1. Fundamentals of Sampling Method
Week 4
Research Methods & Data Analysis
Dr. Mario Mazzocchi
Research Methods & Data Analysis

2. Research Methods & Data Analysis
Tutorials
Thursday 30th October AG GL 20 (M. Mazzocchi)
Tuesday 4th November 11-1pm (H.Neeliah)
You may attend:
One (the most convenient for you)
Both (it may be very useful)
None (not really advised…)
Dr. Mario Mazzocchi
Research Methods & Data Analysis

3. Research Methods & Data Analysis
Lecture outline
Key notions of statistics
Simple random sampling
Sampling error
Sampling size
Other sampling methods
Dr. Mario Mazzocchi
Research Methods & Data Analysis

4. Research Methods & Data Analysis
Distributions
A set of values of a set of data together with their
Absolute frequencies
Relative frequencies (probabilities)
Dr. Mario Mazzocchi
Research Methods & Data Analysis

5. Relative and cumulate frequencies
fi=ni/N
Dr. Mario Mazzocchi
Research Methods & Data Analysis

6. Distributions of random variables
The distribution of possible values together with their probabilities (probability density function, p.d.f.)
Dr. Mario Mazzocchi
Research Methods & Data Analysis

7. The normal (Gaussian) distribution
…is the distribution representing perfect randomness around a mean value
In statistics, the normal distribution play a key role in the theory of errors
The central limit theorem implies that “averaging” almost always give origin to a normal distribution (error on the average is random), provided that the number of observation is large (>40)
Dr. Mario Mazzocchi
Research Methods & Data Analysis

8. The normal distributionp
95% of values
0,025
0,025
m-1.96s m
m+1.96s
Dr. Mario Mazzocchi
Research Methods & Data Analysis

9. The student-t distribution
When the parameter in the population has a normal distribution (with unknown variance), within the sample the parameter assumes a t distribution
The t-distribution is similar to the normal distribution, apart from having higher tail-probabilities
The bigger is the sample, the more similar the t-distribution is to the normal distribution
For samples with more than units, the difference between the two distributions is negligible
Dr. Mario Mazzocchi
Research Methods & Data Analysis

10. Research Methods & Data Analysis
The t-distribution
x-ta/2sx x
x+ta/2sx
Dr. Mario Mazzocchi
Research Methods & Data Analysis

11. ta/2 and za/2 – tabled values
Dr. Mario Mazzocchi
Research Methods & Data Analysis

12. Population parameters (in a population of N elements)
Mean
Variance
Standard deviation
Dr. Mario Mazzocchi
Research Methods & Data Analysis

13. Research Methods & Data Analysis
Sampling
A sample is a subgroup of the population selected for the study
Sample statistics allow to make inference about the population parameters, through estimation and hypothesis testing
The sample space is a complete set of all possible results of the sampling procedure
Dr. Mario Mazzocchi
Research Methods & Data Analysis

14. Simple random sampling
Each element of the population has a known and equal probability of selection
Every element is selected independently from other elements
The probability of selecting a given sample of n elements is computable (known)
The Central Limit Theorem guarantees that for simple random samples with sample size (n) sufficiently large (>40), the sample mean in a S.R.S. follows the normal distribution
Dr. Mario Mazzocchi
Research Methods & Data Analysis

15. Research Methods & Data Analysis
Sample statistics
Sample mean
Sample variance
Sample standard deviation
unbiasedness
Dr. Mario Mazzocchi
Research Methods & Data Analysis

16. Standard deviation and standard error
The standard deviation measures the variability of a given variable (e.g. X) within the population or sample
The standard error refers to the accuracy (variability) of the sample statistics (e.g. mean), i.e. the error due to the fact that the statistic is computed on a sample rather than on the population (sampling error)
Dr. Mario Mazzocchi
Research Methods & Data Analysis

17. Basic SRS sample statistics (unknown pop. variance)
Mean case
Proportion case (p)
Sample standard deviation of X
Standard error of the mean/proportion
ACCURACY of sample estimates
Dr. Mario Mazzocchi
Research Methods & Data Analysis

18. Finite population correction factor
For finite population (…i.e. all in social research), large samples (more than 10% of N) tend to overestimate the standard error of the sample mean (proportion)
In order to account for that, the following correction is necessary
Dr. Mario Mazzocchi
Research Methods & Data Analysis

19. Level of confidence a and z parameter
The level of confidence a refers to the probability that the true population mean falls in the identified confidence interval
For the normal distribution, given a value of a, the corresponding za/2 values is tabulated
a=0.05
za/2 =1.96
a/2
a/2 x
Confidence interval for x at a level of confidence a
Dr. Mario Mazzocchi
Research Methods & Data Analysis

20. Research Methods & Data Analysis
The t-distribution
x-ta/2sx x
x+ta/2sx
Dr. Mario Mazzocchi
Research Methods & Data Analysis

21. Research Methods & Data Analysis
Confidence intervals
Calculate the sample mean
Decide a level of confidence (usually 95% or 99%)
Choose whether using the Student-t distribution or the Normal distribution
Compute the sample standard error
Define the lower and upper bound of the confidence interval
Dr. Mario Mazzocchi
Research Methods & Data Analysis

22. Research Methods & Data Analysis
Exercise
Suppose that you have interviewed 20 students out of 200 in the agricultural building, asking them how much they paid for lunch yesterday
You get an average of £ 3.67
The standard deviation is 1.25
Compute the 95% confidence interval
Compute the 99% confidence interval
Dr. Mario Mazzocchi
Research Methods & Data Analysis

23. Determining sample size
Factors influencing sample size (n):
Size of the population (N)
Variability of the population (s)
Desired level of accuracy (q)
Level of confidence (a)
Budget constraint
Dr. Mario Mazzocchi
Research Methods & Data Analysis

24. Simple random sampling: determining sample size
Relative sampling error (r.s.e)
Determining sampling size for a given r.s.e. (approximate formula)
Dr. Mario Mazzocchi
Research Methods & Data Analysis

25. The sampling design process
Define the target population, its elements and the sampling units
Determine the sampling frame (list)
Select a sampling technique
Sampling with/without replacement
Probability/Nonprobability sampling
Determine the sample size
Precision versus costs
The marginal value in terms of precision of additional sampling units is decreasing
Execute the sampling process
Dr. Mario Mazzocchi
Research Methods & Data Analysis

26. The sampling techniques
Probabilistic samples
Simple random sampling
Systematic sampling
Stratified sampling
Cluster sampling
Other sampling techniques
Nonprobabilistic samples
Convenience sampling
Judgmental sampling
Quota sampling
Snowball sampling
Dr. Mario Mazzocchi
Research Methods & Data Analysis

27. Research Methods & Data Analysis
Representativeness
A sample can be considered as “representative” when it is expected to exhibit the average properties of the population
Dr. Mario Mazzocchi
Research Methods & Data Analysis

28. Research Methods & Data Analysis
Selection bias
Improper selection of sample units (ignoring a relevant “control variable” that generate bias), so that the values observed in the sample are biased and the sample is not representative.
Example:
A survey is conducted for measuring goat milk consumption, but the interviewers just select people in urban areas, that on average drink less goat milk.
Dr. Mario Mazzocchi
Research Methods & Data Analysis

29. Simple random sampling
Each element of the population has a known and equal probability of selection
Every element is selected independently from other elements
The probability of selecting a given sample of n elements is computable (known)
Statistical inference is possible
It is easily understood
Representative samples are large and expensive
Standard errors are larger than in other probabilistic sampling techniques
Sometimes it is difficult to execute a really random sampling
Dr. Mario Mazzocchi
Research Methods & Data Analysis

30. Research Methods & Data Analysis
Systematic sampling
A list of N elements in the population is compiled, ordered according to a specified variable
Unrelated to the target variable (similar to SRS)
Related to the target variable (increased representativeness)
A sampling size n is chosen
A systematic step of k=N/n is set
A random number s between 1 and N is extracted and represents the first element to be included
Then the other elements selected are s+k, s+2k, s+3k…
Cheaper and easier than SRS
More representative if order is related to the interest variable (monotone)
Sampling frame not always necessary
Less representative (biased) if the order is cyclical
Dr. Mario Mazzocchi
Research Methods & Data Analysis

31. Research Methods & Data Analysis
Stratified sampling
Population is partitioned in strata through control variables (stratification variables), closely related with the target variable, so that there is homogeneity within each stratum and heterogeneity between strata
A simple random sampling frame is applied in each strata of the population
Proportionate sampling: size of the sample from each stratum is proportional to the relative size of the stratum in the total population
Disproportionate sampling: size is also proportional to the standard deviation of the target variable in each stratum
Gains in precision
Include all relevant subpopolation even if small
Stratification variables may not be easily identifiable
Stratification can be expensive
Dr. Mario Mazzocchi
Research Methods & Data Analysis

32. Research Methods & Data Analysis
Cluster sampling
The population is partitioned into clusters
Elements within the cluster should be as heterogeneous as possible with respect to the variable of interests (e.g. area sampling)
A random sample of clusters is extracted through SRS (with probability proportional to the cluster size)
2a. All the elements of the cluster are selected (one-stage)
2b. A probabilistic sample is extracted from the cluster (two-stage cluster sampling)
Reduced costs
Higher feasibility
Less precision
Inference can be difficult
Dr. Mario Mazzocchi
Research Methods & Data Analysis

33. Non probabilistic samples
Dr. Mario Mazzocchi
Research Methods & Data Analysis

34. Research Methods & Data Analysis
Convenience sampling
Only “convenient” elements enter the sample
Cheapest method
Quickest method
Selection bias
Non representativeness
Inference is not possible
Dr. Mario Mazzocchi
Research Methods & Data Analysis

35. Research Methods & Data Analysis
Judgmental sampling
Selection based on the judgment of the researcher
Low cost
Quick
Non representativeness
Inference is not possible
Subjective
Dr. Mario Mazzocchi
Research Methods & Data Analysis

36. Research Methods & Data Analysis
Quota sampling
Define control categories (quotas) for the population elements, such as sex, age…
Apply a “restricted judgmental sampling”, so that quotas in the sample are the same of those in the population
Cheapest method
Quickest method
There is no guarantee that the sample is representative (relevance of control characteristic chosen)
Many sources of selection bias
No assessment of sampling error
Dr. Mario Mazzocchi
Research Methods & Data Analysis

37. Research Methods & Data Analysis
Snowball sampling
A first small sample is selected randomly
Respondents are asked to identify others who belong to the population of interests
The referrals will have demographic and psychographic characteristics similar to the referrers
Lower costs
Low variability
Useful for “rare” populations
Inference is not possible
Dr. Mario Mazzocchi
Research Methods & Data Analysis