Probability Vs Non Probability Sampling

Probability Sampling

• Drawing Samples in Random manner

• Using random numbers

• Writing names on identical cards or slips and then drawing randomly

• Choosing every nth item of the population

• First dividing the population into homogeneous groups and then drawing samples randomly

Probability Vs Non Probability Sampling

Non Probability Sampling

• man-on-the-street interviews

• call-in surveys

• readership surveys

• web surveys

Types of Variables

• Qualitative• Quantitative• Discrete• Continuous

• Categorical • Numerical

Sampling Error

• “Sampling error is simply the difference between the estimates obtained from the sample and the true population value.”

Sampling Error = X - µWhere X = Mean of the Sample µ = Mean of the Population

Validity of Sampling Process

Sampling Distributions

• A distribution of all possible statistics calculated from all possible samples of size n drawn from a population is called a Sampling Distribution.

• Three things we want to know about any distribution?

– Central Tendency

– Dispersion

– Shape

Sampling Distribution of Means

• Suppose a population consists of three numbers 1,2 and 3

• All the possible samples of size 2 are drawn from the population

• Mean of the Pop (µ) = (1 + 2 + 3)/3 = 2

• Variance

• Standard Deviation = 0.82

Distribution of the Population

Sampling distribution of means

n = 2

Sample # Sample Sample Mean

1 1,1 1

2 1,2 1.5

3 1,3 2

4 2,1 1.5

5 2,2 2

6 2,3 2.5

7 3,1 2

8 3,2 2.5

9 3,3 3

Mean of SD 2

= µ

= 0.6

= µ

• The population’s distribution has far more variability than that of sample means

• As the sample size increases the dispersion becomes less and in the SD

<

0.6 < 0.8

• The mean of the sampling distribution of ALL the sample means is equal to the true population mean.

• The standard deviation of a sampling distribution called Standard Error is calculated as

Central Limit Theorem ……

• The variability of a sample mean decreases as the sample size increases

• If the population distribution is normal, so is the sampling distribution

• For ANY population (regardless of its shape) the distribution of sample means will approach a normal distribution as n increases

• It can be demonstrated with the help of simulation.

Central Limit Theorem ……

• How large is a “large sample”?

• It depends upon the form of the distribution from which the samples were taken

• If the population distribution deviates greatly from normality larger samples will be needed to approximate normality.

Implications of CLT

• A light bulb manufacturer claims that the life span of its light bulbs has a mean of 54 months and a standard deviation of 6 months. A consumer advocacy group tests 50 of them. Assuming the manufacturer’s claims are true, what is the probability that it finds a mean lifetime of less than 52 months?

Implications of CLT Cont

• From the data we know that

• µ = 54 Months = 6 Months

• By Central Limit Theorem

= µ = 54

=

54

o-2.35

0.0094

52

• To find ,we need to convert to z-scores:

• From the Area table = 0.4906

• Hence, the probability of this happening is 0.0094.

• We are 99.06% certain that this will not happen

What can go wrong

• Statistics can be manipulated by taking biased samples intentionally

Examples

• Asking leading questions in Interviews and questionnaires

• A survey which showed that 2 out 3 dentists recommend a particular brand of tooth paste

• Some time there is non response from particular portion of population effecting the sampling design

How to do it rightly

• Need to make sure that sample truly represents the population

• Use Random ways where possible

• Avoid personal bias

• Avoid measurement bias

• Do not make any decisions about the population based on the samples until you have applied statistical inferential techniques to the sample.