# BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I

Save this PDF as:

Size: px
Start display at page:

## 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 Further Thoughts on Experimental Design 16 Individuals (8 each from two populations) with replicates Pop 1 Pop 2 Randomly sample 4 individuals

### 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

### 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,

### 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

### We will use the following data sets to illustrate measures of center. DATA SET 1 The following are test scores from a class of 20 students:

MODE The mode of the sample is the value of the variable having the greatest frequency. Example: Obtain the mode for Data Set 1 77 For a grouped frequency distribution, the modal class is the class having

### 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

### 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

### The Big 50 Revision Guidelines for S1

The Big 50 Revision Guidelines for S1 If you can understand all of these you ll do very well 1. Know what is meant by a statistical model and the Modelling cycle of continuous refinement 2. Understand

### 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

### Chapter 3: Data Description Numerical Methods

Chapter 3: Data Description Numerical Methods Learning Objectives Upon successful completion of Chapter 3, you will be able to: Summarize data using measures of central tendency, such as the mean, median,

### 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

### Numerical Measures of Central Tendency

Numerical Measures of Central Tendency Often, it is useful to have special numbers which summarize characteristics of a data set These numbers are called descriptive statistics or summary statistics. A

### 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),

### Chapter 3 Descriptive Statistics: Numerical Measures. Learning objectives

Chapter 3 Descriptive Statistics: Numerical Measures Slide 1 Learning objectives 1. Single variable Part I (Basic) 1.1. How to calculate and use the measures of location 1.. How to calculate and use the

### 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

### Data Mining Part 2. Data Understanding and Preparation 2.1 Data Understanding Spring 2010

Data Mining Part 2. and Preparation 2.1 Spring 2010 Instructor: Dr. Masoud Yaghini Introduction Outline Introduction Measuring the Central Tendency Measuring the Dispersion of Data Graphic Displays References

### 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.

### 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

### 1 Measures for location and dispersion of a sample

Statistical Geophysics WS 2008/09 7..2008 Christian Heumann und Helmut Küchenhoff Measures for location and dispersion of a sample Measures for location and dispersion of a sample In the following: Variable

### 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

### 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

### 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

### Statistical Foundations: Measures of Location and Central Tendency and Summation and Expectation

Statistical Foundations: and Central Tendency and and Lecture 4 September 5, 2006 Psychology 790 Lecture #4-9/05/2006 Slide 1 of 26 Today s Lecture Today s Lecture Where this Fits central tendency/location

### Sampling, frequency distribution, graphs, measures of central tendency, measures of dispersion

Statistics Basics Sampling, frequency distribution, graphs, measures of central tendency, measures of dispersion Part 1: Sampling, Frequency Distributions, and Graphs The method of collecting, organizing,

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### 13.2 Measures of Central Tendency

13.2 Measures of Central Tendency Measures of Central Tendency For a given set of numbers, it may be desirable to have a single number to serve as a kind of representative value around which all the numbers

### Chapter 3: Central Tendency

Chapter 3: Central Tendency Central Tendency In general terms, central tendency is a statistical measure that determines a single value that accurately describes the center of the distribution and represents

### 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

### 1. 2. 3. 4. Find the mean and median. 5. 1, 2, 87 6. 3, 2, 1, 10. Bellwork 3-23-15 Simplify each expression.

Bellwork 3-23-15 Simplify each expression. 1. 2. 3. 4. Find the mean and median. 5. 1, 2, 87 6. 3, 2, 1, 10 1 Objectives Find measures of central tendency and measures of variation for statistical data.

### 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

### 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

### 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

### Numerical Summarization of Data OPRE 6301

Numerical Summarization of Data OPRE 6301 Motivation... In the previous session, we used graphical techniques to describe data. For example: While this histogram provides useful insight, other interesting

### 10-3 Measures of Central Tendency and Variation

10-3 Measures of Central Tendency and Variation So far, we have discussed some graphical methods of data description. Now, we will investigate how statements of central tendency and variation can be used.

### 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

### 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

### Chapter 15 Multiple Choice Questions (The answers are provided after the last question.)

Chapter 15 Multiple Choice Questions (The answers are provided after the last question.) 1. What is the median of the following set of scores? 18, 6, 12, 10, 14? a. 10 b. 14 c. 18 d. 12 2. Approximately

### 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

### Describe what is meant by a placebo Contrast the double-blind procedure with the single-blind procedure Review the structure for organizing a memo

Readings: Ha and Ha Textbook - Chapters 1 8 Appendix D & E (online) Plous - Chapters 10, 11, 12 and 14 Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability

### 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

### 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

### 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

### Treatment and analysis of data Applied statistics Lecture 3: Sampling and descriptive statistics

Treatment and analysis of data Applied statistics Lecture 3: Sampling and descriptive statistics Topics covered: Parameters and statistics Sample mean and sample standard deviation Order statistics and

### 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

### 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

### Biostatistics: A QUICK GUIDE TO THE USE AND CHOICE OF GRAPHS AND CHARTS

Biostatistics: A QUICK GUIDE TO THE USE AND CHOICE OF GRAPHS AND CHARTS 1. Introduction, and choosing a graph or chart Graphs and charts provide a powerful way of summarising data and presenting them in

### 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

### 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

### 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

### 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

### 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

### 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

### Histogram. Graphs, and measures of central tendency and spread. Alternative: density (or relative frequency ) plot /13/2004

Graphs, and measures of central tendency and spread 9.07 9/13/004 Histogram If discrete or categorical, bars don t touch. If continuous, can touch, should if there are lots of bins. Sum of bin heights

### GCSE Statistics Revision notes

GCSE Statistics Revision notes Collecting data Sample This is when data is collected from part of the population. There are different methods for sampling Random sampling, Stratified sampling, Systematic

### CHINHOYI UNIVERSITY OF TECHNOLOGY

CHINHOYI UNIVERSITY OF TECHNOLOGY SCHOOL OF NATURAL SCIENCES AND MATHEMATICS DEPARTMENT OF MATHEMATICS MEASURES OF CENTRAL TENDENCY AND DISPERSION INTRODUCTION From the previous unit, the Graphical displays

### 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

### 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

### 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

### MINITAB ASSISTANT WHITE PAPER

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

### Each exam covers lectures from since the previous exam and up to the exam date.

Sociology 301 Exam Review Liying Luo 03.22 Exam Review: Logistics Exams must be taken at the scheduled date and time unless 1. You provide verifiable documents of unforeseen illness or family emergency,

### x Measures of Central Tendency for Ungrouped Data Chapter 3 Numerical Descriptive Measures Example 3-1 Example 3-1: Solution

Chapter 3 umerical Descriptive Measures 3.1 Measures of Central Tendency for Ungrouped Data 3. Measures of Dispersion for Ungrouped Data 3.3 Mean, Variance, and Standard Deviation for Grouped Data 3.4

### 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.

### 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

### 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)

### 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

### 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

### 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

### 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

### Models for Discrete Variables

Probability Models for Discrete Variables Our study of probability begins much as any data analysis does: What is the distribution of the data? Histograms, boxplots, percentiles, means, standard deviations

### STATISTICS FOR PSYCH MATH REVIEW GUIDE

STATISTICS FOR PSYCH MATH REVIEW GUIDE ORDER OF OPERATIONS Although remembering the order of operations as BEDMAS may seem simple, it is definitely worth reviewing in a new context such as statistics formulae.

### 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

### 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

### 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

### 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?

### 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

### 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

### 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

### 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

### 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,

### 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,

### MAT 12O ELEMENTARY STATISTICS I

LAGUARDIA COMMUNITY COLLEGE CITY UNIVERSITY OF NEW YORK DEPARTMENT OF MATHEMATICS, ENGINEERING, AND COMPUTER SCIENCE MAT 12O ELEMENTARY STATISTICS I 3 Lecture Hours, 1 Lab Hour, 3 Credits Pre-Requisite:

### 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

### 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

### 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

### 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

### 2. A is a subset of the population. 3. Construct a frequency distribution for the data of the grades of 25 students taking Math 11 last

Math 111 Chapter 12 Practice Test 1. If I wanted to survey 50 Cabrini College students about where they prefer to eat on campus, which would be the most appropriate way to conduct my survey? a. Find 50

### 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

### 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

### 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