Sampling, Making Data Usable, Understanding Data

1. DEVAPRAKASAM DEIVASAGAYAM
Professor of Mechanical Engineering
Room:11, LW, 2nd Floor
School of Mechanical and Building Sciences
Email: devaprakasam.d@vit.ac.in, dr.devaprakasam@gmail.com
RES701: RESEARCH METHODOLOGY (3:0:0:3)
Devaprakasam D, Email: devaprakasam.d@vit.ac.in, Ph: +91 9786553933

2. Design in the Research Process

3. 14-3
Small Samples Can Enlighten
““The proof of the pudding is in the eating.The proof of the pudding is in the eating.
ByBy a small samplea small sample we may judge of thewe may judge of the
whole piece.”whole piece.”
Miguel de Cervantes SaavedraMiguel de Cervantes Saavedra
authorauthor

4. 14-4
The Nature of Sampling
•Population
•Population Element
•Census
•Sample
•Sampling frame

5. Sampling Terminology
• Sample
– A subset, or some part, of a larger population.
• Population (universe)
– Any complete group of entities that share some
common set of characteristics.
• Population Element
– An individual member of a population.
• Census
– An investigation of all the individual elements that
make up a population.

6. 14-6
Why Sample?
GreaterGreater
accuracyaccuracy
AvailabilityAvailability
of elementsof elements
GreaterGreater
speedspeed
SamplingSampling
providesprovides
Lower costLower cost

7. Why Sample?
• Pragmatic Reasons
– Budget and time constraints.
– Limited access to total population.
• Accurate and Reliable Results
– Samples can yield reasonably accurate information.information.
–– Strong similaritiesStrong similarities in population elements makesin population elements makes
sampling possible.sampling possible.
–– Sampling may beSampling may be more accuratemore accurate than a census.than a census.
•• Destruction of Test UnitsDestruction of Test Units
–– SamplingSampling reduces the costsreduces the costs of research in finiteof research in finite
populations.populations.

8. A Photographic Example of How Sampling Works

9. 14-9
Steps in Sampling Design
What is the target population?What is the target population?
What are the parameters ofWhat are the parameters of
interest?interest?
What are the parameters ofWhat are the parameters of
interest?interest?
What is the sampling frame?What is the sampling frame?
What is the appropriateWhat is the appropriate
sampling method?sampling method?
What is the appropriateWhat is the appropriate
sampling method?sampling method?
What size sample is needed?What size sample is needed?

10. 14-10
When to Use Larger Sample?
DesiredDesired
precisionprecision
Number ofNumber of
subgroupssubgroups
ConfidenceConfidence
levellevel
PopulationPopulation
variancevariance
Small errorSmall error
rangerange

11. 14-11
Simple Random
Advantages
• Easy to implement with
random dialing
Disadvantages
• Requires list of
population elements
• Time consuming
• Larger sample needed
• Produces larger errors
• High cost

12. 14-12
Systematic
Advantages
• Simple to design
• Easier than simple
random
• Easy to determine
sampling distribution of
mean or proportion
Disadvantages
• Periodicity within
population may skew
sample and results
• Trends in list may bias
results
• Moderate cost

13. Statistical estimation
Population
Random sample
Parameters
Statistics
Every member of the
population has the
same chance of being
selected in the sample
estimation

14. Statistical inference. Role of chance.
Reason and intuition Empirical observation
Scientific knowledge
Formulate
hypotheses
Collect data to
test hypotheses

15. Statistical inference. Role of chance.
Formulate
hypotheses
Collect data to
test hypotheses
Accept hypothesis Reject hypothesis
C H A N C E
Random error (chance) can be controlled by statistical significance
or by confidence interval
Systematic error

16. Making Data Usable
• To make data usable, this information must be
organized and summarized.
• Methods for doing this include:
–frequency distributions
–proportions
–measures of central tendency and
dispersion

17. Population Mean
Making Data Usable (cont’d)
• Proportion
– The percentage of elements that meet some
criterion
• Measures of Central Tendency
– Mean: the arithmetic average.
– Median: the midpoint; the value below which half
the values in a distribution fall.
– Mode: the value that occurs most often.
Sample Mean

18. Statistics and Research Design
• Statistics: Theory and method of analyzing
quantitative data from samples of
observations … to help make decisions about
hypothesized relations.
– Tools used in research design
• Research Design: Plan and structure of the
investigation so as to answer the research
questions (or hypotheses)

19. Frequency
• Frequency Distributions
– In tables, the frequency distribution is constructed
by summarizing data in terms of the number or
frequency of observations in each category, score,
or score interval
– In graphs, the data can be concisely summarized
into bar graphs, histograms, or frequency
polygons

20. Measures of Dispersion
• The Range
–The distance between the
smallest and the largest values
of a frequency distribution.

21. Descriptive Statistics
• Measures of Central Tendency
– Mode
• The most frequently occurring score
• 3 3 3 4 4 4 5 5 5 6 6 6 6: Mode is 6
• 3 3 3 4 4 4 5 5 6 6 7 7 8: Mode is 3 and 4
– Median
• The score that divides a group of scores in half with 50% falling above and
50% falling below the median.
• 3 3 3 5 8 8 8: The median is 5
• 3 3 5 6: The median is 4 (Average of two middle numbers)
– Mean
• Preferred whenever possible and is the only measure of central tendency
that is used in advanced statistical calculations:
– More reliable and accurate
– Better suited to arithmetic calculations
• Basically, and average of all scores. Add up all scores and divide by total
number of scores.
• 2 3 4 6 10: Mean is 5 (25/5)

22. Measure of Dispersion
• Measures of Variability (Dispersion)
– Range
• Calculated by subtracting the lowest score from the highest score.
• Used only for Ordinal, Interval, and Ratio scales as the data must
be ordered
– Example: 2 3 4 6 8 11 24 (Range is 22)
– Variance
• The extent to which individual scores in a distribution of scores
differ from one another
– Standard Deviation
• The square root of the variance
• Most widely used measure to describe the dispersion among a set
of observations in a distribution.

23. Low Dispersion versus High Dispersion

24. Descriptive Statistics
– Normal Curve – Bimodal Curve

25. Descriptive Statistics
– Positively Skewed – Negatively
Skewed

26. Measures of Dispersion (cont’d)
• Why Use the Standard Deviation?
– Variance
• A measure of variability or dispersion.
• Its square root is the standard deviation.
– Standard deviation
• A quantitative index of a distribution’s spread, or variability;
the square root of the variance for a distribution.
• The average of the amount of variance for a distribution.
• Used to calculate the likelihood (probability) of an event
occurring.

27. Calculating Deviation
Standard Deviation =

28. Calculating a Standard Deviation: Number of Sales Calls per Day for Eight
Salespeople

29. 17–29
Population Distribution, Sample
Distribution, and Sampling
Distribution
• Population Distribution
– A frequency distribution of the elements of a
population.
• Sample Distribution
– A frequency distribution of a sample.
• Sampling Distribution
– A theoretical probability distribution of sample means
for all possible samples of a certain size drawn from a
particular population.
• Standard Error of the Mean
– The standard deviation of the sampling distribution.

30. EXHIBIT 17.13
Fundamental
Types of
Distributions

31. Three Important Distributions

32. Central-limit Theorem
• Central-limit Theorem
– The theory that, as sample size increases, the
distribution of sample means of size n, randomly
selected, approaches a normal distribution.

33. The Mean Distribution of Any Distribution Approaches Normal as n Increases

34. The Normal Distribution
• Normal Distribution
– A symmetrical, bell-shaped distribution (normal curve)
that describes the expected probability distribution of
many chance occurrences.
– 99% of its values are within ± 3 standard deviations
from its mean.
• Standardized Normal Distribution
– A purely theoretical probability distribution that
reflects a specific normal curve for the standardized
value, z.

35. EXHIBIT 17.8 Normal Distribution: Distribution of Intelligence Quotient (IQ) Scores

36. The Normal Distribution (cont’d)
• Characteristics of a Standardized Normal
Distribution
1. It is symmetrical about its mean; the tails on both sides
are equal.
2. The mean identifies the normal curve’s highest point
(the mode) and the vertical line about which this
normal curve is symmetrical.
3. The normal curve has an infinite number of cases (it is a
continuous distribution), and the area under the curve
has a probability density equal to 1.0.
4. The standardized normal distribution has a mean of 0
and a standard deviation of 1.

37. Standardized Normal Distribution

38. The Normal Distribution (cont’d)
• Standardized Values, Z
– Used to compare an individual value to the population
mean in units of the standard deviation
– The standardized normal distribution can be used to
translate/transform any normal variable, X, into the
standardized value, Z.
– Researchers can evaluate the probability of the
occurrence of many events without any difficulty.