2 Correlation key concepts: Types of correlation Methods of studying correlation a) Scatter diagram b) Karl pearson s coefficient of correlation c) Spearman s Rank correlation coefficient d) Method of least squares
3 Correlation Correlation: The degree of relationship between the variables under consideration is measure through the correlation analysis. The measure of correlation called the correlation coefficient The degree of relationship is expressed by coefficient which range from correlation ( -1 r +1) The direction of change is indicated by a sign. The correlation analysis enable us to have an idea about the degree & direction of the relationship between the two variables under study.
4 Correlation Correlation is a statistical tool that helps to measure and analyze the degree of relationship between two variables. Correlation analysis deals with the association between two or more variables.
5 Correlation & Causation Causation means cause & effect relation. Correlation denotes the interdependency among the variables for correlating two phenomenon, it is essential that the two phenomenon should have cause-effect relationship,& if such relationship does not exist then the two phenomenon can not be correlated. If two variables vary in such a way that movement in one are accompanied by movement in other, these variables are called cause and effect relationship. Causation always implies correlation but correlation does not necessarily implies causation.
6 Types of Correlation Type I Correlation Positive Correlation Negative Correlation
7 Types of Correlation Type I Positive Correlation: The correlation is said to be positive correlation if the values of two variables changing with same direction. Ex. Pub. Exp. & sales, Height & weight. Negative Correlation: The correlation is said to be negative correlation when the values of variables change with opposite direction. Ex. Price & qty. demanded.
8 Direction of the Correlation Positive relationship Variables change in the same direction. As X is increasing, Y is increasing As X is decreasing, Y is decreasing E.g., As height increases, so does weight. Negative relationship Variables change in opposite directions. As X is increasing, Y is decreasing As X is decreasing, Y is increasing E.g., As TV time increases, grades decrease Indicated by sign; (+) or (-).
9 More examples Positive relationships water consumption and temperature. study time and grades. Negative relationships: alcohol consumption and driving ability. Price & quantity demanded
10 Types of Correlation Type II Correlation Simple Multiple Partial Total
11 Types of Correlation Type II Simple correlation: Under simple correlation problem there are only two variables are studied. Multiple Correlation: Under Multiple Correlation three or more than three variables are studied. Ex. Q d = f ( P,P C, P S, t, y ) Partial correlation: analysis recognizes more than two variables but considers only two variables keeping the other constant. Total correlation: is based on all the relevant variables, which is normally not feasible.
12 Types of Correlation Type III Correlation LINEAR NON LINEAR
13 Types of Correlation Type III Linear correlation: Correlation is said to be linear when the amount of change in one variable tends to bear a constant ratio to the amount of change in the other. The graph of the variables having a linear relationship will form a straight line. Ex X = 1, 2, 3, 4, 5, 6, 7, 8, Y = 5, 7, 9, 11, 13, 15, 17, 19, Y = 3 + 2x Non Linear correlation: The correlation would be non linear if the amount of change in one variable does not bear a constant ratio to the amount of change in the other variable.
14 Methods of Studying Correlation Scatter Diagram Method Graphic Method Karl Pearson s Coefficient of Correlation Method of Least Squares
15 Scatter Diagram Method Scatter Diagram is a graph of observed plotted points where each points represents the values of X & Y as a coordinate. It portrays the relationship between these two variables graphically.
16 A perfect positive correlation Weight Weight of B Weight of A A linear relationship Height of A Height of B Height
17 High Degree of positive correlation Positive relationship r = +.80 Weight Height
18 Degree of correlation Moderate Positive Correlation Shoe Size r = Weight
19 Degree of correlation Perfect Negative Correlation TV watching per week r = -1.0 Exam score
20 Degree of correlation Moderate Negative Correlation TV watching per week r = -.80 Exam score
21 Degree of correlation Weak negative Correlation Shoe Size r = Weight
22 Degree of correlation No Correlation (horizontal line) IQ r = 0.0 Height
23 Degree of correlation (r) r = +.80 r = +.60 r = +.40 r = +.20
24 2) Direction of the Relationship Positive relationship Variables change in the same direction. As X is increasing, Y is increasing As X is decreasing, Y is decreasing E.g., As height increases, so does weight. Negative relationship Variables change in opposite directions. As X is increasing, Y is decreasing As X is decreasing, Y is increasing E.g., As TV time increases, grades decrease Indicated by sign; (+) or (-).
25 Advantages of Scatter Diagram Simple & Non Mathematical method Not influenced by the size of extreme item First step in investing the relationship between two variables
26 Disadvantage of scatter diagram Can not adopt the an exact degree of correlation
27 Karl Pearson's Coefficient of Correlation Pearson s r is the most common correlation coefficient. Karl Pearson s Coefficient of Correlation denoted by- r The coefficient of correlation r measure the degree of linear relationship between two variables say x & y.
28 Karl Pearson's Coefficient of Correlation Karl Pearson s Coefficient of Correlation denoted by- r -1 r +1 Degree of Correlation is expressed by a value of Coefficient Direction of change is Indicated by sign ( - ve) or ( + ve)
29 Karl Pearson's Coefficient of Correlation When deviation taken from actual mean: r(x, y)= Σxy / Σx² Σy² When deviation taken from an assumed mean: r = N Σdxdy - Σdx Σdy N Σdx²-(Σdx)² N Σdy²-(Σdy)²
30 Procedure for computing the correlation coefficient Calculate the mean of the two series x & y Calculate the deviations x & y in two series from their respective mean. Square each deviation of x & y then obtain the sum of the squared deviation i.e. x 2 &. y 2 Multiply each deviation under x with each deviation under y & obtain the product of xy.then obtain the sum of the product of x, y i.e. xy Substitute the value in the formula.
31 Interpretation of Correlation Coefficient (r) The value of correlation coefficient r ranges from -1 to +1 If r = +1, then the correlation between the two variables is said to be perfect and positive If r = -1, then the correlation between the two variables is said to be perfect and negative If r = 0, then there exists no correlation between the variables
32 Properties of Correlation coefficient The correlation coefficient lies between -1 & +1 symbolically ( - 1 r 1 ) The correlation coefficient is independent of the change of origin & scale. The coefficient of correlation is the geometric mean of two regression coefficient. r = bxy * byx The one regression coefficient is (+ve) other regression coefficient is also (+ve) correlation coefficient is (+ve)
33 Assumptions of Pearson s Correlation Coefficient There is linear relationship between two variables, i.e. when the two variables are plotted on a scatter diagram a straight line will be formed by the points. Cause and effect relation exists between different forces operating on the item of the two variable series.
34 Advantages of Pearson s Coefficient It summarizes in one value, the degree of correlation & direction of correlation also.
35 Limitation of Pearson s Coefficient Always assume linear relationship Interpreting the value of r is difficult. Value of Correlation Coefficient is affected by the extreme values. Time consuming methods
36 Coefficient of Determination The convenient way of interpreting the value of correlation coefficient is to use of square of coefficient of correlation which is called Coefficient of Determination. The Coefficient of Determination = r 2. Suppose: r = 0.9, r 2 = 0.81 this would mean that 81% of the variation in the dependent variable has been explained by the independent variable.
37 Coefficient of Determination The maximum value of r 2 is 1 because it is possible to explain all of the variation in y but it is not possible to explain more than all of it. Coefficient of Determination = Explained variation / Total variation
38 Coefficient of Determination: An example Suppose: r = 0.60 r = 0.30 It does not mean that the first correlation is twice as strong as the second the r can be understood by computing the value of r 2. When r = 0.60 r 2 = (1) r = 0.30 r 2 = (2) This implies that in the first case 36% of the total variation is explained whereas in second case 9% of the total variation is explained.
39 Spearman s Rank Coefficient of Correlation When statistical series in which the variables under study are not capable of quantitative measurement but can be arranged in serial order, in such situation pearson s correlation coefficient can not be used in such case Spearman Rank correlation can be used. R = 1- (6 D 2 ) / N (N 2 1) R = Rank correlation coefficient D = Difference of rank between paired item in two series. N = Total number of observation.
40 Interpretation of Rank Correlation Coefficient (R) The value of rank correlation coefficient, R ranges from -1 to +1 If R = +1, then there is complete agreement in the order of the ranks and the ranks are in the same direction If R = -1, then there is complete agreement in the order of the ranks and the ranks are in the opposite direction If R = 0, then there is no correlation
41 Rank Correlation Coefficient (R) a) Problems where actual rank are given. 1) Calculate the difference D of two Ranks i.e. (R1 R2). 2) Square the difference & calculate the sum of the difference i.e. D 2 3) Substitute the values obtained in the formula.
42 Rank Correlation Coefficient b) Problems where Ranks are not given :If the ranks are not given, then we need to assign ranks to the data series. The lowest value in the series can be assigned rank 1 or the highest value in the series can be assigned rank 1. We need to follow the same scheme of ranking for the other series. Then calculate the rank correlation coefficient in similar way as we do when the ranks are given.
43 Rank Correlation Coefficient (R) Equal Ranks or tie in Ranks: In such cases average ranks should be assigned to each individual. R = 1- (6 D 2 ) + AF / N (N 2 1) AF = 1/12(m 13 m 1 ) + 1/12(m 23 m 2 ) +. 1/12(m 23 m 2 ) m = The number of time an item is repeated
44 Merits Spearman s Rank Correlation This method is simpler to understand and easier to apply compared to karl pearson s correlation method. This method is useful where we can give the ranks and not the actual data. (qualitative term) This method is to use where the initial data in the form of ranks.
45 Limitation Spearman s Correlation Cannot be used for finding out correlation in a grouped frequency distribution. This method should be applied where N exceeds 30.
46 Advantages of Correlation studies Show the amount (strength) of relationship present Can be used to make predictions about the variables under study. Can be used in many places, including natural settings, libraries, etc. Easier to collect co relational data
47 Regression Analysis Regression Analysis is a very powerful tool in the field of statistical analysis in predicting the value of one variable, given the value of another variable, when those variables are related to each other.
48 Regression Analysis Regression Analysis is mathematical measure of average relationship between two or more variables. Regression analysis is a statistical tool used in prediction of value of unknown variable from known variable.
49 Advantages of Regression Analysis Regression analysis provides estimates of values of the dependent variables from the values of independent variables. Regression analysis also helps to obtain a measure of the error involved in using the regression line as a basis for estimations. Regression analysis helps in obtaining a measure of the degree of association or correlation that exists between the two variable.
50 Assumptions in Regression Analysis Existence of actual linear relationship. The regression analysis is used to estimate the values within the range for which it is valid. The relationship between the dependent and independent variables remains the same till the regression equation is calculated. The dependent variable takes any random value but the values of the independent variables are fixed. In regression, we have only one dependant variable in our estimating equation. However, we can use more than one independent variable.
51 Regression line Regression line is the line which gives the best estimate of one variable from the value of any other given variable. The regression line gives the average relationship between the two variables in mathematical form. The Regression would have the following properties: a) ( Y Y c ) = 0 and b) ( Y Y c ) 2 = Minimum
52 Regression line For two variables X and Y, there are always two lines of regression Regression line of X on Y : gives the best estimate for the value of X for any specific given values of Y X = a + b Y a = X - intercept b = Slope of the line X = Dependent variable Y = Independent variable
53 Regression line For two variables X and Y, there are always two lines of regression Regression line of Y on X : gives the best estimate for the value of Y for any specific given values of X Y = a + bx a = Y - intercept b = Slope of the line Y = Dependent variable x= Independent variable
54 The Explanation of Regression Line In case of perfect correlation ( positive or negative ) the two line of regression coincide. If the two R. line are far from each other then degree of correlation is less, & vice versa. The mean values of X &Y can be obtained as the point of intersection of the two regression line. The higher degree of correlation between the variables, the angle between the lines is smaller & vice versa.
55 Regression Equation / Line & Method of Least Squares Regression Equation of y on x Y = a + bx In order to obtain the values of a & b y = na + b x xy = a x + b x 2 Regression Equation of x on y X = c + dy In order to obtain the values of c & d x = nc + d y xy = c y + d y 2
56 Regression Equation / Line when Deviation taken from Arithmetic Mean Regression Equation of y on x: Y = a + bx In order to obtain the values of a & b a = Y bx b = xy / x 2 Regression Equation of x on y: X = c + dy c = X dy d = xy / y 2
57 Regression Equation / Line when Deviation taken from Arithmetic Mean Regression Equation of y on x: Regression Equation of x on y: Y Y = b yx (X X) b yx = xy / x 2 b yx = r (σy / σx ) X X = b xy (Y Y) b xy = xy / y 2 b xy = r (σx / σy )
58 Properties of the Regression Coefficients The coefficient of correlation is geometric mean of the two regression coefficients. r = b yx * b xy If b yx is positive than b xy should also be positive & vice versa. If one regression coefficient is greater than one the other must be less than one. The coefficient of correlation will have the same sign as that our regression coefficient. Arithmetic mean of b yx & b xy is equal to or greater than coefficient of correlation. b yx + b xy / 2 r Regression coefficient are independent of origin but not of scale.
59 Standard Error of Estimate. Standard Error of Estimate is the measure of variation around the computed regression line. Standard error of estimate (SE) of Y measure the variability of the observed values of Y around the regression line. Standard error of estimate gives us a measure about the line of regression. of the scatter of the observations about the line of regression.
60 Standard Error of Estimate. Standard Error of Estimate of Y on X is: S.E. of Yon X (SE xy ) = (Y Y e ) 2 / n-2 Y = Observed value of y Y e = Estimated values from the estimated equation that correspond to each y value e = The error term (Y Y e ) n = Number of observation in sample. The convenient formula: (SE xy ) = Y 2 _ a Y _ b YX / n 2 X = Value of independent variable. Y = Value of dependent variable. a = Y intercept. b = Slope of estimating equation. n = Number of data points.
61 Correlation analysis vs. Regression analysis. Regression is the average relationship between two variables Correlation need not imply cause & effect relationship between the variables understudy.- R A clearly indicate the cause and effect relation ship between the variables. There may be non-sense correlation between two variables.- There is no such thing like non-sense regression.
62 Correlation analysis vs. Regression analysis. Regression is the average relationship between two variables R A.
65 What is regression? Fitting a line to the data using an equation in order to describe and predict data Simple Regression Uses just 2 variables (X and Y) Other: Multiple Regression (one Y and many X s) Linear Regression Fits data to a straight line Other: Curvilinear Regression (curved line) We re doing: Simple, Linear Regression
66 From Geometry: Any line can be described by an equation For any point on a line for X, there will be a corresponding Y the equation for this is y = mx + b m is the slope, b is the Y-intercept (when X = 0) Slope = change in Y per unit change in X Y-intercept = where the line crosses the Y axis (when X = 0)
67 Regression equation Find a line that fits the data the best, = find a line that minimizes the distance from all the data points to that line ^ Regression Equation: Y(Y-hat) = bx + a Y(hat) is the predicted value of Y given a certain X b is the slope a is the y-intercept
68 Regression Equation: Y =.823X We can predict a Y score from an X by plugging a value for X into the equation and calculating Y What would we expect a person to get on quiz #4 if they got a 12.5 on quiz #3? Y =.823(12.5) = 6.049
69 Advantages of Correlation studies Show the amount (strength) of relationship present Can be used to make predictions about the variables studied Can be used in many places, including natural settings, libraries, etc. Easier to collect correlational data
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,
Section 3 Part 1 Relationships between two numerical variables 1 Relationship between two variables The summary statistics covered in the previous lessons are appropriate for describing a single variable.
Lesson 4 Part 1 Relationships between two numerical variables 1 Correlation Coefficient The correlation coefficient is a summary statistic that describes the linear relationship between two numerical variables
Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis
Figure 9-6 Example: Boats and Manatees Slide 1 Given the sample data in Table 9-1, find the value of the linear correlation coefficient r, then refer to Table A-6 to determine whether there is a significant
Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the
Slide 1 Section 14 Simple Linear Regression: Introduction to Least Squares Regression There are several different measures of statistical association used for understanding the quantitative relationship
Grade 8 Overview View unit yearlong overview here The unit design was created in line with the areas of focus for grade 8 Mathematics as identified by the Common Core State Standards and the PARCC Model
CORRELATION AND REGRESSION / 47 CHAPTER EIGHT CORRELATION AND REGRESSION Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables.
Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table co-variation least squares
Statistics Quiz Correlation and Regression -- ANSWERS 1. Temperature and air pollution are known to be correlated. We collect data from two laboratories, in Boston and Montreal. Boston makes their measurements
Click on the links below to jump directly to the relevant section Elements of a graph Linear equations and their graphs What is slope? Slope and y-intercept in the equation of a line Comparing lines on
Demand Forecasting When a product is produced for a market, the demand occurs in the future. The production planning cannot be accomplished unless the volume of the demand known. The success of the business
PLOTTING DATA AND INTERPRETING GRAPHS Fundamentals of Graphing One of the most important sets of skills in science and mathematics is the ability to construct graphs and to interpret the information they
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
Unit 9 Describing Relationships in Scatter Plots and Line Graphs Objectives: To construct and interpret a scatter plot or line graph for two quantitative variables To recognize linear relationships, non-linear
Georgia Standards of Excellence Curriculum Map Mathematics GSE 8 th Grade These materials are for nonprofit educational purposes only. Any other use may constitute copyright infringement. GSE Eighth Grade
Lecture 8 : Coordinate Geometry The coordinate plane The points on a line can be referenced if we choose an origin and a unit of 0 distance on the axis and give each point an identity on the corresponding
7. The Point-Slope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope
The importance of graphing the data: Anscombe s regression examples Bruce Weaver Northern Health Research Conference Nipissing University, North Bay May 30-31, 2008 B. Weaver, NHRC 2008 1 The Objective
Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between
Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear
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
The verbal answers to all of the following questions should be memorized before completion of algebra. Answers that are not memorized will hinder your ability to succeed in geometry and algebra. (Everything
Brunswick High School has reinstated a summer math curriculum for students Algebra 1, Geometry, and Algebra 2 for the 2014-2015 school year. Goal The goal of the summer math program is to help students
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
We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical
AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks
Chapter 3 Student Lecture Notes 3- Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing
Pennsylvania System of School Assessment The Assessment Anchors, as defined by the Eligible Content, are organized into cohesive blueprints, each structured with a common labeling system that can be read
Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations
Temperature Scales INTRODUCTION The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. The unit of temperature in the metric system is
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools
Academic Content Standards Grade Eight Ohio Pre-Algebra 2008 STANDARDS Number, Number Sense and Operations Standard Number and Number Systems 1. Use scientific notation to express large numbers and small
Elasticity I. What is Elasticity? The purpose of this section is to develop some general rules about elasticity, which may them be applied to the four different specific types of elasticity discussed in
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
CORRELATION ANALYSIS Learning Objectives Understand how correlation can be used to demonstrate a relationship between two factors. Know how to perform a correlation analysis and calculate the coefficient
Linear Regression Chapter 5 Regression Objective: To quantify the linear relationship between an explanatory variable (x) and response variable (y). We can then predict the average response for all subjects
1.2 GRAPHS OF EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch graphs of equations. Find x- and y-intercepts of graphs of equations. Use symmetry to sketch graphs
LAGUARDIA COMMUNITY COLLEGE CITY UNIVERSITY OF NEW YORK DEPARTMENT OF MATHEMATICS, ENGINEERING, AND COMPUTER SCIENCE MAT 119 STATISTICS AND ELEMENTARY ALGEBRA 5 Lecture Hours, 2 Lab Hours, 3 Credits Pre-
Linear Approximations ACADEMIC RESOURCE CENTER Table of Contents Linear Function Linear Function or Not Real World Uses for Linear Equations Why Do We Use Linear Equations? Estimation with Linear Approximations
Core Florida Math for College Readiness Florida Math for College Readiness provides a fourth-year math curriculum focused on developing the mastery of skills identified as critical to postsecondary readiness
Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.
Chapter 9. Systems of Linear Equations 9.1. Solve Systems of Linear Equations by Graphing KYOTE Standards: CR 21; CA 13 In this section we discuss how to solve systems of two linear equations in two variables
A Correlation of 2013 To the to the Introduction This document demonstrates how Mathematics Accelerated Grade 7, 2013, meets the. Correlation references are to the pages within the Student Edition. Meeting
Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.
Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a real-valued
Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H
Algebra 1, Quarter 2, Unit 2.1 Creating, Solving, and Graphing Systems of Linear Equations and Linear Inequalities Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned
COWLEY COUNTY COMMUNITY COLLEGE REVIEW GUIDE Compass Algebra Level This study guide is for students trying to test into College Algebra. There are three levels of math study guides. 1. If x and y 1, what
17. SIMPLE LINEAR REGRESSION II The Model In linear regression analysis, we assume that the relationship between X and Y is linear. This does not mean, however, that Y can be perfectly predicted from X.
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
Prentice Hall Connected Mathematics 2, 7th Grade Units 2009 Grade 7 C O R R E L A T E D T O from March 2009 Grade 7 Problem Solving Build new mathematical knowledge through problem solving. Solve problems
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
2013 MBA Jump Start Program Module 1: Statistics Thomas Gilbert Part 3 Statistics Module Part 3 Hypothesis Testing (Inference) Regressions 2 1 Making an Investment Decision A researcher in your firm just
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
Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general
Correlational Research Stephen E. Brock, Ph.D., NCSP California State University, Sacramento 1 Correlational Research A quantitative methodology used to determine whether, and to what degree, a relationship
CURVE FITTING LEAST SQUARES APPROXIMATION Data analysis and curve fitting: Imagine that we are studying a physical system involving two quantities: x and y Also suppose that we expect a linear relationship
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
Section. Linear Equations: Slope and Equations of Lines Slope The measure of the steepness of a line is called the slope of the line. It is the amount of change in y, the rise, divided by the amount of
IV. ALGEBRAIC CONCEPTS Algebra is the language of mathematics. Much of the observable world can be characterized as having patterned regularity where a change in one quantity results in changes in other
Introduction to Quantitative Methods October 15, 2009 Contents 1 Definition of Key Terms 2 2 Descriptive Statistics 3 2.1 Frequency Tables......................... 4 2.2 Measures of Central Tendencies.................
In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,
Regression Introduction In the last lesson, we saw how to aggregate data from different sources, identify measures and dimensions, to build data marts for business analysis. Some techniques were introduced
Graphing Linear Equations I. Graphing Linear Equations a. The graphs of first degree (linear) equations will always be straight lines. b. Graphs of lines can have Positive Slope Negative Slope Zero slope
Part Three Cost Behavior Analysis Cost Behavior Cost behavior is the manner in which a cost changes as some related activity changes An understanding of cost behavior is necessary to plan and control costs
McDougal Littell California: Pre-Algebra Algebra 1 correlated to the California Math Content s Grades 7 8 McDougal Littell California Pre-Algebra Components: Pupil Edition (PE), Teacher s Edition (TE),
Physics Lab Report Guidelines Summary The following is an outline of the requirements for a physics lab report. A. Experimental Description 1. Provide a statement of the physical theory or principle observed
Name: Score: / Homework 11 Part 1 null 1 For which of the following correlations would the data points be clustered most closely around a straight line? A. r = 0.50 B. r = -0.80 C. r = 0.10 D. There is
parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL HS America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does
Econometrics Simple Linear Regression Burcu Eke UC3M Linear equations with one variable Recall what a linear equation is: y = b 0 + b 1 x is a linear equation with one variable, or equivalently, a straight
Univariate Regression Correlation and Regression The regression line summarizes the linear relationship between 2 variables Correlation coefficient, r, measures strength of relationship: the closer r is
1 Functions, Graphs and Limits 1.1 The Cartesian Plane In this course we will be dealing a lot with the Cartesian plane (also called the xy-plane), so this section should serve as a review of it and its
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
Linear Equations Domain and Range Domain refers to the set of possible values of the x-component of a point in the form (x,y). Range refers to the set of possible values of the y-component of a point in
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,
Grade 6 Mathematics, Quarter 2, Unit 2.1 Measurement with Ratios Overview Number of instructional days: 15 (1 day = 45 minutes) Content to be learned Use ratio reasoning to solve real-world and mathematical
Lecture 11: Chapter 5, Section 3 Relationships between Two Quantitative Variables; Correlation Display and Summarize Correlation for Direction and Strength Properties of Correlation Regression Line Cengage