BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I

Size: px
Start display at page:

Download "BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I"

Transcription

1 BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I

2 Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential statistical methods, with a focus on using MATLAB for implementation. The four modules are: Introduction and descriptive statistics Probability distributions Hypothesis testing Correlation and regression Each lecture will be supplemented with a MATLAB tutorial on the same topic We will work through part or all of the tutorial after reviewing the concepts; anything we don t get to should be reviewed outside of class! 2

3 Statistics a (very brief) introduction 1663 Natural and Political Observations upon the Bills of Mortality by John Graunt is published Motivated by the desire to base policy on demographic data 1700s Laplace introduces the normal distribution and regression via his study of astronomy 1800s Quetelet applies statistical analysis to human biology The central purpose of statistics is to learn more about some population of interest (e.g., all humans in the world) However, we very rarely, if ever, have access to every individual in the population! sample a subset of the entire population??? compilation of data about the entire population With a sample in hand, we seek to either summarize that data (using descriptive statistics) or use the data to make some prediction or statement about the population (using inferential statistics) 3

4 The Central Dogma of Statistics used to summarize data; (this is the focus for today) used to make inferences about the population

5 Dimensionality of data sets Univariate: measurements made on one variable per subject. This will be the focus for modules 1-3 Bivariate: measurement made on two variables per subject. Multivariate: measurement made on many variables per subject.

6 Types of descriptive statistics Central tendency measures: computed to give a center around which the measurements in the data are distributed (also called measures of location. (mean, median, mode, quartiles) Variation or variability measures: describe data spread or how far the measurements are away from the center. (variance, standard dev.) Relative standing measures: describe the relative position of specific measurements in the data. (percentiles)

7 Central tendency measures: mean The sample mean (a.k.a. average): To calculate the average x of a set of observations, add their values and divide by the number of observations: n x x n 1 x = = Σ x n n i i = 1

8 Central tendency measures: median Median the exact middle value Calculation If there are an odd number of observations, find the middle value If there are an even number of observations, find the mean of the middle two values Example: Median = ave(22,23) = 22.5 Age of students:

9 Which central tendency measure is better? In other words, which measure better approximates the center of a data set? Mean is best for symmetric distributions w/o outliers Median is useful for skewed distributions or data with (one-directional) outliers mean = median = 3 mean = median = 4

10 Scale: variance The sample variance is the average of squared deviations of values from the mean n s 2 = Σ(x i x) 2 i = 1 n 1 Square the deviations to get rid of the negatives The result is that the contribution to the variance increases as you go farther from the mean in either direction

11 Scale: standard deviation s = i n Σ(x i x) 2 = 1 n 1 Procedure to obtain the sample standard deviation: Score/measure observations (in the units that are meaningful, let s say m/s) Find the mean of the observations (m/s) Find each score s deviation from the mean (m/s) Square all those deviations (m/s) 2 Divide by n 1 (m/s) 2 (note that this is the variance) square root (m/s) now we have the starting units! Let s do a simple example problem!

12 Central tendency measures: mode The mode is the observation that takes place most frequently in a data set Unlike the mean or median, the mode is not necessarily unique the same maximum frequency may occur at different values. Based on the previous slide, is the mode a parametric statistic? (hint: remember that it is a measure of central tendency)

13 Scale: quartiles and IQR Q 2 is the same as the median The first quartile (Q 1 ) and third quartile (Q 3 ) are the medians of the data sets that would be created if all of the values below and above Q 2, respectively, were chosen. The interquartile range (IQR) is Q 3 Q 1

14 Quartiles example problem Find the three quartiles and IQR of the following two datasets: Q 1 = 4.5 Median = 11 Q 3 = 14.5 IQR = 10 Q 1 = 7 Median = 13 Q 3 = 20 IQR = 13 Note from this example that the 25% rule from the previous slide isn t precisely correct It is easiest to first insert the median the lower and upper halves from which to find Q 1 and Q 3 should then be obvious

15 Percentiles (aka quantiles) Generally, the n th percentile is a value such that n% of the observations fall at or below it: Q 1 = 25 th percentile Median = 50 th percentile Q 2 = 75 th percentile

16 Univariate data: histograms and bar plots What s the differences between a histogram and bar plot? Bar plot Used for categorical variables to show frequency or proportion in each category. Translate the data from frequency tables into a pictoral representation... Histogram Used to visualize distribution (shape, center, range, variation) of continuous variables bin size is important

17 Effect of bin size on histogram Simulated 1000 N(0,1) 1000 random numbers from the standard normal distribution with mean 0 and st. dev. 1

18 More on histograms What s the difference between a frequency histogram and a density histogram?

19 More on histograms What s the difference between a frequency histogram and a density histogram?

20 More on histograms So, for our roughly gaussian distribution from earlier, the density histogram looks like this: mean = relative frequency mean = median = median = observation

21 Stem and leaf plots

22 Box and whisker plots An outlier is a score either 1.5 IQR above the upper quartile or below the lower quartile

23 Example problem Two different classes take a quiz and gets the following scores. Class 1: 2, 4, 6, 8, 10, 12, 14 Class 2: 2, 2, 3, 8, 8, 10, 23 What the mean and median of each set? The same! Will making a box and whisker plot of each set of data give us a better picture of their distributions? (let s do the second one together)

24 Box plot procedure Steps to make our box plot: Find the median, Q1, Q3, and IQR Draw 3 horizontal lines, at Q1, median, and Q3 Draw the corresponding vertical lines to make the boxes Compute the lower inner fence (Q1 1.5*IQR) and the upper inner fence (Q *IQR) Draw a whisker downward from Q1 to lower inner fence or minimum, whichever comes first Draw a whisker upward from Q3 to upper inner fence or maximum, whichever comes first Compute the lower outer fence (Q1 3*IQR) and the upper outer fence (Q3 + 3*IQR) Mild outliers fall between the inner and outer fences, mark with O Extreme outliers fall outside outer fences, mark with *

25 Now let s switch over and do some work in MATLAB! 25

26 Probability Distributions

27 The Central Dogma of Statistics (this is the focus for today) i.e., the probability distribution

28 Probability distributions We ve discussed that data can be normally distributed (a.k.a. Gaussian or bell-shaped ) in fact, many reallife variables are, including: But what does this mean mathematically?

29 Probability distributions A probability distribution which can either be discrete or continuous is a table (discrete) or mathematical function (continuous) of one or more variables that describes the likelihood that any given value (discrete) or set of values (continuous) will occur Because the entire population is characterized, the main usefulness is in calculating the probability that certain values (discrete) or a range of values (continuous) will occur First, let s see examine a couple discrete cases (we ll then move to the continuous case)

30 What is the probability distribution of rolling a die? If all outcomes are equally likely (i.e., if the die is fair ), then: probability distribution: P(1) =? Note the total probability is 1! We use X (upper case) to denote an individual from the population For example, P(X = 2) = 1/6 x i P(x i ) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 If written as a function, we call it the probability mass function (pmf)

31 What is the probability distribution of a random number generator? Say you have a program (e.g., rand in MATLAB) that picks a real number between 0 and 1 (the uniform distribution ): f(x) 1 f(x) = 1; 0 x 1 0; otherwise x Since we still need our total area to equal 1, what must the value of f(x) be (i.e., at what y-axis value is the upper line in the graph)? f(x) is the probability density function (pdf) it is the continuous analog of the pmf. Here, we run into a problem: if x can be any real #, what must be the probability of observing a given value i.e., what is P(X = x) for any continuous distribution? Unlike in the discrete case, P(X = x) = 0 in the continuous case

32 What is the probability distribution of a random number generator (cont.)? In the continuous case, we instead care about the probability of a randomly selected variable X from the distribution being within a certain range of values 1 f(x) f(x) = 1; 0 x 1 0; otherwise x F(x) = x; 0 x 1 0; otherwise Look at f(x) above; if we want P(0.25 < X < 0.75), how can we evaluate this mathematically? Integrating the pdf gives the cumulative distribution function (cdf), or F(x), which is evaluated over the desired limits! Let s do an example what is P(0.25 < X < 0.75)?

33 The mean and variance of continuous random variables There are many different continuous probability distributions (here are a few examples we will see): Normal Uniform Exponential Parabolic Every distribution has a unique: Expected value: E(X) = μ a weighted average of all the possible values that this random variable can take on Variance: V(X) = σ 2 a measure of the spread, or the extent to which values in the distribution are dispersed If we know the pdf of a given distribution, we can calculate its mean and variance! 33

34 Expected value of random variables Let s re-visit our die problem; on average, what is the expected value of a roll, given the die goes from 1-6 (hint: it s not one of the integers)? Mathematically, how would you fill in the parentheses below to arrive at the same answer? E(X) = Σ ( )( ) How can we express the same concept in the continuous case? 6 i = 1 E(X) = x f(x) dx - Note that these limits will vary depending on the distribution, according to where f(x) is non-zero Let s try this out for our uniform distribution example and the generalized case!

35 References Lecture 2 Descriptive Statistics and Exploratory Data Analysis University of Washington School of Medicine. s/math_300/final/p14/default.html

Lecture 2: Descriptive Statistics and Exploratory Data Analysis

Lecture 2: Descriptive Statistics and Exploratory Data Analysis Lecture 2: Descriptive Statistics and Exploratory Data Analysis Further Thoughts on Experimental Design 16 Individuals (8 each from two populations) with replicates Pop 1 Pop 2 Randomly sample 4 individuals

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

Exploratory data analysis (Chapter 2) Fall 2011

Exploratory data analysis (Chapter 2) Fall 2011 Exploratory data analysis (Chapter 2) Fall 2011 Data Examples Example 1: Survey Data 1 Data collected from a Stat 371 class in Fall 2005 2 They answered questions about their: gender, major, year in school,

More information

Lecture 1: Review and Exploratory Data Analysis (EDA)

Lecture 1: Review and Exploratory Data Analysis (EDA) Lecture 1: Review and Exploratory Data Analysis (EDA) Sandy Eckel seckel@jhsph.edu Department of Biostatistics, The Johns Hopkins University, Baltimore USA 21 April 2008 1 / 40 Course Information I Course

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

STATS8: Introduction to Biostatistics. Data Exploration. Babak Shahbaba Department of Statistics, UCI

STATS8: Introduction to Biostatistics. Data Exploration. Babak Shahbaba Department of Statistics, UCI STATS8: Introduction to Biostatistics Data Exploration Babak Shahbaba Department of Statistics, UCI Introduction After clearly defining the scientific problem, selecting a set of representative members

More information

Geostatistics Exploratory Analysis

Geostatistics Exploratory Analysis Instituto Superior de Estatística e Gestão de Informação Universidade Nova de Lisboa Master of Science in Geospatial Technologies Geostatistics Exploratory Analysis Carlos Alberto Felgueiras cfelgueiras@isegi.unl.pt

More information

Descriptive statistics Statistical inference statistical inference, statistical induction and inferential statistics

Descriptive statistics Statistical inference statistical inference, statistical induction and inferential statistics Descriptive statistics is the discipline of quantitatively describing the main features of a collection of data. Descriptive statistics are distinguished from inferential statistics (or inductive statistics),

More information

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

How To Write A Data Analysis

How To Write A Data Analysis 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

Week 1. Exploratory Data Analysis

Week 1. Exploratory Data Analysis Week 1 Exploratory Data Analysis Practicalities This course ST903 has students from both the MSc in Financial Mathematics and the MSc in Statistics. Two lectures and one seminar/tutorial per week. Exam

More information

Exercise 1.12 (Pg. 22-23)

Exercise 1.12 (Pg. 22-23) Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.

More information

Center: Finding the Median. Median. Spread: Home on the Range. Center: Finding the Median (cont.)

Center: Finding the Median. Median. Spread: Home on the Range. Center: Finding the Median (cont.) Center: Finding the Median When we think of a typical value, we usually look for the center of the distribution. For a unimodal, symmetric distribution, it s easy to find the center it s just the center

More information

3: Summary Statistics

3: Summary Statistics 3: Summary Statistics Notation Let s start by introducing some notation. Consider the following small data set: 4 5 30 50 8 7 4 5 The symbol n represents the sample size (n = 0). The capital letter X denotes

More information

Data Exploration Data Visualization

Data Exploration Data Visualization Data Exploration Data Visualization What is data exploration? A preliminary exploration of the data to better understand its characteristics. Key motivations of data exploration include Helping to select

More information

Exploratory Data Analysis. Psychology 3256

Exploratory Data Analysis. Psychology 3256 Exploratory Data Analysis Psychology 3256 1 Introduction If you are going to find out anything about a data set you must first understand the data Basically getting a feel for you numbers Easier to find

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

Module 4: Data Exploration

Module 4: Data Exploration Module 4: Data Exploration Now that you have your data downloaded from the Streams Project database, the detective work can begin! Before computing any advanced statistics, we will first use descriptive

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

Random Variables. Chapter 2. Random Variables 1

Random Variables. Chapter 2. Random Variables 1 Random Variables Chapter 2 Random Variables 1 Roulette and Random Variables A Roulette wheel has 38 pockets. 18 of them are red and 18 are black; these are numbered from 1 to 36. The two remaining pockets

More information

The right edge of the box is the third quartile, Q 3, which is the median of the data values above the median. Maximum Median

The right edge of the box is the third quartile, Q 3, which is the median of the data values above the median. Maximum Median CONDENSED LESSON 2.1 Box Plots In this lesson you will create and interpret box plots for sets of data use the interquartile range (IQR) to identify potential outliers and graph them on a modified box

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

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

WEEK #22: PDFs and CDFs, Measures of Center and Spread

WEEK #22: PDFs and CDFs, Measures of Center and Spread WEEK #22: PDFs and CDFs, Measures of Center and Spread Goals: Explore the effect of independent events in probability calculations. Present a number of ways to represent probability distributions. Textbook

More information

Lecture 2. Summarizing the Sample

Lecture 2. Summarizing the Sample Lecture 2 Summarizing the Sample WARNING: Today s lecture may bore some of you It s (sort of) not my fault I m required to teach you about what we re going to cover today. I ll try to make it as exciting

More information

Diagrams and Graphs of Statistical Data

Diagrams and Graphs of Statistical Data Diagrams and Graphs of Statistical Data One of the most effective and interesting alternative way in which a statistical data may be presented is through diagrams and graphs. There are several ways in

More information

A Correlation of. to the. South Carolina Data Analysis and Probability Standards

A Correlation of. to the. South Carolina Data Analysis and Probability Standards A Correlation of to the South Carolina Data Analysis and Probability Standards INTRODUCTION This document demonstrates how Stats in Your World 2012 meets the indicators of the South Carolina Academic Standards

More information

Manhattan Center for Science and Math High School Mathematics Department Curriculum

Manhattan Center for Science and Math High School Mathematics Department Curriculum Content/Discipline Algebra 1 Semester 2: Marking Period 1 - Unit 8 Polynomials and Factoring Topic and Essential Question How do perform operations on polynomial functions How to factor different types

More information

Intro to Statistics 8 Curriculum

Intro to Statistics 8 Curriculum Intro to Statistics 8 Curriculum Unit 1 Bar, Line and Circle Graphs Estimated time frame for unit Big Ideas 8 Days... Essential Question Concepts Competencies Lesson Plans and Suggested Resources Bar graphs

More information

THE BINOMIAL DISTRIBUTION & PROBABILITY

THE BINOMIAL DISTRIBUTION & PROBABILITY REVISION SHEET STATISTICS 1 (MEI) THE BINOMIAL DISTRIBUTION & PROBABILITY The main ideas in this chapter are Probabilities based on selecting or arranging objects Probabilities based on the binomial distribution

More information

A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution

A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution A Primer on Mathematical Statistics and Univariate Distributions; The Normal Distribution; The GLM with the Normal Distribution PSYC 943 (930): Fundamentals of Multivariate Modeling Lecture 4: September

More information

Introduction to Statistics for Psychology. Quantitative Methods for Human Sciences

Introduction to Statistics for Psychology. Quantitative Methods for Human Sciences Introduction to Statistics for Psychology and Quantitative Methods for Human Sciences Jonathan Marchini Course Information There is website devoted to the course at http://www.stats.ox.ac.uk/ marchini/phs.html

More information

Summarizing and Displaying Categorical Data

Summarizing and Displaying Categorical Data Summarizing and Displaying Categorical Data Categorical data can be summarized in a frequency distribution which counts the number of cases, or frequency, that fall into each category, or a relative frequency

More information

1.3 Measuring Center & Spread, The Five Number Summary & Boxplots. Describing Quantitative Data with Numbers

1.3 Measuring Center & Spread, The Five Number Summary & Boxplots. Describing Quantitative Data with Numbers 1.3 Measuring Center & Spread, The Five Number Summary & Boxplots Describing Quantitative Data with Numbers 1.3 I can n Calculate and interpret measures of center (mean, median) in context. n Calculate

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

EXPLORING SPATIAL PATTERNS IN YOUR DATA

EXPLORING SPATIAL PATTERNS IN YOUR DATA EXPLORING SPATIAL PATTERNS IN YOUR DATA OBJECTIVES Learn how to examine your data using the Geostatistical Analysis tools in ArcMap. Learn how to use descriptive statistics in ArcMap and Geoda to analyze

More information

4.1 Exploratory Analysis: Once the data is collected and entered, the first question is: "What do the data look like?"

4.1 Exploratory Analysis: Once the data is collected and entered, the first question is: What do the data look like? Data Analysis Plan The appropriate methods of data analysis are determined by your data types and variables of interest, the actual distribution of the variables, and the number of cases. Different analyses

More information

7 CONTINUOUS PROBABILITY DISTRIBUTIONS

7 CONTINUOUS PROBABILITY DISTRIBUTIONS 7 CONTINUOUS PROBABILITY DISTRIBUTIONS Chapter 7 Continuous Probability Distributions Objectives After studying this chapter you should understand the use of continuous probability distributions and the

More information

Introduction to Exploratory Data Analysis

Introduction to Exploratory Data Analysis Introduction to Exploratory Data Analysis A SpaceStat Software Tutorial Copyright 2013, BioMedware, Inc. (www.biomedware.com). All rights reserved. SpaceStat and BioMedware are trademarks of BioMedware,

More information

Dongfeng Li. Autumn 2010

Dongfeng Li. Autumn 2010 Autumn 2010 Chapter Contents Some statistics background; ; Comparing means and proportions; variance. Students should master the basic concepts, descriptive statistics measures and graphs, basic hypothesis

More information

determining relationships among the explanatory variables, and

determining relationships among the explanatory variables, and Chapter 4 Exploratory Data Analysis A first look at the data. As mentioned in Chapter 1, exploratory data analysis or EDA is a critical first step in analyzing the data from an experiment. Here are the

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

Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs

Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs Types of Variables Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs Quantitative (numerical)variables: take numerical values for which arithmetic operations make sense (addition/averaging)

More information

Data Mining: Exploring Data. Lecture Notes for Chapter 3. Slides by Tan, Steinbach, Kumar adapted by Michael Hahsler

Data Mining: Exploring Data. Lecture Notes for Chapter 3. Slides by Tan, Steinbach, Kumar adapted by Michael Hahsler Data Mining: Exploring Data Lecture Notes for Chapter 3 Slides by Tan, Steinbach, Kumar adapted by Michael Hahsler Topics Exploratory Data Analysis Summary Statistics Visualization What is data exploration?

More information

Visualizing Data. Contents. 1 Visualizing Data. Anthony Tanbakuchi Department of Mathematics Pima Community College. Introductory Statistics Lectures

Visualizing Data. Contents. 1 Visualizing Data. Anthony Tanbakuchi Department of Mathematics Pima Community College. Introductory Statistics Lectures Introductory Statistics Lectures Visualizing Data Descriptive Statistics I Department of Mathematics Pima Community College Redistribution of this material is prohibited without written permission of the

More information

List of Examples. Examples 319

List of Examples. Examples 319 Examples 319 List of Examples DiMaggio and Mantle. 6 Weed seeds. 6, 23, 37, 38 Vole reproduction. 7, 24, 37 Wooly bear caterpillar cocoons. 7 Homophone confusion and Alzheimer s disease. 8 Gear tooth strength.

More information

COMMON CORE STATE STANDARDS FOR

COMMON CORE STATE STANDARDS FOR COMMON CORE STATE STANDARDS FOR Mathematics (CCSSM) High School Statistics and Probability Mathematics High School Statistics and Probability Decisions or predictions are often based on data numbers in

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

How To Check For Differences In The One Way Anova

How To Check For Differences In The One Way Anova MINITAB ASSISTANT WHITE PAPER This paper explains the research conducted by Minitab statisticians to develop the methods and data checks used in the Assistant in Minitab 17 Statistical Software. One-Way

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

II. DISTRIBUTIONS distribution normal distribution. standard scores

II. DISTRIBUTIONS distribution normal distribution. standard scores Appendix D Basic Measurement And Statistics The following information was developed by Steven Rothke, PhD, Department of Psychology, Rehabilitation Institute of Chicago (RIC) and expanded by Mary F. Schmidt,

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

business statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar

business statistics using Excel OXFORD UNIVERSITY PRESS Glyn Davis & Branko Pecar business statistics using Excel Glyn Davis & Branko Pecar OXFORD UNIVERSITY PRESS Detailed contents Introduction to Microsoft Excel 2003 Overview Learning Objectives 1.1 Introduction to Microsoft Excel

More information

Using Excel (Microsoft Office 2007 Version) for Graphical Analysis of Data

Using Excel (Microsoft Office 2007 Version) for Graphical Analysis of Data Using Excel (Microsoft Office 2007 Version) for Graphical Analysis of Data Introduction In several upcoming labs, a primary goal will be to determine the mathematical relationship between two variable

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

STAT355 - Probability & Statistics

STAT355 - Probability & Statistics STAT355 - Probability & Statistics Instructor: Kofi Placid Adragni Fall 2011 Chap 1 - Overview and Descriptive Statistics 1.1 Populations, Samples, and Processes 1.2 Pictorial and Tabular Methods in Descriptive

More information

Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010

Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010 Curriculum Map Statistics and Probability Honors (348) Saugus High School Saugus Public Schools 2009-2010 Week 1 Week 2 14.0 Students organize and describe distributions of data by using a number of different

More information

Descriptive statistics parameters: Measures of centrality

Descriptive statistics parameters: Measures of centrality Descriptive statistics parameters: Measures of centrality Contents Definitions... 3 Classification of descriptive statistics parameters... 4 More about central tendency estimators... 5 Relationship between

More information

AMS 7L LAB #2 Spring, 2009. Exploratory Data Analysis

AMS 7L LAB #2 Spring, 2009. Exploratory Data Analysis AMS 7L LAB #2 Spring, 2009 Exploratory Data Analysis Name: Lab Section: Instructions: The TAs/lab assistants are available to help you if you have any questions about this lab exercise. If you have any

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

Introduction to Environmental Statistics. The Big Picture. Populations and Samples. Sample Data. Examples of sample data

Introduction to Environmental Statistics. The Big Picture. Populations and Samples. Sample Data. Examples of sample data A Few Sources for Data Examples Used Introduction to Environmental Statistics Professor Jessica Utts University of California, Irvine jutts@uci.edu 1. Statistical Methods in Water Resources by D.R. Helsel

More information

Interpreting Data in Normal Distributions

Interpreting Data in Normal Distributions Interpreting Data in Normal Distributions This curve is kind of a big deal. It shows the distribution of a set of test scores, the results of rolling a die a million times, the heights of people on Earth,

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

Algebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard

Algebra 1 2008. Academic Content Standards Grade Eight and Grade Nine Ohio. Grade Eight. Number, Number Sense and Operations Standard Academic Content Standards Grade Eight and Grade Nine Ohio Algebra 1 2008 Grade Eight STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express

More information

Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics

Course Text. Required Computing Software. Course Description. Course Objectives. StraighterLine. Business Statistics Course Text Business Statistics Lind, Douglas A., Marchal, William A. and Samuel A. Wathen. Basic Statistics for Business and Economics, 7th edition, McGraw-Hill/Irwin, 2010, ISBN: 9780077384470 [This

More information

Statistics Review PSY379

Statistics Review PSY379 Statistics Review PSY379 Basic concepts Measurement scales Populations vs. samples Continuous vs. discrete variable Independent vs. dependent variable Descriptive vs. inferential stats Common analyses

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics Primer Descriptive statistics Central tendency Variation Relative position Relationships Calculating descriptive statistics Descriptive Statistics Purpose to describe or summarize

More information

Northumberland Knowledge

Northumberland Knowledge Northumberland Knowledge Know Guide How to Analyse Data - November 2012 - This page has been left blank 2 About this guide The Know Guides are a suite of documents that provide useful information about

More information

Algebra I Vocabulary Cards

Algebra I Vocabulary Cards Algebra I Vocabulary Cards Table of Contents Expressions and Operations Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Absolute Value Order of Operations Expression

More information

Density Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:

Density Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve

More information

Section 1.3 Exercises (Solutions)

Section 1.3 Exercises (Solutions) Section 1.3 Exercises (s) 1.109, 1.110, 1.111, 1.114*, 1.115, 1.119*, 1.122, 1.125, 1.127*, 1.128*, 1.131*, 1.133*, 1.135*, 1.137*, 1.139*, 1.145*, 1.146-148. 1.109 Sketch some normal curves. (a) Sketch

More information

HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS

HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 1 of 25 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS

More information

2. Simple Linear Regression

2. Simple Linear Regression Research methods - II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according

More information

DESCRIPTIVE STATISTICS AND EXPLORATORY DATA ANALYSIS

DESCRIPTIVE STATISTICS AND EXPLORATORY DATA ANALYSIS DESCRIPTIVE STATISTICS AND EXPLORATORY DATA ANALYSIS SEEMA JAGGI Indian Agricultural Statistics Research Institute Library Avenue, New Delhi - 110 012 seema@iasri.res.in 1. Descriptive Statistics Statistics

More information

South Carolina College- and Career-Ready (SCCCR) Probability and Statistics

South Carolina College- and Career-Ready (SCCCR) Probability and Statistics South Carolina College- and Career-Ready (SCCCR) Probability and Statistics South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR)

More information

CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

More information

seven Statistical Analysis with Excel chapter OVERVIEW CHAPTER

seven Statistical Analysis with Excel chapter OVERVIEW CHAPTER seven Statistical Analysis with Excel CHAPTER chapter OVERVIEW 7.1 Introduction 7.2 Understanding Data 7.3 Relationships in Data 7.4 Distributions 7.5 Summary 7.6 Exercises 147 148 CHAPTER 7 Statistical

More information

consider the number of math classes taken by math 150 students. how can we represent the results in one number?

consider the number of math classes taken by math 150 students. how can we represent the results in one number? ch 3: numerically summarizing data - center, spread, shape 3.1 measure of central tendency or, give me one number that represents all the data consider the number of math classes taken by math 150 students.

More information

Assignment #03: Time Management with Excel

Assignment #03: Time Management with Excel Technical Module I Demonstrator: Dereatha Cross dac4303@ksu.edu Assignment #03: Time Management with Excel Introduction Success in any endeavor depends upon time management. One of the optional exercises

More information

Descriptive Statistics and Exploratory Data Analysis

Descriptive Statistics and Exploratory Data Analysis Descriptive Statistics and Exploratory Data Analysis Dean s s Faculty and Resident Development Series UT College of Medicine Chattanooga Probasco Auditorium at Erlanger January 14, 2008 Marc Loizeaux,

More information

Using GAISE and NCTM Standards as Frameworks for Teaching Probability and Statistics to Pre-Service Elementary and Middle School Mathematics Teachers

Using GAISE and NCTM Standards as Frameworks for Teaching Probability and Statistics to Pre-Service Elementary and Middle School Mathematics Teachers Using GAISE and NCTM Standards as Frameworks for Teaching Probability and Statistics to Pre-Service Elementary and Middle School Mathematics Teachers Mary Louise Metz Indiana University of Pennsylvania

More information

The Comparisons. Grade Levels Comparisons. Focal PSSM K-8. Points PSSM CCSS 9-12 PSSM CCSS. Color Coding Legend. Not Identified in the Grade Band

The Comparisons. Grade Levels Comparisons. Focal PSSM K-8. Points PSSM CCSS 9-12 PSSM CCSS. Color Coding Legend. Not Identified in the Grade Band Comparison of NCTM to Dr. Jim Bohan, Ed.D Intelligent Education, LLC Intel.educ@gmail.com The Comparisons Grade Levels Comparisons Focal K-8 Points 9-12 pre-k through 12 Instructional programs from prekindergarten

More information

Chapter 2 Data Exploration

Chapter 2 Data Exploration Chapter 2 Data Exploration 2.1 Data Visualization and Summary Statistics After clearly defining the scientific question we try to answer, selecting a set of representative members from the population of

More information

Tutorial 3: Graphics and Exploratory Data Analysis in R Jason Pienaar and Tom Miller

Tutorial 3: Graphics and Exploratory Data Analysis in R Jason Pienaar and Tom Miller Tutorial 3: Graphics and Exploratory Data Analysis in R Jason Pienaar and Tom Miller Getting to know the data An important first step before performing any kind of statistical analysis is to familiarize

More information

DESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses.

DESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE STATISTICS The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE VS. INFERENTIAL STATISTICS Descriptive To organize,

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

AP Statistics: Syllabus 1

AP Statistics: Syllabus 1 AP Statistics: Syllabus 1 Scoring Components SC1 The course provides instruction in exploring data. 4 SC2 The course provides instruction in sampling. 5 SC3 The course provides instruction in experimentation.

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics Descriptive statistics consist of methods for organizing and summarizing data. It includes the construction of graphs, charts and tables, as well various descriptive measures such

More information

Describing Data: Measures of Central Tendency and Dispersion

Describing Data: Measures of Central Tendency and Dispersion 100 Part 2 / Basic Tools of Research: Sampling, Measurement, Distributions, and Descriptive Statistics Chapter 8 Describing Data: Measures of Central Tendency and Dispersion In the previous chapter we

More information

What is Data Analysis. Kerala School of MathematicsCourse in Statistics for Scientis. Introduction to Data Analysis. Steps in a Statistical Study

What is Data Analysis. Kerala School of MathematicsCourse in Statistics for Scientis. Introduction to Data Analysis. Steps in a Statistical Study Kerala School of Mathematics Course in Statistics for Scientists Introduction to Data Analysis T.Krishnan Strand Life Sciences, Bangalore What is Data Analysis Statistics is a body of methods how to use

More information

Description. Textbook. Grading. Objective

Description. Textbook. Grading. Objective EC151.02 Statistics for Business and Economics (MWF 8:00-8:50) Instructor: Chiu Yu Ko Office: 462D, 21 Campenalla Way Phone: 2-6093 Email: kocb@bc.edu Office Hours: by appointment Description This course

More information

Multivariate Normal Distribution

Multivariate Normal Distribution Multivariate Normal Distribution Lecture 4 July 21, 2011 Advanced Multivariate Statistical Methods ICPSR Summer Session #2 Lecture #4-7/21/2011 Slide 1 of 41 Last Time Matrices and vectors Eigenvalues

More information

Chapter 4 - Lecture 1 Probability Density Functions and Cumul. Distribution Functions

Chapter 4 - Lecture 1 Probability Density Functions and Cumul. Distribution Functions Chapter 4 - Lecture 1 Probability Density Functions and Cumulative Distribution Functions October 21st, 2009 Review Probability distribution function Useful results Relationship between the pdf and the

More information

a. mean b. interquartile range c. range d. median

a. mean b. interquartile range c. range d. median 3. Since 4. The HOMEWORK 3 Due: Feb.3 1. A set of data are put in numerical order, and a statistic is calculated that divides the data set into two equal parts with one part below it and the other part

More information

Mean = (sum of the values / the number of the value) if probabilities are equal

Mean = (sum of the values / the number of the value) if probabilities are equal Population Mean Mean = (sum of the values / the number of the value) if probabilities are equal Compute the population mean Population/Sample mean: 1. Collect the data 2. sum all the values in the population/sample.

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

Module 2: Introduction to Quantitative Data Analysis

Module 2: Introduction to Quantitative Data Analysis Module 2: Introduction to Quantitative Data Analysis Contents Antony Fielding 1 University of Birmingham & Centre for Multilevel Modelling Rebecca Pillinger Centre for Multilevel Modelling Introduction...

More information

Def: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.

Def: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1. Lecture 6: Chapter 6: Normal Probability Distributions A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve.

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

Organizing Your Approach to a Data Analysis

Organizing Your Approach to a Data Analysis Biost/Stat 578 B: Data Analysis Emerson, September 29, 2003 Handout #1 Organizing Your Approach to a Data Analysis The general theme should be to maximize thinking about the data analysis and to minimize

More information