EMPIRICAL FREQUENCY DISTRIBUTION

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "EMPIRICAL FREQUENCY DISTRIBUTION"

Transcription

1 INTRODUCTION TO MEDICAL STATISTICS: Mirjana Kujundžić Tiljak EMPIRICAL FREQUENCY DISTRIBUTION observed data DISTRIBUTION - described by mathematical models 2 1

2 when some empirical distribution approximates a particular probability distribution theoretical knowledge of that distribution could be used answer questions about data evaluation of probabilities is required 3 PROBABILITY (P) measures uncertainty measures the chance of a given event occurring 0 P 1 P = 0 event cannot occur P = 1 event must occur Q = 1-P probability of the complementary event (the event not occurring) 4 2

3 PROBABILITY (P) Various approaches in probability calculations: Subjective personal degree of belief that the event will occur (e.g. the world sill come to an end in the year 2050) Frequentist the proportion of times the event would occur if the experiment will be repeated a large number of times (e.g. the number of times we would get a head") A priori requires knowledge of the theoretical model probability distribution which describes the probabilities of all possible outcomes of the experiment (e.g. genetic theory allows us to describe the probability distribution for eye color in a baby born t a blue-eyed women and brown-eyed man by initially specifying all possible genotypes of eye color in the baby and their probabilities) 5 PROBABILITY (P) The addition rule: if two events (A and B) are mutually exclusive the probability that either one or the other occurs (A or B) is equal to the sum of their probabilities Prob (A or B) = Prob (A) + Prob (B) The multiplication rule: if two events (A and B) are independent the probability that both events occur (A and B) is equal to the product of the probability of each Prob (A and B) = Prob (A) Prob (B) 6 3

4 RANDOM VARIABLES random variable a quantity that can take any one of a set of mutally excluseve values with a given probability discrete or discontinuous random variable = numerical values are integer E.g. number of children in family 0, 1, 2, 3, k continuus random variable = numerical values are real numbers E.g. body weight 72,35 kg, blood glucose level 7,2 mmol/l 7 PROBABILITY DISTRIBUTION Probability distribution shows the probabilities of all possible values of the random variable a theoretical distribution that is expressed mathematically has a mean and variance that are analogous to those of and empirical distribution parameters summary measures (e.g. mean, variance) characterizing that distribution are estimated in the sample by relevant statistics depending on whether the random variable is discrete or continuous the probability distribution can be either discrete or continuous 8 4

5 PROBABILITY DISCRETE (Binomial, Poisson) the probability can be derived corresponding to every possible value of the random variable the sum of all such probabilitis is 1 9 PROBABILITY CONTINUOUS (Normal, Chi-squared, t, F) the probability of the random variable, x, taking values in certain ranges, could be derived if the horizontal axis represents the values of x the curve from the equation of the distribution could be drawn (= probability density function) Total area under the curve = 1 represents the probability of all possible events Probability that x lies between two limits is equal to the area under the curve between these values 10 5

6 PROBABILITY Probability that x lies between two limits? 11 PROBABILITY Probability that x lies between two limits? 12 6

7 THE NORMAL (GAUSSIAN) DISTRIBUTION one of the most important distributions in statistics german mathematician C.F. Gauss the most biological measurements follow normal distribution it is used in many analytical models 13 THE NORMAL (GAUSSIAN) DISTRIBUTION Probability density function: f (x) = (1/σ 2π) e a a = -1/2 ((x-µ)/σ)

8 THE NORMAL (GAUSSIAN) DISTRIBUTION Completely described by two parameters: - mean (µ ) -variance(σ 2 ) X~ N (µ,σ 2 ) 15 THE NORMAL (GAUSSIAN) DISTRIBUTION 16 8

9 THE NORMAL (GAUSSIAN) DISTRIBUTION normal distribution curve: area under curve = 1 bell-shaped (unimodal= symmetrical about its mean apsolute maximum for x = µ shifted to the right if the mean is increased and to the left if the mean is decreased (assuming constant variance) flattened as the variance is increased but becomes more peaked as the variance is decreased (for a ficed mean) 17 THE NORMAL (GAUSSIAN) DISTRIBUTION the mean and median and mode of a Normal distribution are equal the probability (P) that a normally distributed random variable, x, with mean, µ, and standard deviation, σ, lies between: (µ - σ) and (µ + σ) = 0,68 (µ σ) and (µ σ) = 0.95 (µ 2.58σ) and (µ σ) = 0.99 these intervals may be used to define reference intervals 18 9

10 THE NORMAL (GAUSSIAN) DISTRIBUTION changing µ, constant σ: 19 THE NORMAL (GAUSSIAN) DISTRIBUTION changing µ, constant σ: 20 10

11 THE NORMAL (GAUSSIAN) DISTRIBUTION changing σ, constant µ: 21 THE NORMAL (GAUSSIAN) DISTRIBUTION changing σ, constant µ: 22 11

12 THE NORMAL (GAUSSIAN) DISTRIBUTION changing σ, constant µ: 23 THE STANDARD NORMAL DISTRIBUTION transformation of original value (x) to Standardized Normal Deviate (SND) (z i ): z i = (x 1 - µ)/σ sample: = random variable that has a Standard Normal distribution z i = (x 1 - x)/s mean (µ) = 0; variance (σ 2 ) = 1; N (0,1) 24 12

13 THE STANDARD NORMAL DISTRIBUTION X 1 Z 1 X 2 Z 2 X 3 Z 3 X n Z n, s =?, s z =? 25 THE STANDARD NORMAL DISTRIBUTION X 1 Z 1 X 2 Z 2 X 3 Z 3 X n Z n, s =0, s z =

14 THE STANDARD NORMAL DISTRIBUTION X 1 Z 1 X 2 Z 2 X 3 Z 3 X n Z n, s =0, s z =1 Z~N(0,1) 27 THE STANDARD NORMAL DISTRIBUTION 28 14

15 THE STANDARD NORMAL DISTRIBUTION 29 THE STANDARD NORMAL DISTRIBUTION 30 15

16 THE STANDARD NORMAL DISTRIBUTION 31 THE STANDARD NORMAL DISTRIBUTION 32 16

17 THE STUDENT S t-distribution W.S. Gossett (pseudonym Student) parameter that characterizes the t-distribution = the degrees of freedom Similar shape as normal distribution (more spread out with longer tails) as the degrees of freedom increase its shape approaches Normality Useful for calculating confidence intervals for testing hypotheses about one or two means 33 THE STUDENT S t-distribution 34 17

18 THE CHI-SQUARE (χ 2 ) DISTRIBUTION a right skewed distribution taking positive values characterized by its degrees of freedom its shape depends on the degrees of freedom it becomes more symmetrical and approaches Normality as they increase useful for analysing categorical data 35 THE CHI-SQUARE (χ 2 ) DISTRIBUTION 36 18

19 THE F-DISTRIBUTION skewed to the right defined by a ratio the distribution of a ratio of two estimated variances calculated from Normal dana approximates the F-distritution characterized by degrees of freedom of the numerator and the denominator of the ratio useful for comparing two variances, and more than two means using the analysis of variance 37 THE LOGNORMAL DISTRIBUTION the probability distribution of a random variable whose log (to base 10 or e) follows the Normal distribution highly skewed to the right logs of row data skewed to the right an empirical distribution that is nearly Normal = data approximate Log-normal distribution geometric mean = a summary measure of location 38 19

20 THE LOGNORMAL DISTRIBUTION 39 THE BINOMIAL DISTRIBUTION theoretical distribution for discrete random variable definition: Jacob Bernuolli, two outcomes: success i failure n events E.g. n = 100 unrelated women undergoing IVF outcome = success (pregnancy) or failure 40 20

21 THE BINOMIAL DISTRIBUTION Two parameters that describe the Binomial distribution: n = number of indivudial in the sample (or repetitions of a trial) π = the true probability of success for each individual (or in each trial) X~B(n,p) 41 THE BINOMIAL DISTRIBUTION Mean = nπ (the value for the random variable that we expect if we look at n individuals, or repeat the trial n times) Variance = nπ (1- π) small n the distribution is skewed to the right if π <0.5 the distribution is skewed to dhe right if π >

22 THE BINOMIAL DISTRIBUTION the distribution becomes more symmetrical as the sample size increases and approximates to the Normal distribution if both nπ and nπ (1 π) are greater than 5 the properties of the Binomial distribution could be use when making inferences about proportions the Normal approximation of the Binomial distribution when analyzing proportions is often used 43 THE BINOMIAL DISTRIBUTION Example: gene recombination Chromosomal locus: 2 allels: A and a p = probability of A Q = 1 p = probability of a P(A) = p, P(a) = q, (p+q = 1) 44 22

23 THE BINOMIAL DISTRIBUTION conception outcame space:{aa, Aa, aa} P(AA) = P(A) * P(A)= p 2 P(aa) = P(a) * P(a) = q 2 P(Aa) = P(A) * P(a) = pq P(aA) = P(a) * P(A)= qp 1,0 2pq p 2 + 2pq + q 2 = (p+q) 2 = 1 2 = 1 45 THE BINOMIAL DISTRIBUTION 46 23

24 THE BINOMIAL DISTRIBUTION Example probability of genotypes: frequency of gene A = 0,33 frequency of gene a = 0,67 (p+q) 2 = (0,33 + 0,67) 2 = 0, * 0,33 * 0,67 + 0,67 2 P (AA)= 0,33 2 = P (Aa) = 0,33 * 0,67 = 0,2211 P (aa) = 0,67 * 0,33 = 0,2211 P (aa) = 0,67 2 = 0, THE BINOMIAL DISTRIBUTION Graphical presentatnion probabilities of different genotypes 0,5 0,45 0,4 0,35 0,3 P 0,25 0,2 0,15 0,1 0,05 0 AA Aa aa 48 24

25 THE BINOMIAL DISTRIBUTION Example death outcome as binomial distribution: Letality od neke bolesti = 0,30..(30/100) Survival probability = 0,70 n = 5 Binom: (0,30 + 0,70) 5 Number of death examinees Binom Probability 5 (everybody) (nobody) P 5 5p 4 q 10p 3 q 2 10p 2 q 3 5pq 4 q 5 0, , , , , ,16807 Total 1, THE POISSON DISTRIBUTION Poisson (begining of XIX century) the Poisson random variable = the count or the number of events that occur independently and randomly in time or space at some average rate, µ (0 and all positive integers) example: the number of hospital admissions per day typically follows the Poisson distribution use of the Poisson cistribution to calculate the probability of a certain number of admissions on any particular day 50 25

26 THE POISSON DISTRIBUTION Mean (average rate, µ) = the parameter that describes the Poisson distribution The mean equals the variance in the Poisson distribution Unimodal curve, right skewed if the mean is small, but becomes more symmetrical as the mean increases, when it approximates n Normal distribution 51 26

Exploratory Data Analysis

Exploratory Data Analysis Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction

More information

An Introduction to Basic Statistics and Probability

An Introduction to Basic Statistics and Probability An Introduction to Basic Statistics and Probability Shenek Heyward NCSU An Introduction to Basic Statistics and Probability p. 1/4 Outline Basic probability concepts Conditional probability Discrete Random

More information

PROBABILITY AND SAMPLING DISTRIBUTIONS

PROBABILITY AND SAMPLING DISTRIBUTIONS PROBABILITY AND SAMPLING DISTRIBUTIONS SEEMA JAGGI AND P.K. BATRA Indian Agricultural Statistics Research Institute Library Avenue, New Delhi - 0 0 seema@iasri.res.in. Introduction The concept of probability

More information

Lecture 8. Confidence intervals and the central limit theorem

Lecture 8. Confidence intervals and the central limit theorem Lecture 8. Confidence intervals and the central limit theorem Mathematical Statistics and Discrete Mathematics November 25th, 2015 1 / 15 Central limit theorem Let X 1, X 2,... X n be a random sample of

More information

Confidence intervals, t tests, P values

Confidence intervals, t tests, P values Confidence intervals, t tests, P values Joe Felsenstein Department of Genome Sciences and Department of Biology Confidence intervals, t tests, P values p.1/31 Normality Everybody believes in the normal

More information

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as... HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

More information

Section 5 Part 2. Probability Distributions for Discrete Random Variables

Section 5 Part 2. Probability Distributions for Discrete Random Variables Section 5 Part 2 Probability Distributions for Discrete Random Variables Review and Overview So far we ve covered the following probability and probability distribution topics Probability rules Probability

More information

Chapter 4. Probability and Probability Distributions

Chapter 4. Probability and Probability Distributions Chapter 4. robability and robability Distributions Importance of Knowing robability To know whether a sample is not identical to the population from which it was selected, it is necessary to assess the

More information

STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables

STT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random

More information

12.5: CHI-SQUARE GOODNESS OF FIT TESTS

12.5: CHI-SQUARE GOODNESS OF FIT TESTS 125: Chi-Square Goodness of Fit Tests CD12-1 125: CHI-SQUARE GOODNESS OF FIT TESTS In this section, the χ 2 distribution is used for testing the goodness of fit of a set of data to a specific probability

More information

CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction

CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous

More information

Section 7.2 Confidence Intervals for Population Proportions

Section 7.2 Confidence Intervals for Population Proportions Section 7.2 Confidence Intervals for Population Proportions 2012 Pearson Education, Inc. All rights reserved. 1 of 83 Section 7.2 Objectives Find a point estimate for the population proportion Construct

More information

STAT 155 Introductory Statistics. Lecture 5: Density Curves and Normal Distributions (I)

STAT 155 Introductory Statistics. Lecture 5: Density Curves and Normal Distributions (I) The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 5: Density Curves and Normal Distributions (I) 9/12/06 Lecture 5 1 A problem about Standard Deviation A variable

More information

4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions 4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

More information

Descriptive Statistics. Understanding Data: Categorical Variables. Descriptive Statistics. Dataset: Shellfish Contamination

Descriptive Statistics. Understanding Data: Categorical Variables. Descriptive Statistics. Dataset: Shellfish Contamination Descriptive Statistics Understanding Data: Dataset: Shellfish Contamination Location Year Species Species2 Method Metals Cadmium (mg kg - ) Chromium (mg kg - ) Copper (mg kg - ) Lead (mg kg - ) Mercury

More information

Chapter 2, part 2. Petter Mostad

Chapter 2, part 2. Petter Mostad Chapter 2, part 2 Petter Mostad mostad@chalmers.se Parametrical families of probability distributions How can we solve the problem of learning about the population distribution from the sample? Usual procedure:

More information

Lecture 8: More Continuous Random Variables

Lecture 8: More Continuous Random Variables Lecture 8: More Continuous Random Variables 26 September 2005 Last time: the eponential. Going from saying the density e λ, to f() λe λ, to the CDF F () e λ. Pictures of the pdf and CDF. Today: the Gaussian

More information

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur

Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce

More information

The Normal distribution

The Normal distribution The Normal distribution The normal probability distribution is the most common model for relative frequencies of a quantitative variable. Bell-shaped and described by the function f(y) = 1 2σ π e{ 1 2σ

More information

6.4 Normal Distribution

6.4 Normal Distribution Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under

More information

Inferential Statistics

Inferential Statistics Inferential Statistics Sampling and the normal distribution Z-scores Confidence levels and intervals Hypothesis testing Commonly used statistical methods Inferential Statistics Descriptive statistics are

More information

MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters

MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters MAT140: Applied Statistical Methods Summary of Calculating Confidence Intervals and Sample Sizes for Estimating Parameters Inferences about a population parameter can be made using sample statistics for

More information

The basics of probability theory. Distribution of variables, some important distributions

The basics of probability theory. Distribution of variables, some important distributions The basics of probability theory. Distribution of variables, some important distributions 1 Random experiment The outcome is not determined uniquely by the considered conditions. For example, tossing a

More information

CHI-SQUARE: TESTING FOR GOODNESS OF FIT

CHI-SQUARE: TESTING FOR GOODNESS OF FIT CHI-SQUARE: TESTING FOR GOODNESS OF FIT In the previous chapter we discussed procedures for fitting a hypothesized function to a set of experimental data points. Such procedures involve minimizing a quantity

More information

Chapter Additional: Standard Deviation and Chi- Square

Chapter Additional: Standard Deviation and Chi- Square Chapter Additional: Standard Deviation and Chi- Square Chapter Outline: 6.4 Confidence Intervals for the Standard Deviation 7.5 Hypothesis testing for Standard Deviation Section 6.4 Objectives Interpret

More information

Descriptive Statistics. Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion

Descriptive Statistics. Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Descriptive Statistics Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Statistics as a Tool for LIS Research Importance of statistics in research

More information

Summary of Probability

Summary of Probability Summary of Probability Mathematical Physics I Rules of Probability The probability of an event is called P(A), which is a positive number less than or equal to 1. The total probability for all possible

More information

Using pivots to construct confidence intervals. In Example 41 we used the fact that

Using pivots to construct confidence intervals. In Example 41 we used the fact that Using pivots to construct confidence intervals In Example 41 we used the fact that Q( X, µ) = X µ σ/ n N(0, 1) for all µ. We then said Q( X, µ) z α/2 with probability 1 α, and converted this into a statement

More information

Comment on the Tree Diagrams Section

Comment on the Tree Diagrams Section Comment on the Tree Diagrams Section The reversal of conditional probabilities when using tree diagrams (calculating P (B A) from P (A B) and P (A B c )) is an example of Bayes formula, named after the

More information

5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.

5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives. The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution

More information

An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476<p<0.

An interval estimate (confidence interval) is an interval, or range of values, used to estimate a population parameter. For example 0.476<p<0. Lecture #7 Chapter 7: Estimates and sample sizes In this chapter, we will learn an important technique of statistical inference to use sample statistics to estimate the value of an unknown population parameter.

More information

Sampling and Hypothesis Testing

Sampling and Hypothesis Testing Population and sample Sampling and Hypothesis Testing Allin Cottrell Population : an entire set of objects or units of observation of one sort or another. Sample : subset of a population. Parameter versus

More information

Week 4: Standard Error and Confidence Intervals

Week 4: Standard Error and Confidence Intervals Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.

More information

PROBABILITIES AND PROBABILITY DISTRIBUTIONS

PROBABILITIES AND PROBABILITY DISTRIBUTIONS Published in "Random Walks in Biology", 1983, Princeton University Press PROBABILITIES AND PROBABILITY DISTRIBUTIONS Howard C. Berg Table of Contents PROBABILITIES PROBABILITY DISTRIBUTIONS THE BINOMIAL

More information

Binomial Distribution n = 20, p = 0.3

Binomial Distribution n = 20, p = 0.3 This document will describe how to use R to calculate probabilities associated with common distributions as well as to graph probability distributions. R has a number of built in functions for calculations

More information

Goodness of Fit. Proportional Model. Probability Models & Frequency Data

Goodness of Fit. Proportional Model. Probability Models & Frequency Data Probability Models & Frequency Data Goodness of Fit Proportional Model Chi-square Statistic Example R Distribution Assumptions Example R 1 Goodness of Fit Goodness of fit tests are used to compare any

More information

BASIC STATISTICAL METHODS FOR GENOMIC DATA ANALYSIS

BASIC STATISTICAL METHODS FOR GENOMIC DATA ANALYSIS BASIC STATISTICAL METHODS FOR GENOMIC DATA ANALYSIS SEEMA JAGGI Indian Agricultural Statistics Research Institute Library Avenue, New Delhi-110 012 seema@iasri.res.in Genomics A genome is an organism s

More information

MBA 611 STATISTICS AND QUANTITATIVE METHODS

MBA 611 STATISTICS AND QUANTITATIVE METHODS MBA 611 STATISTICS AND QUANTITATIVE METHODS Part I. Review of Basic Statistics (Chapters 1-11) A. Introduction (Chapter 1) Uncertainty: Decisions are often based on incomplete information from uncertain

More information

The Normal Curve. The Normal Curve and The Sampling Distribution

The Normal Curve. The Normal Curve and The Sampling Distribution Discrete vs Continuous Data The Normal Curve and The Sampling Distribution We have seen examples of probability distributions for discrete variables X, such as the binomial distribution. We could use it

More information

Chi-Square Test. Contingency Tables. Contingency Tables. Chi-Square Test for Independence. Chi-Square Tests for Goodnessof-Fit

Chi-Square Test. Contingency Tables. Contingency Tables. Chi-Square Test for Independence. Chi-Square Tests for Goodnessof-Fit Chi-Square Tests 15 Chapter Chi-Square Test for Independence Chi-Square Tests for Goodness Uniform Goodness- Poisson Goodness- Goodness Test ECDF Tests (Optional) McGraw-Hill/Irwin Copyright 2009 by The

More information

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as...

HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1. used confidence intervals to answer questions such as... HYPOTHESIS TESTING (ONE SAMPLE) - CHAPTER 7 1 PREVIOUSLY used confidence intervals to answer questions such as... You know that 0.25% of women have red/green color blindness. You conduct a study of men

More information

Statistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013

Statistics I for QBIC. Contents and Objectives. Chapters 1 7. Revised: August 2013 Statistics I for QBIC Text Book: Biostatistics, 10 th edition, by Daniel & Cross Contents and Objectives Chapters 1 7 Revised: August 2013 Chapter 1: Nature of Statistics (sections 1.1-1.6) Objectives

More information

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes together with the number of data values from the set that

More information

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions...

MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo. 3 MT426 Notebook 3 3. 3.1 Definitions... 3. 3.2 Joint Discrete Distributions... MT426 Notebook 3 Fall 2012 prepared by Professor Jenny Baglivo c Copyright 2004-2012 by Jenny A. Baglivo. All Rights Reserved. Contents 3 MT426 Notebook 3 3 3.1 Definitions............................................

More information

H + T = 1. p(h + T) = p(h) x p(t)

H + T = 1. p(h + T) = p(h) x p(t) Probability and Statistics Random Chance A tossed penny can land either heads up or tails up. These are mutually exclusive events, i.e. if the coin lands heads up, it cannot also land tails up on the same

More information

Fairfield Public Schools

Fairfield Public Schools Mathematics Fairfield Public Schools AP Statistics AP Statistics BOE Approved 04/08/2014 1 AP STATISTICS Critical Areas of Focus AP Statistics is a rigorous course that offers advanced students an opportunity

More information

Lecture 5 : The Poisson Distribution

Lecture 5 : The Poisson Distribution Lecture 5 : The Poisson Distribution Jonathan Marchini November 10, 2008 1 Introduction Many experimental situations occur in which we observe the counts of events within a set unit of time, area, volume,

More information

4. Introduction to Statistics

4. Introduction to Statistics Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

More information

A review of the portions of probability useful for understanding experimental design and analysis.

A review of the portions of probability useful for understanding experimental design and analysis. Chapter 3 Review of Probability A review of the portions of probability useful for understanding experimental design and analysis. The material in this section is intended as a review of the topic of probability

More information

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

More information

Chapter 10 Monte Carlo Methods

Chapter 10 Monte Carlo Methods 411 There is no result in nature without a cause; understand the cause and you will have no need for the experiment. Leonardo da Vinci (1452-1519) Chapter 10 Monte Carlo Methods In very broad terms one

More information

Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures.

Introduction to Hypothesis Testing. Point estimation and confidence intervals are useful statistical inference procedures. Introduction to Hypothesis Testing Point estimation and confidence intervals are useful statistical inference procedures. Another type of inference is used frequently used concerns tests of hypotheses.

More information

DATA INTERPRETATION AND STATISTICS

DATA INTERPRETATION AND STATISTICS PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE

More information

fifty Fathoms Statistics Demonstrations for Deeper Understanding Tim Erickson

fifty Fathoms Statistics Demonstrations for Deeper Understanding Tim Erickson fifty Fathoms Statistics Demonstrations for Deeper Understanding Tim Erickson Contents What Are These Demos About? How to Use These Demos If This Is Your First Time Using Fathom Tutorial: An Extended Example

More information

Mendelian Inheritance & Probability

Mendelian Inheritance & Probability Mendelian Inheritance & Probability (CHAPTER 2- Brooker Text) January 31 & Feb 2, 2006 BIO 184 Dr. Tom Peavy Problem Solving TtYy x ttyy What is the expected phenotypic ratio among offspring? Tt RR x Tt

More information

Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization. Learning Goals. GENOME 560, Spring 2012

Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization. Learning Goals. GENOME 560, Spring 2012 Why Taking This Course? Course Introduction, Descriptive Statistics and Data Visualization GENOME 560, Spring 2012 Data are interesting because they help us understand the world Genomics: Massive Amounts

More information

Chi-Square Distribution. is distributed according to the chi-square distribution. This is usually written

Chi-Square Distribution. is distributed according to the chi-square distribution. This is usually written Chi-Square Distribution If X i are k independent, normally distributed random variables with mean 0 and variance 1, then the random variable is distributed according to the chi-square distribution. This

More information

Sampling Distributions

Sampling Distributions Sampling Distributions You have seen probability distributions of various types. The normal distribution is an example of a continuous distribution that is often used for quantitative measures such as

More information

4: Probability. What is probability? Random variables (RVs)

4: Probability. What is probability? Random variables (RVs) 4: Probability b binomial µ expected value [parameter] n number of trials [parameter] N normal p probability of success [parameter] pdf probability density function pmf probability mass function RV random

More information

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS

UNIT I: RANDOM VARIABLES PART- A -TWO MARKS UNIT I: RANDOM VARIABLES PART- A -TWO MARKS 1. Given the probability density function of a continuous random variable X as follows f(x) = 6x (1-x) 0

More information

16. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION

16. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION 6. THE NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTION It is sometimes difficult to directly compute probabilities for a binomial (n, p) random variable, X. We need a different table for each value of

More information

F. Farrokhyar, MPhil, PhD, PDoc

F. Farrokhyar, MPhil, PhD, PDoc Learning objectives Descriptive Statistics F. Farrokhyar, MPhil, PhD, PDoc To recognize different types of variables To learn how to appropriately explore your data How to display data using graphs How

More information

How to Conduct a Hypothesis Test

How to Conduct a Hypothesis Test How to Conduct a Hypothesis Test The idea of hypothesis testing is relatively straightforward. In various studies we observe certain events. We must ask, is the event due to chance alone, or is there some

More information

SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation

SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation SYSM 6304: Risk and Decision Analysis Lecture 3 Monte Carlo Simulation M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu September 19, 2015 Outline

More information

Chapter 3 RANDOM VARIATE GENERATION

Chapter 3 RANDOM VARIATE GENERATION Chapter 3 RANDOM VARIATE GENERATION In order to do a Monte Carlo simulation either by hand or by computer, techniques must be developed for generating values of random variables having known distributions.

More information

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression

More information

Lecture 7: Binomial Test, Chisquare

Lecture 7: Binomial Test, Chisquare Lecture 7: Binomial Test, Chisquare Test, and ANOVA May, 01 GENOME 560, Spring 01 Goals ANOVA Binomial test Chi square test Fisher s exact test Su In Lee, CSE & GS suinlee@uw.edu 1 Whirlwind Tour of One/Two

More information

2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table

2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table 2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations

More information

Basic Statistics. Probability and Confidence Intervals

Basic Statistics. Probability and Confidence Intervals Basic Statistics Probability and Confidence Intervals Probability and Confidence Intervals Learning Intentions Today we will understand: Interpreting the meaning of a confidence interval Calculating the

More information

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers)

Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) Probability and Statistics Vocabulary List (Definitions for Middle School Teachers) B Bar graph a diagram representing the frequency distribution for nominal or discrete data. It consists of a sequence

More information

Variables. Exploratory Data Analysis

Variables. Exploratory Data Analysis Exploratory Data Analysis Exploratory Data Analysis involves both graphical displays of data and numerical summaries of data. A common situation is for a data set to be represented as a matrix. There is

More information

15.0 More Hypothesis Testing

15.0 More Hypothesis Testing 15.0 More Hypothesis Testing 1 Answer Questions Type I and Type II Error Power Calculation Bayesian Hypothesis Testing 15.1 Type I and Type II Error In the philosophy of hypothesis testing, the null hypothesis

More information

Simple Linear Regression Inference

Simple Linear Regression Inference Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

More information

Statistics. Measurement. Scales of Measurement 7/18/2012

Statistics. Measurement. Scales of Measurement 7/18/2012 Statistics Measurement Measurement is defined as a set of rules for assigning numbers to represent objects, traits, attributes, or behaviors A variableis something that varies (eye color), a constant does

More information

Technology Step-by-Step Using StatCrunch

Technology Step-by-Step Using StatCrunch Technology Step-by-Step Using StatCrunch Section 1.3 Simple Random Sampling 1. Select Data, highlight Simulate Data, then highlight Discrete Uniform. 2. Fill in the following window with the appropriate

More information

2 GENETIC DATA ANALYSIS

2 GENETIC DATA ANALYSIS 2.1 Strategies for learning genetics 2 GENETIC DATA ANALYSIS We will begin this lecture by discussing some strategies for learning genetics. Genetics is different from most other biology courses you have

More information

Review of Random Variables

Review of Random Variables Chapter 1 Review of Random Variables Updated: January 16, 2015 This chapter reviews basic probability concepts that are necessary for the modeling and statistical analysis of financial data. 1.1 Random

More information

Chi Square Tests. Chapter 10. 10.1 Introduction

Chi Square Tests. Chapter 10. 10.1 Introduction Contents 10 Chi Square Tests 703 10.1 Introduction............................ 703 10.2 The Chi Square Distribution.................. 704 10.3 Goodness of Fit Test....................... 709 10.4 Chi Square

More information

Quantitative Methods for Finance

Quantitative Methods for Finance Quantitative Methods for Finance Module 1: The Time Value of Money 1 Learning how to interpret interest rates as required rates of return, discount rates, or opportunity costs. 2 Learning how to explain

More information

The Standard Normal distribution

The Standard Normal distribution The Standard Normal distribution 21.2 Introduction Mass-produced items should conform to a specification. Usually, a mean is aimed for but due to random errors in the production process we set a tolerance

More information

Probability Distributions

Probability Distributions Learning Objectives Probability Distributions Section 1: How Can We Summarize Possible Outcomes and Their Probabilities? 1. Random variable 2. Probability distributions for discrete random variables 3.

More information

Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4) Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

More information

Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics.

Business Statistics. Successful completion of Introductory and/or Intermediate Algebra courses is recommended before taking Business Statistics. Business Course Text Bowerman, Bruce L., Richard T. O'Connell, J. B. Orris, and Dawn C. Porter. Essentials of Business, 2nd edition, McGraw-Hill/Irwin, 2008, ISBN: 978-0-07-331988-9. Required Computing

More information

GCSE HIGHER Statistics Key Facts

GCSE HIGHER Statistics Key Facts GCSE HIGHER Statistics Key Facts Collecting Data When writing questions for questionnaires, always ensure that: 1. the question is worded so that it will allow the recipient to give you the information

More information

8. THE NORMAL DISTRIBUTION

8. THE NORMAL DISTRIBUTION 8. THE NORMAL DISTRIBUTION The normal distribution with mean μ and variance σ 2 has the following density function: The normal distribution is sometimes called a Gaussian Distribution, after its inventor,

More information

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution

More information

Mathematics. Probability and Statistics Curriculum Guide. Revised 2010

Mathematics. Probability and Statistics Curriculum Guide. Revised 2010 Mathematics Probability and Statistics Curriculum Guide Revised 2010 This page is intentionally left blank. Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction

More information

The Binomial Probability Distribution

The Binomial Probability Distribution The Binomial Probability Distribution MATH 130, Elements of Statistics I J. Robert Buchanan Department of Mathematics Fall 2015 Objectives After this lesson we will be able to: determine whether a probability

More information

Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement

Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement Chapter 4: Data & the Nature of Graziano, Raulin. Research Methods, a Process of Inquiry Presented by Dustin Adams Research Variables Variable Any characteristic that can take more than one form or value.

More information

MATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators...

MATH4427 Notebook 2 Spring 2016. 2 MATH4427 Notebook 2 3. 2.1 Definitions and Examples... 3. 2.2 Performance Measures for Estimators... MATH4427 Notebook 2 Spring 2016 prepared by Professor Jenny Baglivo c Copyright 2009-2016 by Jenny A. Baglivo. All Rights Reserved. Contents 2 MATH4427 Notebook 2 3 2.1 Definitions and Examples...................................

More information

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k.

REPEATED TRIALS. The probability of winning those k chosen times and losing the other times is then p k q n k. REPEATED TRIALS Suppose you toss a fair coin one time. Let E be the event that the coin lands heads. We know from basic counting that p(e) = 1 since n(e) = 1 and 2 n(s) = 2. Now suppose we play a game

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics Suppose following data have been collected (heights of 99 five-year-old boys) 117.9 11.2 112.9 115.9 18. 14.6 17.1 117.9 111.8 16.3 111. 1.4 112.1 19.2 11. 15.4 99.4 11.1 13.3 16.9

More information

Descriptive Statistics

Descriptive Statistics Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web

More information

Chapter Five: Paired Samples Methods 1/38

Chapter Five: Paired Samples Methods 1/38 Chapter Five: Paired Samples Methods 1/38 5.1 Introduction 2/38 Introduction Paired data arise with some frequency in a variety of research contexts. Patients might have a particular type of laser surgery

More information

THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7.

THE FIRST SET OF EXAMPLES USE SUMMARY DATA... EXAMPLE 7.2, PAGE 227 DESCRIBES A PROBLEM AND A HYPOTHESIS TEST IS PERFORMED IN EXAMPLE 7. THERE ARE TWO WAYS TO DO HYPOTHESIS TESTING WITH STATCRUNCH: WITH SUMMARY DATA (AS IN EXAMPLE 7.17, PAGE 236, IN ROSNER); WITH THE ORIGINAL DATA (AS IN EXAMPLE 8.5, PAGE 301 IN ROSNER THAT USES DATA FROM

More information

Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

More information

Consider a system that consists of a finite number of equivalent states. The chance that a given state will occur is given by the equation.

Consider a system that consists of a finite number of equivalent states. The chance that a given state will occur is given by the equation. Probability and the Chi-Square Test written by J. D. Hendrix Learning Objectives Upon completing the exercise, each student should be able: to determine the chance that a given state will occur in a system

More information

AP Statistics Solutions to Packet 2

AP Statistics Solutions to Packet 2 AP Statistics Solutions to Packet 2 The Normal Distributions Density Curves and the Normal Distribution Standard Normal Calculations HW #9 1, 2, 4, 6-8 2.1 DENSITY CURVES (a) Sketch a density curve that

More information

Data Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1

Data Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1 Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2011 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields

More information

Beta Distribution. Paul Johnson <pauljohn@ku.edu> and Matt Beverlin <mbeverlin@ku.edu> June 10, 2013

Beta Distribution. Paul Johnson <pauljohn@ku.edu> and Matt Beverlin <mbeverlin@ku.edu> June 10, 2013 Beta Distribution Paul Johnson and Matt Beverlin June 10, 2013 1 Description How likely is it that the Communist Party will win the net elections in Russia? In my view,

More information