the statistical analysis of data - UBC Department of Computer Science
the statistical analysis of data
by Dr. Dang Quang A & Dr. Bui The Hong
Hanoi Institute of Information Technology
1
Preface
• Statistics is the science of collecting, organizing and interpreting numerical and nonnumerical facts, which we call data.
• The collection and study of data are important in the work of many professions, so that training in the science of statistics is valuable preparation for variety of careers.
, for example economists and financial advisors, businessmen, engineers, farmers
• Knownedge of probability and statistical methods also are useful for informatic specialists of various fields such as data mining, knowledge discovery, neural network, fuzzy system and so on.
• Whatever else it may be, statistics is, firsrt and foremost, a collection of tools used for converting raw data into information to help decision makers in their works.
The science of data - statistics - is the subject of this course
.
2
Audience and objective
•
Audience
This tutorial as an introductory course to statistics is intended mainly for users such as engineers, economists, managers,...which need to use statistical methods in their work and for students. However, it will be in many aspects useful for computer trainers.
•
Objectives
• Understanding statistical reasoning
• Mastering basic statistical methods for analyzing data such as descriptive and inferential methods
• Ability to use methods of statistics in practice with the help of computer softwares in statistics
• Entry requirements
High school algebra course (+elements of calculus)
Skill of working with computer
3
Contents
Preface
Chapter 1 Introduction ………….
Chapter 2 Data presentation
…...
Chapter 3 Data characteristics ... descriptive summary statistics
Chapter 4 Probability: Basic …...
concepts …………………… .
Chapter 5 Basic Probability distributions ………………...
Chapter 6 Sampling
Distributions
……………….
Chapter 7 Estimation ………….
Chapter 8 General Concepts of
Hypothesis Testing …………..
Chapter 9 Applications of
Hypothesis Testing …………..
Chapter 10 Categorical Data …....
Analysis and Analysis of variance
Chapter 11 Simple Linear ……… regression and correlation ……
Chapter 12 Multiple regression …
Chapter 13 Nonparametric statistics
…………………………
References
Appendix A
Appendix B
Appendix C
Appendix D
Index
4
Chapter 1 Introduction
1.1 What is Statistics
– Whatever else it may be, statistics is, first and foremost, a collection of tools used for converting raw data into information to help decision makers in their works.
1.2. Populations and samples
– A population is a whole, and a sample is a fraction of the whole.
– A population is a collection of all the elements we are studying and about which we are trying to draw conclusions. Such a population is often referred to as the target population.
– A sample is a collection of some, but not all, of the elements of the population
1.3. Descriptive and inferential statistics
– Descriptive statistics is devoted to the summarization and description of data (population or sample) .
– Inferential statistics uses sample data to make an inference about a population .
1.4. Brief history of statistics
5
1.5. Computer softwares for statistical analysis
Chapter 2 Data presentation
2.1 Introduction
• The objective of data description is to summarize the characteristics of a data set.
Ultimately, we want to make the data set more comprehensible and meaningful. In this chapter we will show how to construct charts and graphs that convey the nature of a data set. The procedure that we will use to accomplish this objective depends on the type of data
.
2.2 Types of data
• Quantitative data are observations measured on a numerical scale.
Nonnumerical data that can only be classified into categories are said to be
qualitative data..
2.3 Qualitative data presentation
• Category frequency = the number of observations that fall in that category.
• Relative frequency = the proportion of the total number of observations that fall in that category
• Percentage for a category = Relative frequency for the category x 100%
2.4 Graphical description of qualitative data
Bar graphs and pie charts
6
Chapter 2 (continued 1)
2.5 Graphical description of quantitative data: Stem and
Leaf displays
• Stem and leaf display is widely used in exploratory data analysis when the data set is small
• Steps to follow in constructing a Stem and Leaf Display
• Advantages and disdvantage of a stem and leaf display
•
2.6 Tabulating quantitative data: Relative frequency distributions
Frequency distribution is a table that organizes data into classes
• Class frequency = the number of observations that fall into the class.
• Class relative frequency = Class frequency/ Total number of observations
Relative class percentage = Class relative frequency x 100%
2.7 Graphical description of quantitative data: histogram and polygon
• frequency histogram, relative frequency histogram and percentage histogram.
• frequency polygon, relative frequency polygon and percentage polygon
2.8 Cumulative distributions and cumulative polygons
2.9 Exercises
7
Chapter 3 Data characteristics: descriptive summary statistics
3.1 Introduction
Types of numerical descriptive measures
3.3 Measures of central tendency
3.4 Measures of data variation
Measures of relative standing
3.6 Shape
3.7 Methods for detecting outlier
3.8 Calculating some statistics from grouped data
3.9 Computing descriptive summary statistics using computer softwares
3.10 Exercises
8
Chapter 3
(continued 1)
3.2 Types of numerical descriptive measures:
Location, Dispersion, Relative standing and Shape
3.3 Measures of location ( or central tendency)
3.3.1 Mean
3.3.2 Median
3.3.3 Mode
3.3.4 Geometric mean
3.4 Measures of data variation
3.4.1 Range
3.1.2 Variance and standard deviation
Uses of the standard deviation: Chebyshev’s Theorem,
The Empirical Rule
3.4.3 Relative dispersion: The coefficient of variation
9
Chapter 3
(continued 2)
3.5 Measures of relative standing
Descriptive measures that locate the relative position of an observation in relation to the other observations are called measures of relative standing
The pth percentile is a number such that p% of the observations of the data set fall below and (100-p)% of the observations fall above it.
– Lower quartile = 25 th percentile Mid- quartile ,= 50 th percentile.
– Upper quartile = 75 th percentile, Interquartile range, z-score
3.6 Shape
3.6.1 Skewness
3.6.2 Kurtosis
3.7 Methods for detecting outlier
3.8 Calculating some statistics from grouped data
10
Chapter 4. Probability: Basic concepts
]
4.1 Experiment, Events and Probability of an Event
4.2 Approaches to probability
4.3 The field of events
4.4 Definitions of probability
4.5 Conditional probability and independence
4.6 Rules for calculating probability
4.7 Exercises
11
Chapter 4
(continued 1)
4.1 Experiment, Events and Probability of an Event
The process of making an observation or recording a measurement under a given set of conditions is a trial or experiment.
Outcomes of an experiment are called events.
We denote events by capital letters A, B, C,…
The probability of an event A, denoted by P(A), in general, is the chance A will happen
.
4.2 Approaches to probability
.Definitions of probability as a quantitative measure of the
“degree of certainty” of the observer of experiment.
.Definitions that reduce the concept of probability to the more primitive notion of “equal likelihood” (the so-called “classical definition “).
.Definitions that take as their point of departure the “relative frequency” of occurrence of the event in a large number of trials
(“statistical” definition).
12
]
Chapter 4
(continued 2)
4.3 The field of events
• Definitions and relations between the events: A implies B, A and
B are equivalent (A=B), product or intersection of the events A and
B (AB), sum or union of A and B (A+B), difference of A and (A-B or
A\B), certain (or sure) event , impossible event, complement of A, mutually exclusive events, simple (or elementary), sample space.
• Ven diagrams
• Field of events
4.4 Definitions of probability
4.4.1 The classical definition of probability
4.4.2 The statistical definition of probability
4.4.3 Axiomatic construction of the theory of probability (optional)
4.5 Conditional probability and independence
Definition, formula, multiplicative theorem, independent and dependent events
13
Chapter 4
(continued 3)
4.5 Conditional probability and independence
4.6 Rules for calculating probability
4.6.1 The addition rule
– for pairwise mutually exclusive events
P(A
1
+ A
2
+ ...+A n
)= P(A
1
)+P(A
2
)+ ...+P(A n
)
– for two nonmutually exclusive events A and B
P(A+B) = P(A) + P(B) – P(AB).
4.6.2 Multiplicative rule
P(AB) = P(A) P(B|A) = P(B) P(A|B).
4.6.3 Formula of total probability
P(B)= P(A
1
)P(B|A
1
)+P(A
2
)P(B|A
2
)+ ...+P(A n
)P(B|A n
).
14
Chapter 5 Basic Probability distributions
5.1 Random variables
5.2 The probability distribution for a discrete random variable
5.3 Numerical characteristics of a discrete random variable
5.4 The binomial probability distribution
5.5 The Poisson distribution
5.6 Continuous random variables: distribution function and density function
5.7 Numerical characteristics of a continuous random variable
5.8 The normal distribution
5.9 Exercises 15
•[
Chapter 5
(continued 1)
5.1 Random variables
• A random variable is a variable that assumes numerical values associated with events of an experiment.
• Classification of random variables: A discrete random variable and continuous random variable
5.2 The probability distribution for a discrete random variable
• The probability distribution for a discrete random variable x is a table, graph, or formula that gives the probability of observing each value of x.
• Properties of the probability distribution
16
•[
Chapter 5
(continued 2)
5.3 Numerical characteristics of a discrete random variable
5.3.1 Mean or expected value:
=E(X)=
xp(x)
5.3.2 Variance and standard deviation
2 =E[(X-
) 2 ]
5.4 The binomial probability distribution
• Model (or characteristics) of a binomial random variable
• The probability distribution
• mean and variance for a binomial random variable
5.5 The Poisson distribution
• Model (or characteristics) of a Poisson random variable
• The probability distribution
• mean and variance for a Poisson random variable
17
Chapter 5
(continued 3)
5.6 Continuous random variables: distribution function and density function
• Cumulative distribution function F(x)=P(X<x)
• Density probability function f(x) = F’(x)
5.7 Numerical characteristics of a continuous random variable
• Mean or expected value
=E(X)=
xp(x)dx
• Variance and standard deviation
5.8 The normal distribution
• The density function, mean and variance for a normal random variable
•
, 2
and 3
rules
• The normal distribution as an approximation to binomial probability distribution
18
Chapter 6 Sampling Distributions
6.1 Why the method of sampling is important
6.2 Obtaining a Random Sample
6.3 Sampling Distribution
6.4 The sampling distribution of sample mean: the
Central Limit Theorem
6.5 Summary
6.6 Exercises
19
•[
]
Chapter 6
(continued 1)
6.1 Why the method of sampling is important
• two samples from the same population can provide contradictory information about the population
• Random sampling eliminates the possibility of bias in selecting a sample and, in addition, provides a probabilistic basic for evaluating the reliability of an inference
6.2 Obtaining a Random Sample
• A random sample of n experimental units is one selected in such a way that every different sample of size n has an equal probability of selection
• procedures for generating a random sample
20
Chapter 6
(continued 2)
6.3 Sampling Distribution
• A numerical descriptive measure of a population is called a parameter.
A quantity computed from the observations in a random sample is called a statistic.
• A sampling distribution of a sample statistic (based on n observations) is the relative frequency distribution of the values of the statistic theoretically generated by taking repeated random samples of size n and computing the value of the statistic for each sample.
• Examples of computer-generated random samples
6.4 The sampling distribution of sample mean: the
Central Limit Theorem
• If the size is sufficiently large, the mean of a random sample from a population has a sampling distribution that is approximately normal, regardless of the shape of the relative frequency distribution of the target population
• Mean and standard deviation of the sampling distribution
6.5 Summary
21
]
Chapter 7. Estimation
7.1 Introduction
7.2 Estimation of a population mean: Large-sample case
7.3 Estimation of a population mean: small sample case
7.4 Estimation of a population proportion
7.5 Estimation of the difference between two population means: Independent samples
7.6 Estimation of the difference between two population means: Matched pairs
7.7 Estimation of the difference between two population proportions
7.8 Choosing the sample size
7.9 Estimation of a population variance
7.10 Summary 22
Chapter 7
(continued 1)
]
7.2 Estimation of a population mean: Large-sample case
• Point estimate for a population mean:
• Large-sample (1
) 100% Confidence interval for a population mean ( use the fact that For sufficient large sample size n>=30, the sampling distribution of the sample mean, , is approximately normal).
7.3 Estimation of a population mean: small sample case
(n<30)
• Problems arising for small sample sizes and Assumption: the population has an approximate normal distribution.
• (1
) 100% Confidence interval using t-distribution.
7.4 Estimation of a population proportion
• For sufficiently large samples, the sampling distribution of the proportion p-hat is approximately normal.
• Large-sample (1
) 100% Confidence interval for a population proportion
23
Chapter 7
(continued 2)
]
7.5 Estimation of the difference between two population means: Independent samples
• For sufficiently large sample size (n
1 sampling distribution of
1
-
and n
2
>= 30), the
2 based on independent random samples from two populations, is approximately normal
• Small sample sizes under some assumptions on populations
7.6 Estimation of the difference between two population means: Matched pairs
Assumption: the population of paired differences is normally distributed
Procedure
7.7 Estimation of the difference between two population proportions
• For sufficiently large sample size (n
1 and n
2
>= 30), the sampling distribution of p
1
- p
2 based on independent random samples from two populations, is approximately normal
• (1
) 100% Confidence interval for p
1
- p
2
24
Chapter 8. General Concepts of Hypothesis Testing
8.1 Introduction
The procedures to be discussed are useful in situations, where we are interested in making a decision about a parameter value rather then obtaining an estimate of its value
8.2 Formulation of Hypotheses
• A null hypothesis H
0 is the hypothesis against which we hope to gather evidence. The hypothesis for which we wish to gather supporting evidence is called the alternative hypothesises H a
• One-tailed (directional) test and two-tailed test
8.3 Conclusions and Consequences for a Hypothesis Test
• The goal of any hypothesis-testing is to make a decision based on sample information: whether to reject H one of two types of error.
0 in favor of H a
we make
• A Type I error occurs if we reject H
0 when it is true. The probability of committing a Type I error is denoted by
(also called significance level)
• A Type II error occurs if we do not reject H
0 when it is false. The probability of committing a Type II error is denoted by
.
25
Chapter 8
(continued 1)
]
8.4 Test statistics and rejection regions
• The test statistic is a sample ststistic, upon which the decision concerning the null and alternative hypotheses is based .
• The rejection region is the set of possible values of the test statistic for which the null hypotheses will be rejected.
• Steps for testing hypothesis
• Critical value =boundary value of the rejection region
8.5 Summary
8.6 Exercises
26
Chapter 9. Applications of Hypothesis Testing
9.1 Diagnosing a hypothesis test
9.2 Hypothesis test about a population mean
9.3 Hypothesis test about a population proportion
9.4 Hypothesis tests about the difference between two population means
9.5 Hypothesis tests about the difference between two proportions
9.6 Hypothesis test about a population variance
9.7 Hypothesis test about the ratio of two population variances
9.8 Summary
9.9 Exercises
27
Chapter 9
(continued 1)
]
9.2 Hypothesis test about a population mean
• Large- sample test (n>=30):
– the sampling distribution of is approximately normal and s is a good approximation of
.
– Procedure for large- sample test
• Small- sample test :
– Assumption: the population ha aaprox. Normal distribution.
– Procedure for small- sample test (using t-distribution)\
9.3 Hypothesis test about a population proportion
Large- sample test
9.4 Hypothesis tests about the difference between two population means
•
• Large- sample test :
– Assumptions: n
1
>=30, n
2
>=30; samples are selected randomly and independently from the populations
28
Small- sample test
Chapter 9
(continued 2)
9.5 Hypothesis tests about the difference between two proportions:
Assumptions, Procedure
9.6 Hypothesis test about a population variance
– Assumption: the population has an approx. nornal distr.
– Procudure using chi-square distribution
9.7 Hypothesis test about the ratio of two population variances (optional)
– Assumptions: Populations has approx. nornal distr., random samples are independent.
– Procudure using F- distribution
29
Chapter 10. Categorical Data Analysis and Analysis of
Variance
10.1 Introduction
10.2 Tests of goodness of fit
10.3 The analysis of contingency tables
10.4 Contingency tables in statistical software packages
10.5 Introduction to analysis of variance
10.6 Design of experiments
10.7 Completely randomized designs
10.8 Randomized block designs
10.9 Multiple comparisons of means and confidence regions
10.10 Summary
10.11 Exercises
30
]
Chapter 10
(continued 1)
10.1 Introduction
10.2 Tests of goodness -of- fit
– Purpose : to test for a dependence on a qualitative variable that allow for more than two categorires for a response.Namely, it test there is a significant difference between observed frequency distribution and a theoretical frequency distribution .
– Procedure for a Chi-square goodness -of- fit test
10.3 The analysis of contingency tables
– Purpose :to determine whether a dependence exists between to qualitative variables
– Procedure for a Chi-square Test for independence of two directions of Classification
10.4 Contingency tables in statistical software packages
31
Chapter 10
(continued 2)
10.5 Introduction to analysis of variance
Purpose: Comparison of more than two means
10.6 Design of experiments
• Concepts of experiment, design of the experiment, response variable, factor, treatment
• Concepts of Between-sample variation, Within-sample variation
10.7 Completely randomized designs
• This design involves a comparison of the means of k treatments, based on independent random samples of n
1
, n
2
,…, n k from populations.
observations drawn
• Assumptions: All k populations are normal, have equal variances
• F-test for comparing k population means
10.8 Randomized block designs
• Concept of randomized block design
• Tests to compare k Treatment and b Block Means
• 10.9 Multiple comparisons of means and confidence
32
Chapter 11. Simple Linear regression and correlation
11.1 Introduction: Bivariate relationships
11.2 Simple Linear regression: Assumptions
11.3 Estimating A and B: the method of least squares
11.4 Estimating
2
11.5 Making inferences about the slope, B
11.6. Correlation analysis
11.7 Using the model for estimation and prediction
11.8. Simple Linear Regression: An Overview Example
11.9 Exercises
33
•[
Chapter 11
(continued 1)
11.1 Introduction: Bivariate relationships
• Subject is to determine the relationship between two variables.
• Types of relationships: direct and inverse
• Scattergram
11.2 Simple Linear regression: Assumptions
• a simple linear regression model y = A + B x + e
• assumptions required for a linear regression model: E(e) = 0, e is normal,
2 is equal a constant for all value of x.
11.3 Estimating A and B: the method of least squares
• the least squares estimators a and b , formula for a and b
11.4 Estimating
2
• Formula for s 2 , an estimator for
2
• interpretation of s, the estimated standard deviation of e
34
Chapter 11
(continued 2)
•[
]
11.5 Making inferences about the slope, B
• Problem about making an inference of the population regression line
E(y)=A+Bx based on the sample regression line y^=a+bx
• Sampling distribution of the least square estimator of slope b
• Test of the utility of the model: H
0:
B =0 against H
• A (1
) 100% Confidence interval for B a
: B
0 or B>0, B<0
11.6. Correlation analysis
• Is the statistical tool for describing the degree to which one variable is linearly related to another.
• The coefficient of correlation r is a measure of the strength of the linear relationship between two variables
• The coefficient of determination
11.7 Using the model for estimation and prediction
• A (1
) 100% Confidence interval for the mean value of y for x = x p
• A (1
) 100% Confidence interval for an individual y for x = x p
11.8. Simple Linear Regression: An Example
35
Chapter 12. Multiple regression
12.1. Introduction: the general linear model
12.2 Model assumptions
12.3 Fitting the model: the method of least squares
12.4 Estimating
2
12.5 Estimating and testing hypotheses about the B parameters
12.6. Checking the utility of a model
12.7. Using the model for estimating and prediction
12.8 Multiple linear regression: An overview example
12.8. Model building: interaction models
12.9. Model building: quadratic models
12.10 Exercises
36
]
Chapter 12
(continued 1)
12.1. Introduction: the general linear model y = B
0
+ B
1 x
1
+ ... + B k x k
+ e , where y - dependent., x independent variables, e - random error.
1
, x
2
, ..., x k
-
12.2 Model assumptions
– For any given set of values x
1
, x
2
, ..., x k
, the random error e has a normal probability distribution with the mean equal 0 and variance equal
2 .
– The random errors are independent
.
12.3 Fitting the model: the method of least squares
Least square prediction equation: y^= b
0
12.4 Estimating
2
+b
1 x
1
+….+ b k x k
12.5 Estimating and testing hypotheses about the B parameters
• Sampling distributions of b
0
, b
1
, ..., b k
• A (1
) 100% Confidence interval for B i
(i =0, 1,.., k)
• Test of an individual parameter coefficient B i
37
Chapter 12
(continued 1)
•[
]
12.6. Checking the utility of a model
• Finding a measure of how well a linear model fits a set of data: the multiple coefficient of determination
• testing the overall utility of the model
12.7. Using the model for estimating and prediction
• A (1
) 100% confidence interval for the mean value of y for a given x
• A (1
) 100% confidence interval for an individual y for for a given x
12.8 Multiple linear regression: An overview example
12.8. Model building: interaction models
• Interaction model with two independent variables: E(y) = B
0
B
2 x
2
+ B
3 x
1 x
2
• procedure to build an interaction model
+ B
1 x
1
+
12.9. Model building: quadratic models
• Quadratic model in a single variable: E(y) = B
0
• procedure to build a quadratic model
+ B
1 x + B
2 x 2
38
Chapter 13. Nonparametric statistics
13.1. Introduction
• Situations where t and F test are unsuitable
• What do nonparametric methods use?
13.2. The sign test for a single population
• Purpose: to test hypotheses about median of any populations
• Procedure for the sign test for a population median
• Sign test based on a large sample (n>=10)
13.3 Comparing two populations based on independent random samples:Wilcoxon rank sum test
• Nonparametric test about the difference between two populations is the test to detect whether distribution 1 is shifted to the right of distribution 2 or vice versa.
• wilcoxon rank sum test for a shift in population locations
• The case of large samples (n
1
10, n
2
10)
39
Chapter 13
(continued 1)
•[
13.4. Comparing two populations based on matched pairs
• Wilcoxon signed ranks test for a shift in population locations
• Wilcoxon signed ranks test for large samples (n
25)
13.5. Comparing populations using a completely randomized design: The Kruskal-Wallis H test
• The Kruskal-Wallis H test is the nonparam. Equivalent of ANOVA F test when the assumptions that populations are normally distributed with common variance are not satisfied.
• The Kruskal-Wallis H test for comparing k population probability distributions
13.6 Rank Correlation: Spearman’s r s statistic
• Is stastistic developed to measure and to test for correlation between two random variables .
• Formula for computing Spearman’s rank correlation coefficient rs
• Spearman’s nonparametric test for rank correlation
40
13.7 Exercises
