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DEVAPRAKASAM DEIVASAGAYAMProfessor of Mechanical Engineering
Room:11, LW, 2nd FloorSchool of Mechanical and Building Sciences
Email: [email protected], [email protected]
RES701: RESEARCH METHODOLOGY (3:0:0:3)
Devaprakasam D, Email: [email protected], Ph: +91 9786553933
Design in the Research Process
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Small Samples Can Enlighten
““The proof of the pudding is in the eating.The proof of the pudding is in the eating.By By a small sample a small sample we may judge of thewe may judge of thewhole piece.”whole piece.”
Miguel de Cervantes Saavedra Miguel de Cervantes Saavedra authorauthor
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The Nature of Sampling
•Population•Population Element•Census•Sample•Sampling frame
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.
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Why Sample?
Greater Greater accuracyaccuracy
Availability Availability of elementsof elements
Greater Greater speedspeed
Sampling Sampling providesprovides
Lower costLower cost
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 similarities Strong similarities in population elements makes in population elements makes
sampling possible.sampling possible.–– Sampling may be Sampling may be more accurate more accurate than a census.than a census.
•• Destruction of Test UnitsDestruction of Test Units–– Sampling Sampling reduces the costs reduces the costs of research in finite of research in finite
populations.populations.
A Photographic Example of How Sampling Works
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Steps in Sampling Design
What is the target population?What is the target population?
What are the parameters of What are the parameters of interest?interest?
What are the parameters of What are the parameters of interest?interest?
What is the sampling frame?What is the sampling frame?
What is the appropriate What is the appropriate sampling method?sampling method?
What is the appropriate What is the appropriate sampling method?sampling method?
What size sample is needed?What size sample is needed?
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When to Use Larger Sample?
Desired Desired precisionprecision
Number of Number of subgroupssubgroups
Confidence Confidence levellevel
Population Population variancevariance
Small error Small error rangerange
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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
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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
Statistical estimation
Population
Random sample
Parameters
Statistics
Every member of the population has the same chance of beingselected in the sample
estimation
Statistical inference. Role of chance.
Reason and intuition Empirical observation
Scientific knowledge
Formulate hypotheses
Collect data to test hypotheses
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 significanceor by confidence interval
Systematic error
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
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
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)
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
Measures of Dispersion• The Range
–The distance between the smallest and the largest values of a frequency distribution.
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)
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.
Low Dispersion versus High Dispersion
Descriptive Statistics
– Normal Curve – Bimodal Curve
Descriptive Statistics
– Positively Skewed – Negatively Skewed
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.
Calculating Deviation
Standard Deviation =
Calculating a Standard Deviation: Number of Sales Calls per Day for Eight Salespeople
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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.
EXHIBIT 17.13Fundamental
Types of Distributions
Three Important Distributions
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.
The Mean Distribution of Any Distribution Approaches Normal as n Increases
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.
EXHIBIT 17.8 Normal Distribution: Distribution of Intelligence Quotient (IQ) Scores
The Normal Distribution (cont’d)• Characteristics of a Standardized Normal
Distribution1. 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.
Standardized Normal Distribution
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.