Statistical Techniques for Analyzing Quantitative Data

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Statistical Techniques for Analyzing Quantitative
Data

• Maryam Ramezani
• Values in Computer Technology
• CSC 426

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Outline
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Role of Statistics in Research

• With Statistics , we can summarize large bodies
  of data, make predictions about future trends
  ,and determine when different experimental
  treatments have led to significantly different
  outcomes.
• Statistics are among the most powerful tools in
  the research's toolbox.

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How statistics come to research?

• In quantitative research we use numbers to
  represent physical or nonphysical phenomena
• We use statistics to summarize and interpret
  numbers

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Exploring and Organizing a Data Set

• Look at your data and find the ways of organizing
  them
• example Scores of test for 11 children
• What do you see?

Ruth 96, Robert 60, chuck 68, Margaret 88 Tom
56, Mary 92,Ralph 64, Bill 72,Alice 80 Adam
76,Kathy 84
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Exploring and Organizing a Data Set
Alphabetical Order
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Using Computer Spreadsheets to Organize and
Analyze Data

• Sorting
• Graphing
• Formulas
• What Ifs
• Save, Store, recall, update information

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Functions of Statistics

• Descriptive Statistics
• describes what the data look like
• Inferential Statistics
• inference about a large population by collecting
  small samples.

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Considering the Nature of the Data

• Continuous or discrete
• Nominal, ordinal, interval or ratio scale
• Normal or non-normal distribution

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Continuous versus Discrete Variables

• Continuous Data takes on any value within a
  finite or infinite interval. You can count, order
  and measure continuous data.
• Example height, weight, temperature, the amount
  of sugar in an orange, the time required to run a
  mile.
• Discrete Data values / observations belong are
  distinct and separate, i.e. they can be counted
  (1,2,3,....).
• Example the number of kittens in a litter the
  number of patients in a doctors surgery the
  number of flaws in one metre of cloth gender
  (male, female) blood group (O, A, B, AB).

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Nominal Data

• the numbers are simply labels. You can count but
  not order or measure nominal data
• Example males could be coded as 0, females as 1
  marital status of an individual could be coded as
  Y if married, N if single.
• classification data, e.g. m/f
• no ordering, e.g. it makes no sense to state that
  M gt F
• arbitrary labels, e.g., m/f, 0/1, etc

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Ordinal Data

• ordered but differences between values are not
  important
• e.g., Like scales, rank on a scale of 1..5 your
  degree of satisfaction
• rating of 2 rather than 1 might be much less than
  the difference in enjoyment expressed by giving a
  rating of 4 rather than 3.
• You can count and order, but not measure, ordinal
  data.

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Interval Data

• ordered, constant scale, but no natural zero
• differences make sense, but ratios do not
• e.g. 30-2020-10, but 20/10 is not twice
  as hot!
• e.g. Dates the time interval between the starts
  of years 1981 and 1982 is the same as that
  between 1983 and 1984, namely 365 days. The zero
  point, year 1 AD, is arbitrary time did not
  begin then

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Ratio Data

• Like interval data but has true zero
• Ordered, Constant scale, natural zero
• e.g., height, weight, age, length

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Normal and Non-Normal Distributions
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Normal Distribution
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Non-Normal Distributions
Skewed to the Left(Negatively Skewed)
Skewed to the Right (Positevely Skewed)
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Leptokurtic and Platykurtic Distributions
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Descriptive Statistics

• Descriptive Statistics describes data
• Points of Central Tendency
• Amount of Variability
• Relation of different variables to each other

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Points Of Central Tendency Mean

• Measuring center If the n observations are x1,
  x2,, xn, arithmetic mean is

Geometric Mean
e.x. Biological growth, Population growth
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Measure of Central Tendency
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Measures of Variability
How great is the Spread? RangeHighest
Score-Lowest score the quartiles The pth
percentile of a distribution is the value such
that p percent of the observations fall at or
below it. The 50th percentile median, M The
25th percentile first quartile, Q1 The 75th
percentile third quartile, Q3 Interquartile
Quartile 3- Quartile 1

• Example
• 13 13 16 19 21 21 23 23 24 26 26 27 27
  27 28 28 30 30
• M?, Q1?, Q3?

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Measures of Variability
Standard Devastation
standardized score
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Measure of Relationship Correlation

• correlation indicates the strength and direction
  of a linear relationship between two variables.
• See page 266 for other examples or correlation
  statistics

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Notes about Correlation

• Substantial correlations between two
  characteristics needs reasonable Validity and
  Reliability in measuring
• Correlation does not indicate causation

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Examples of using Statistics in Computer Science

• Conceptual Representation of User Transactions or
  Sessions

Pageview/objects
Session/user data
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Inferential Statistics

• We use the samples as estimate of population
  parameter.
• The quality of all statistical analysis depends
  on the quality of the sample data

Random Sampling every unit in the population
has an equal chance to be Chosen A random sample
should represent the population well, so sample
statistics from a random sample should provide
reasonable estimates of population parameters
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Some definitions

• Parameter describes a population
• Statistic describes a sample

A parameter is a characteristic or quality of a
population that in concept is constant ,however,
its value is variable. example radius is a
parameter in a circle
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Inferential Statistics

• Estimate a population parameter from a random
  sample
• Test statistically hypotheses

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Inferential Statistics Estimate a Population
Parameter from Sample

• All sample statistics have some error in
  estimating population parameters
• Example estimate mean height of 10 year old boys
  in Chicago, Sample200 boys
• How close the sample mean is to the population
  mean?
• we dont know but we know
• The mean from an infinite number of samples form
  a normal distribution.
• The population mean equals the average (mean) of
  all samples.
• The Standard deviation of sample distribution (
  standard error) is directly related to the std
  of the characteristic in question for the overall
  population.

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Standard Error

• Standard error tell us how much the particular
  mean vary from one sample to another when all
  samples are the same size and drawn randomly from
  the sample population.
• Standard Error
• n is size of all samples and s is the population
  std which we dont have!
• We use the std of sample

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Accuracy of the Estimator
As in many problems, there is a trade off between
accuracy and dollars.
What we will get from our money if we
invest dollars in obtaining a larger size?
n 100? n 200?
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Point versus Interval Estimate

• A point estimate is a single value--a
  point--taken from a sample and used to estimate
  the corresponding parameter of a population
• , s, s2 and r estimate µ, s, s2, ?
  respectively
• An interval estimate is a range of values--an
  interval within whose limits a population
  parameter probably lies.
• we say that we are 95 confident that the unknown
  population mean lies in the interval

95 confidence interval for µ.
(x -2?/(n1/2), x2 ?/(n1/2))

• In only 5 of all samples,
• the sample mean x is not in the above interval,
• that is 5 of all samples give inaccurate results.

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Testing Hypothesis

• Confidence intervals are used when the goal of
  our analysis is to estimate an unknown parameter
  in the population.
• A second goal of a statistical analysis is to
  verify some claim about the population on the
  basis of the data.
• Research Hypothesis /Statistical hypothesis
• A test of significance is a procedure to assess
  the truth about a hypothesis using the observed
  data. The results of the test are expressed in
  terms of a probability that measures how well the
  data support the hypothesis.

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Example To determine whether the mean nicotine
content of a brand of cigarettes is greater than
the advertised value of 1.4 milligrams, a health
advocacy group takes a sample of 500 cigarettes
and measures the amount of nicotine in the
sample.
Sample values The sample average of nicotine
1.51 mlg The standard deviation 1.016.
The estimated amount of nicotine is 1.51mlg,
based on the sample values. The standard error
of the sample average is S.E.s.d./sqrt(n-1)0.04
5 Is there an actual difference between the
sample value (1.51mlg) and the advertised value
(1.4 mlg)? Or is it just due to sampling
error? To answer this question we need a Test of
Significance
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Stating an hypotheses
The null hypothesis H0 expresses the idea that
the observed difference is due to chance. It is a
statement of no effect or no difference,
and is expressed in terms of the population
parameter.
Let ? denote the true average amount of
nicotine. H0 ? 1.4mlg
The alternative hypothesis Ha represents the idea
that the difference is real. It is expressed as
the statement we hope or suspect is true instead
of the null hypothesis.
The alternative hypothesis states that the
cigarettes contain a higher amount of nicotine,
that is Ha ? gt 14mlg
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General comments on stating hypotheses

• It is not easy to state the null and the
  alternative hypothesis!
• The hypotheses are statements on the population
  values.
• The alternative hypothesis Ha is often called
  researcher hypothesis, because it is the
  hypothesis we are interested about.
• A significance test is a test against the null
  hypothesis
• Often we set Ha first and then Ho is defined as
  the opposite statement!

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Errors in Hypothesis testing

• Type I Error the null hypothesis is rejected
  when it is in fact true that is, H0 is wrongly
  rejected.
• Type II Error the null hypothesis H0, is not
  rejected when it is in fact false

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Meta- Analysis

• Meta-analysis refers to the analysis of
  analyses...the statistical analysis of a large
  collection of analysis results from individual
  studies for the purpose of integrating the
  findings. (Glass, 1976, p. 3)
• Conduct a fairly extensive search for relevant
  studies
• Identify appropriate studies to include in
  meta-analysis
• Convert each studys results to a common
  statistical index

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Using Statistical Software Packages

• SPSS
• SAS
• Matlab Statistics toolbox
• SYSTAT, Minitab, Stat View, Statistica

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Interpreting the Data

• Relating the findings to the original research
  problem and to the specific research questions
  and hypothesis
• Relating the findings to preexisting literature,
  concepts, theories and research results.
• Determining whether the findings have practical
  significance as well as statistical significance
• Identifying limitations of the study