PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 5
|
|
- Jessie Warner
- 5 years ago
- Views:
Transcription
1 6. Mathematical background PSYCHOLOGICAL RESEARCH (PYC 34-C) Lecture 5 Numbers and quantification offer us a very special language which enables us to express ourselves in exact terms. This language is called Mathematics. We will now learn the basic rules of Mathematics in order to communicate effectively with figures. A huge part of psychological research deals with statistical analysis so that one needs an adequate mathematical background to understand statistical computations. 6.1 Pocket calculator In both modules, you will need a scientific calculator, that is, one which has statistical functions and, more preferably, one having the regression mode. The most cost-effective calculator for this course is the CASIO FX-82 TL (it costs about Rs 3). This will save you a tremendous amount of time in the examinations once statistical data entered, statistics like the number of observations, mean, standard deviation, correlation and regression coefficients can be readily obtained by just pressing buttons. Note The study guide advises students to buy a programmable calculator (which, in my opinion, is not worth it for these modules). 6.2 Summation notation The summation notation is used to summarise a series, that is, the sum of the terms of a sequence. It is denoted by Greek capital letter sigma,, as opposed to small letter sigma, σ, which, in Statistics, stands for standard deviation. Sigma is most of the time seen in the following form: b r = a f ( r) where r is known as the index, a and b are the lower and upper limits of summation respectively and f (r) is known as the general term. r, just like a counter, starts at a and increases by steps of 1 until it reaches b. Each term of the series is obtained by substituting successive values of r in the general term. The following example illustrates the mechanism. 1
2 6.2.1 Example 6 = k 2 (2k + 1) = [ 2(2) + 1] + [ 2(3) + 1] [ 2(6) + 1] = = 45. Here, the index (counter) is k. It can be observed that k takes on an initial value of 2 (the lower limit) and increases by steps of 1 until it reaches the upper limit 6. Every value that k assumes is substituted in the general term (2k + 1) in order to generate a term of the series. Obviously, the terms are added up since Sigma stands for summation. In Statistics, however, we do not actually evaluate such expressions numerically but rather use the summation notation strictly for summarisation purposes. This is because the upper limit is generally non-numerical, that is, a n variable. We deal mostly with expressions of the form. If expanded, this = summation cannot be evaluated since it only gives the expression x 1 + x2 + x x n 1 + x n. Such expressions are found in the formulae for arithmetic mean and standard deviation. In this module, students are simply required to recognise the summation notation and understand its meaning so that they can at least use relevant statistical functions on calculators. i x i 1 7. Presentation of data Once information has been collected, it has to be classified and organised in such a way that it becomes easily readable, that is, converted to data. Before calculation of descriptive statistics, it is sometimes a good idea to present it on charts, diagrams or graphs. Most people find diagrams more helpful than figures in the sense that these present data more meaningfully. In this module, we will only consider the presentation of data in the form of histograms and frequency polygons (read the properties of histograms and frequency polygons in Sections 7.3 and 7.4). 7.1 Ungrouped data This type of information occurs as individual observations, usually as a table or array of disorderly values. These observations are to be firstly arranged in some order (ascending or descending if they are numerical) or simply grouped together in the form of a frequency table before proper presentation on diagrams is possible. 2
3 The following will be used as an example of ungrouped data throughout section 6.1 of the notes Example The following data represent the age of students attending full-time B Sc. courses at De Chazal Du Mée Business School: Table (The above information has been collected from the list of B Sc. Students from DCDMBS administration so that the ages are in random order.) 3
4 Once the observations are arranged in ascending order, for example, they can be more easily manipulable in terms of better arrangement and, hence, can be treated more efficiently. Given the relatively large amount of values, 399 to be more precise, a discrete frequency table (see Table below) is a much more appropriate way of classifying them without loss of information. The identity of each value is preserved so that exact calculation of statistics still remains possible (to be dealt with further). Age Frequency Total 399 Table Presentation of ungrouped data on a histogram Histogram of ungrouped data 16 Number of students (frequency) <=18 (18, 19] (19, 2] (2, 21] (21, 22] (22, 23] (23, 24] (24, 25] >25 Age of students Fig
5 7.1.3 Presentation of ungrouped data on a frequency polygon Frequency polygon for ungrouped data 16 Number of students (frequency) Age of students Fig Grouped data When the range of values (not observations) is too wide, a discrete frequency table starts to become quite lengthy and cumbersome. Observations are then grouped into cells or classes in order to compress the set of data for more suitable tabulation. In this case, Example would not be a good illustration, given the little variation in ages of students (from 19 to 24). The main drawback in grouping of data is that the identity (value) of each observation is lost so that important descriptive statistics like the mean and standard deviation can only be estimated and not exactly calculated. For example, if the age group has frequency 5, nothing can be said about the values of these 5 observations. Besides, a lot of new quantities have to be calculated in order to satisfy statistical calculations and analyses as will be explained in the following sections Limits and real limits (or boundaries) A class is bounded by a lower and an upper limit in the previous paragraph, the lower and upper limits of the age group are 21 and 25 5
6 respectively. A real limit is obtained by making a continuity correction to a limit (explained below). In a frequency distribution, we differentiate between limits and real limits by the fact that the upper limit of a cell can never be equal to the lower limit of the next cell. Real limits are fictitious values if the values recorded are discrete. However, they are useful not only for the purpose of calculations but also for presentation of data on histograms as well as several other types of charts and diagrams. For instance, if we have a frequency distribution of ages in which we have the two neighbouring cells and 26 3, then drawing a histogram for this distribution will require that the limits 25 and 26 be equal, the reason being that there is no gap between any two successive rectangles of a histogram! We therefore make a continuity correction of ±.5, the equivalent of half a gap. Note The gap between any pair of successive cells in a frequency distribution is equal to the degree of accuracy to which the original observations were recorded. In the above example, it is easy to deduce that age was recorded to the nearest unit since the gap between the cells and 26 3 is 1. The real limits of these 2 will now be and Note that the following relationships hold: Lower real limit = Lower limit continuity correction Upper real limit = Upper limit + continuity correction Mid-class values (MCV) The mid-class value, MCV, of a cell is defined as its midpoint, that is, the average of its limits or real limits. Thus, the MCV of the cell is 23. The MCV of a cell is the representative of that cell in the sense that, since the values of all the observations in the cell are unknown individually, it is assumed that they are all equal to the MCV. This assumption is not fortuitous and neither is it unjustified. It has the logical implication that if observations are unknown, the best way of estimating statistics more accurately would be to assume that, at least, they are uniformly distributed within the cell (which could be untrue, of course!). Mathematically, the sum of the observations would be equal to the number of observations multiplied by the MCV (think about it!). The importance of the midclass value can thus never be underestimated, especially for the calculation of the crucial statistics like the mean and standard deviation. 6
7 7.2.3 Class interval or cell width The cell width is simply the length of the cell, that is, the difference between its lower and upper real limits. Note Do not make the mistake of subtracting the lower limit from the upper limit since this will not give the exact cell width. This can be easily verified by taking the cell Its cell width is 5 (21, 22, 23, 24 and 25), which is obtained by subtracting 2.5 from We therefore use the following formula: Cell width = Upper real limit Lower real limit Example Consider the following set of data, which represents the ages of workers of a private company. The real limits and mid-class values have already been computed. Age group Real limits Mid-class value Frequency Total 143 Table The data is presented on the histogram in Fig and the frequency polygon in Fig
8 Presentation of ungrouped data (uniform class interval) on a histogram 45 4 Histogram for grouped data Number of workers (frequency) [2.5, 25.5) [25.5, 3.5) [3.5, 35.5) [35.5, 4.5) [4.5, 45.5) [45.5, 5.5) [5.5, 55.5) [55.5, 6.5) Age group of workers Fig Presentation of ungrouped data on a frequency polygon Frequency polygon for grouped data Number of students (frequency) Age of students Fig
9 7.3 Histograms Out of several methods of presenting a frequency distribution graphically, the histogram is the most popular and widely used in practice. A histogram is a set of vertical bars whose areas are proportional to the frequencies of the classes that they represent. While constructing a histogram, the variable is always taken on the x-axis while the frequencies are on the y-axis. Each class is then represented by a distance on the scale that is proportional to its class interval (see Section 7.2.3). The distance for each rectangle on the x-axis shall remain the same in the case that the class intervals are uniform throughout the distribution. If the classes have different class intervals, they will obviously vary accordingly on the x-axis. The y- axis represents the frequencies of each class which constitute the height of the rectangle. When class intervals are unequal, a correction must be made. This consists of finding the frequency density for each class, which is the ratio of the frequency to the class interval. The frequency densities now become the actual heights of the rectangles since the areas of the rectangles should be proportional to the frequencies Example (unequal class intervals) The temperatures (in degrees Fahrenheit) were simultaneously recorded in various cities in the world at a specific moment. Table below gives the thermometer readings. Temperature Class intervals Frequency Frequency density Total 54 Table Note 2 3 means from 2 to 3, including 2 but excluding 3 9
10 Presentation of grouped data (unequal class intervals) on a histogram Histogram (unequal class intervals) using frequency density Frequency density Temparature (degrees Fahrenheit) Fig The histogram should be clearly distinguished from the bar chart. The most striking physical difference between these two diagrams is that, unlike the bar chart, there are no gaps between successive rectangles of a histogram. A bar chart is one-dimensional since only the length, and not the width, matters whereas a histogram is two-dimensional since both length and width are important. A histogram is mainly used to display data for continuous variables but can also be adjusted so as to present discrete data by making an appropriate continuity correction (see Section 7.2.1). Moreover, it can be quite misleading if the distribution has unequal class intervals. 7.4 Frequency polygons A frequency polygon is a graph of frequency distribution. There is a very effective in which a frequency polygon may be constructed: Draw a histogram of the given data and then join, by means of straight lines, the midpoints of the upper horizontal side of each rectangle with the adjacent ones. It is an accepted practice to close the polygon at both ends of the distribution by extending them to the base line. When this is done, two 1
11 hypothetical classes with zero frequencies must be included at each end. This extension is made with the objective of making the area under the polygon equal to the area under the corresponding histogram. A frequency polygon sketches an outline of the data pattern more clearly. In fact, it is the refinement of a histogram, as it does not assume that the frequencies of observations within a class are equal. The polygon becomes increasingly smooth and curve-like as we increase the number of classes in a distribution. Frequency polygon and histogram Frequency density Fig
additionalmathematicsstatisticsadditi onalmathematicsstatisticsadditionalm athematicsstatisticsadditionalmathem aticsstatisticsadditionalmathematicsst
additionalmathematicsstatisticsadditi onalmathematicsstatisticsadditionalm athematicsstatisticsadditionalmathem aticsstatisticsadditionalmathematicsst STATISTICS atisticsadditionalmathematicsstatistic
More informationGrade 3. Grade 3 K 8 Standards 23
Grade 3 In grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; (2) developing
More informationMathematics Grade 3. grade 3 21
Mathematics Grade 3 In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within
More informationCHAPTER 8 INTRODUCTION TO STATISTICAL ANALYSIS
CHAPTER 8 INTRODUCTION TO STATISTICAL ANALYSIS LEARNING OBJECTIVES: After studying this chapter, a student should understand: notation used in statistics; how to represent variables in a mathematical form
More informationBiostatistics Presentation of data DR. AMEER KADHIM HUSSEIN M.B.CH.B.FICMS (COM.)
Biostatistics Presentation of data DR. AMEER KADHIM HUSSEIN M.B.CH.B.FICMS (COM.) PRESENTATION OF DATA 1. Mathematical presentation (measures of central tendency and measures of dispersion). 2. Tabular
More informationPSYCHOLOGICAL STATISTICS
PSYCHOLOGICAL STATISTICS B Sc. Counselling Psychology 011 Admission onwards II SEMESTER COMPLIMETARY COURSE UIVERSITY OF CALICUT SCHOOL OF DISTACE EDUCATIO CALICUT UIVERSITY.P.O., MALAPPURAM, KERALA, IDIA
More informationAuthor : Dr. Pushpinder Kaur. Educational Statistics: Mean Median and Mode
B.ED. PART- II ACADEMIC SESSION : 2017-2018 PAPER XVIII Assessment for Learning Lesson No. 8 Author : Dr. Pushpinder Kaur Educational Statistics: Mean Median and Mode MEAN : The mean is the average value
More informationDescriptive Statistics
Contents 36 Descriptive Statistics 36.1 Describing Data 2 36.2 Exploring Data 26 Learning outcomes In the first Section of this Workbook you will learn how to describe data sets and represent them numerically
More informationMEASURES OF CENTRAL TENDENCY
MAT001-Statistics for Engineers MEASURES OF CENTRAL TENDENCY DESCRIPTIVE STATISTICAL MEASURES Graphical representation summarizes information in the data. In addition to the diagrammatic and graphic representations
More informationClass 11 Maths Chapter 15. Statistics
1 P a g e Class 11 Maths Chapter 15. Statistics Statistics is the Science of collection, organization, presentation, analysis and interpretation of the numerical data. Useful Terms 1. Limit of the Class
More informationF78SC2 Notes 2 RJRC. If the interest rate is 5%, we substitute x = 0.05 in the formula. This gives
F78SC2 Notes 2 RJRC Algebra It is useful to use letters to represent numbers. We can use the rules of arithmetic to manipulate the formula and just substitute in the numbers at the end. Example: 100 invested
More informationGranite School District Parent Guides Utah Core State Standards for Mathematics Grades K-6
Granite School District Parent Guides Grades K-6 GSD Parents Guide for Kindergarten The addresses Standards for Mathematical Practice and Standards for Mathematical Content. The standards stress not only
More informationCHAPTER 14 STATISTICS Introduction
238 MATHEMATICS STATISTICS CHAPTER 14 14.1 Introduction Everyday we come across a wide variety of informations in the form of facts, numerical figures, tables, graphs, etc. These are provided by newspapers,
More informationThe science of learning from data.
STATISTICS (PART 1) The science of learning from data. Numerical facts Collection of methods for planning experiments, obtaining data and organizing, analyzing, interpreting and drawing the conclusions
More informationPre-Algebra (6/7) Pacing Guide
Pre-Algebra (6/7) Pacing Guide Vision Statement Imagine a classroom, a school, or a school district where all students have access to high-quality, engaging mathematics instruction. There are ambitious
More informationMIDTERM EXAMINATION (Spring 2011) STA301- Statistics and Probability
STA301- Statistics and Probability Solved MCQS From Midterm Papers March 19,2012 MC100401285 Moaaz.pk@gmail.com Mc100401285@gmail.com PSMD01 MIDTERM EXAMINATION (Spring 2011) STA301- Statistics and Probability
More informationAssessment in Mathematics Year 6 and KS3. Nigel Bufton MATHSEDUCATIONAL LTD and the London Borough of Camden
Year 6 Number Identify common factors, common multiples, and prime, square and cube numbers; use negative numbers Calculate mentally with mixed operations; use formal written methods to multiply and divide
More informationKCP e-learning. test user - ability basic maths revision. During your training, we will need to cover some ground using statistics.
During your training, we will need to cover some ground using statistics. The very mention of this word can sometimes alarm delegates who may not have done any maths or statistics since leaving school.
More informationCorrelation to the Common Core State Standards
Correlation to the Common Core State Standards Go Math! 2011 Grade 6 Common Core is a trademark of the National Governors Association Center for Best Practices and the Council of Chief State School Officers.
More informationDCSD Common Core State Standards Math Pacing Guide 3rd Grade. Trimester 1
Trimester 1 CCSS Mathematical Practices 1.Make sense of problems and persevere in solving them. 2.Reason abstractly and quantitatively. 3.Construct viable arguments and critique the reasoning of others.
More informationLecture 1 : Basic Statistical Measures
Lecture 1 : Basic Statistical Measures Jonathan Marchini October 11, 2004 In this lecture we will learn about different types of data encountered in practice different ways of plotting data to explore
More informationNew Paltz Central School District Mathematics Third Grade
September - Unit 1: Place Value and Numeration/Addition and Use hundred charts and number lines. Place Value October Subtraction Read and write numbers to 1,000. Pre- What is place value? Order numbers
More informationMath 6 Common Core. Mathematics Prince George s County Public Schools
Math 6 Common Core Mathematics Prince George s County Public Schools 2014-2015 Course Code: Prerequisites: Successful completion of Math 5 Common Core This course begins the transition from the heavy emphasis
More informationThe Not-Formula Book for C2 Everything you need to know for Core 2 that won t be in the formula book Examination Board: AQA
Not The Not-Formula Book for C Everything you need to know for Core that won t be in the formula book Examination Board: AQA Brief This document is intended as an aid for revision. Although it includes
More informationAgile Mind Mathematics 6 Scope and Sequence, Common Core State Standards for Mathematics
In the three years preceding Grade 6, students have acquired a strong foundation in numbers and operations, geometry, measurement, and data. They are fluent in multiplication of multi-digit whole numbers
More informationChapter 2: Tools for Exploring Univariate Data
Stats 11 (Fall 2004) Lecture Note Introduction to Statistical Methods for Business and Economics Instructor: Hongquan Xu Chapter 2: Tools for Exploring Univariate Data Section 2.1: Introduction What is
More information(Refer Slide Time 02:20)
Soil Mechanics Prof. B.V.S. Viswanathan Department of Civil Engineering Indian Institute of Technology, Bombay Lecture 33 Stress Distribution in Soils Lecture No. 6 Students once again we meet. Today s
More informationAgile Mind Mathematics 6 Scope and Sequence, Common Core State Standards for Mathematics
In the three years preceding Grade 6, students have acquired a strong foundation in numbers and operations, geometry, measurement, and data. They are fluent in multiplication of multi- digit whole numbers
More informationMath 2 Variable Manipulation Part 6 System of Equations
Name: Date: 1 Math 2 Variable Manipulation Part 6 System of Equations SYSTEM OF EQUATIONS INTRODUCTION A "system" of equations is a set or collection of equations that you deal with all together at once.
More informationChapter 9: Roots and Irrational Numbers
Chapter 9: Roots and Irrational Numbers Index: A: Square Roots B: Irrational Numbers C: Square Root Functions & Shifting D: Finding Zeros by Completing the Square E: The Quadratic Formula F: Quadratic
More informationSequence of Grade 6 Modules Aligned with the Standards
Sequence of Grade 6 Modules Aligned with the Standards Module 1: Ratios and Unit Rates Module 2: Arithmetic Operations Including Dividing by a Fraction Module 3: Rational Numbers Module 4: Expressions
More informationThese standards are grouped by concepts and are not necessarily arranged in any specific order for presentation.
Transitional Math for Seniors prepares students for their entry-level credit-bearing liberal studies mathematics course at the post-secondary level. This course will solidify their quantitative literacy
More informationChapter 1. ANALYZE AND SOLVE LINEAR EQUATIONS (3 weeks)
Chapter 1. ANALYZE AND SOLVE LINEAR EQUATIONS (3 weeks) Solve linear equations in one variable. 8EE7ab In this Chapter we review and complete the 7th grade study of elementary equations and their solution
More informationMathematics Grade 6. grade 6 39
Mathematics Grade 6 In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to
More informationWhy It s Important. What You ll Learn
How could you solve this problem? Denali and Mahala weed the borders on the north and south sides of their rectangular yard. Denali starts first and has weeded m on the south side when Mahala says he should
More informationGrade 3 Unit Standards ASSESSMENT #1
ASSESSMENT #1 3.NBT.1 Use place value understanding to round whole numbers to the nearest 10 or 100 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties
More informationGrade 3 Yearlong Mathematics Map
Grade 3 Yearlong Mathematics Map Resources: Approved from Board of Education Assessments: PARCC Assessments, Performance Series, District Benchmark Assessments Common Core State Standards Standards for
More informationSupplemental Resources: Engage New York: Lesson 1-21, pages 1.A.3-1.F.45 3 rd Grade Math Folder Performance Task: Math By All Means (Multiplication
Unit 1: Properties of Multiplication and Division and Solving Problems with Units of 2, 3, 4, 5, and 10 (25 days) Unit Description: This unit begins the year by building on students fluency with repeated
More informationSequence Units for the CCRS in Mathematics Grade 3
Sequence Units for the CCRS in Mathematics Grade 3 You must begin explicit instruction of the eight s of Mathematical Practice. Students are expected to know and be able to use them daily. First 9 Weeks
More informationLinear Algebra. The analysis of many models in the social sciences reduces to the study of systems of equations.
POLI 7 - Mathematical and Statistical Foundations Prof S Saiegh Fall Lecture Notes - Class 4 October 4, Linear Algebra The analysis of many models in the social sciences reduces to the study of systems
More informationUNIT 4 RANK CORRELATION (Rho AND KENDALL RANK CORRELATION
UNIT 4 RANK CORRELATION (Rho AND KENDALL RANK CORRELATION Structure 4.0 Introduction 4.1 Objectives 4. Rank-Order s 4..1 Rank-order data 4.. Assumptions Underlying Pearson s r are Not Satisfied 4.3 Spearman
More informationOasis Academy Arena Curriculum Term Plan: Mathematics
Oasis Academy Arena Curriculum Term Plan: Mamatics Reviewed/Updated 25.06.2018 by Alice Cairns and T. Phuntsok Title of Scheme of Learning: Negative, sequences and solving equations (algebra) Subject:
More informationGRADE 6 OVERVIEW. Ratios and Proportional Relationships [RP] Understand ratio concepts and use ratio reasoning to solve problems.
GRADE 6 OVERVIEW Grade 6 content is organized into five domains of focused study as outlined below in the column to the left. The Grade 6 domains listed in bold print on the shaded bars are Ratios and
More informationChapter 5: Exploring Data: Distributions Lesson Plan
Lesson Plan Exploring Data Displaying Distributions: Histograms Interpreting Histograms Displaying Distributions: Stemplots Describing Center: Mean and Median Describing Variability: The Quartiles The
More informationAlgebra. Mathematics Help Sheet. The University of Sydney Business School
Algebra Mathematics Help Sheet The University of Sydney Business School Introduction Terminology and Definitions Integer Constant Variable Co-efficient A whole number, as opposed to a fraction or a decimal,
More informationStatistics. Industry Business Education Physics Chemistry Economics Biology Agriculture Psychology Astronomy, etc. GFP - Sohar University
Statistics اإلحصاء تعاريف 3-1 Definitions Statistics is a branch of Mathematics that deals collecting, analyzing, summarizing, and presenting data to help in the decision-making process. Statistics is
More informationSection-A. Short Questions
Section-A Short Questions Question1: Define Problem? : A Problem is defined as a cultural artifact, which is especially visible in a society s economic and industrial decision making process. Those managers
More informationModule 4 MULTI- RESOLUTION ANALYSIS. Version 2 ECE IIT, Kharagpur
Module 4 MULTI- RESOLUTION ANALYSIS Lesson Theory of Wavelets Instructional Objectives At the end of this lesson, the students should be able to:. Explain the space-frequency localization problem in sinusoidal
More informationThird Grade One-Page Math Curriculum Map for
Third Grade One-Page Math Curriculum Map for 2015-16 First Nine Weeks Second Nine Weeks Third Nine Weeks Fourth Nine Weeks OA.1 OA.2 OA.3 OA.5 (Only use Commutative Property of Multiplication) OA.8 (No
More informationScope and Sequence: National Curriculum Mathematics from Haese Mathematics (7 10A)
Scope and Sequence: National Curriculum Mathematics from Haese Mathematics (7 10A) Updated 06/05/16 http://www.haesemathematics.com.au/ Note: Exercises in red text indicate material in the 10A textbook
More informationAlgebra I. Mathematics Curriculum Framework. Revised 2004 Amended 2006
Algebra I Mathematics Curriculum Framework Revised 2004 Amended 2006 Course Title: Algebra I Course/Unit Credit: 1 Course Number: Teacher Licensure: Secondary Mathematics Grades: 9-12 Algebra I These are
More information6th Grade Pacing Guide st Nine Weeks
Numerical Expressions and Factors NS.2 Fluently divide multi-digit numbers using the standard algorithm. Big Ideas Chapter 1 EE.1 Write and evaluate numerical expressions involving whole-number exponents.
More informationMadison County Schools Suggested 3 rd Grade Math Pacing Guide,
Madison County Schools Suggested 3 rd Grade Math Pacing Guide, 2016 2017 The following Standards have changes from the 2015-16 MS College- and Career-Readiness Standards: Significant Changes (ex: change
More informationUNIT 2 MEAN, MEDIAN AND MODE
Central Tendencies and Dispersion UNIT 2 MEAN, MEDIAN AND MODE Me Structure 2.0 Introduction 2.1 Objectives 2.2 Symbols Used in Calculation of Measures of Central Tendency 2.3 The Arithmetic Mean 2.3.1
More information1.1.1 Algebraic Operations
1.1.1 Algebraic Operations We need to learn how our basic algebraic operations interact. When confronted with many operations, we follow the order of operations: Parentheses Exponentials Multiplication
More informationMath 2 Variable Manipulation Part 7 Absolute Value & Inequalities
Math 2 Variable Manipulation Part 7 Absolute Value & Inequalities 1 MATH 1 REVIEW SOLVING AN ABSOLUTE VALUE EQUATION Absolute value is a measure of distance; how far a number is from zero. In practice,
More information1 Implication and induction
1 Implication and induction This chapter is about various kinds of argument which are used in mathematical proofs. When you have completed it, you should know what is meant by implication and equivalence,
More informationQUADRATIC EQUATIONS M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier
Mathematics Revision Guides Quadratic Equations Page 1 of 8 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier QUADRATIC EQUATIONS Version: 3.1 Date: 6-10-014 Mathematics Revision Guides
More informationFoundations 5 Curriculum Guide
1. Review: Natural Numbers...3 2. Reading and Writing Natural Numbers...6 3. Lines, Rays, and Line Segments...8 4. Comparing Natural Numbers... 12 5. Rounding Numbers... 15 6. Adding Natural Numbers...
More informationCHAPTER 1: Functions
CHAPTER 1: Functions 1.1: Functions 1.2: Graphs of Functions 1.3: Basic Graphs and Symmetry 1.4: Transformations 1.5: Piecewise-Defined Functions; Limits and Continuity in Calculus 1.6: Combining Functions
More informationFunctions and graphs - Grade 10 *
OpenStax-CNX module: m35968 1 Functions and graphs - Grade 10 * Free High School Science Texts Project Heather Williams This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution
More informationCommunication Engineering Prof. Surendra Prasad Department of Electrical Engineering Indian Institute of Technology, Delhi
Communication Engineering Prof. Surendra Prasad Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 41 Pulse Code Modulation (PCM) So, if you remember we have been talking
More informationUnit 1: Ratios and Proportional Relationships
Common Core State Standards for Mathematics Grade 6 This chart correlates the Grade 6 Common Core State Standards in Mathematics to the lessons in Review, Practice, and Mastery. Unit 1: Ratios and Proportional
More informationMath 6 Course Guide
Math 6 Course Guide #620E G6: Math #621E G6: Math Honors #625E G6: ELL Math #626E G6: FS Math #695E G6: Math Spanish #720E Combo: Math #622E Combo: FS Math #771E ACCEL Math 6: Magnet MS Critical Areas
More informationNorth Carolina 6 th GRADE MATH Pacing Guide
North Carolina 6 th GRADE MATH 2018-2019 Pacing Guide The composition of all CASE benchmarks will be adjusted to be in accordance with the NC blueprint when it is released by the state. Note: The eight
More informationMultiple Choice. Chapter 2 Test Bank
Straightforward Statistics 1st Edition Bowen Test Bank Full Download: https://testbanklive.com/download/straightforward-statistics-1st-edition-bowen-test-bank/ Chapter 2 Test Bank Multiple Choice 1. Data
More informationCalifornia CCSS Mathematics Grades 1-3
Operations and Algebraic Thinking Represent and solve problems involving addition and subtraction. 1.OA.1. Use addition and subtraction within 20 to solve word problems involving situations of adding to,
More informationGrade 6: Mathematics Curriculum (2010 Common Core) Warren Hills Cluster (K 8)
Focus Topic:RP Ration & Proportional Relationships TSW = The Student Will TSW understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. (For example,
More information1. For which of these would you use a histogram to show the data? (a) The number of letters for different areas in a postman s bag.
Data Handling 1. For which of these would you use a histogram to show the data? (a) The number of letters for different areas in a postman s bag. (b) The height of competitors in an athletics meet. (c)
More informationOverview of Dispersion. Standard. Deviation
15.30 STATISTICS UNIT II: DISPERSION After reading this chapter, students will be able to understand: LEARNING OBJECTIVES To understand different measures of Dispersion i.e Range, Quartile Deviation, Mean
More informationVirginia Unit-Specific Learning Pathways. Grades 6-Algebra I: Standards of Learning
BUI L T F O VIR R G INIA 2014 2015 Virginia -Specific Learning Pathways Grades 6-Algebra I: Standards of Learning Table of Contents Grade 6...3 Grade 7...6 Grade 8...9 Algebra I... 11 Grade 6 Virginia
More informationYEAR 10 PROGRAM TERM 1 TERM 2 TERM 3 TERM 4
YEAR 10 PROGRAM TERM 1 1. Revision of number operations 3 + T wk 2 2. Expansion 3 + T wk 4 3. Factorisation 7 + T wk 6 4. Algebraic Fractions 4 + T wk 7 5. Formulae 5 + T wk 9 6. Linear Equations 10 +T
More informationPhysics 2A Chapter 1: Introduction and Mathematical Concepts
Physics 2A Chapter 1: Introduction and Mathematical Concepts Anyone who has never made a mistake has never tried anything new. Albert Einstein Experience is the name that everyone gives to his mistakes.
More informationTypes of Symmetry. We will be concerned with two types of symmetry.
Chapter 7: Symmetry Types of Symmetry We will be concerned with two types of symmetry. Types of Symmetry We will be concerned with two types of symmetry. Reflective symmetry Types of Symmetry We will be
More information56 CHAPTER 3. POLYNOMIAL FUNCTIONS
56 CHAPTER 3. POLYNOMIAL FUNCTIONS Chapter 4 Rational functions and inequalities 4.1 Rational functions Textbook section 4.7 4.1.1 Basic rational functions and asymptotes As a first step towards understanding
More information4 th Grade Hinojosa Math Vocabulary Words
Topic 1 Word Definition Picture Digit A symbol used to make numerals. These are the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Place value The value of where the digit is in the number, such as units(ones),
More informationMath Scope and Sequence
Math Scope and Sequence Domain Quarter 1 Quarter 2 Quarter 3 Quarter 4 Standard Standard Standard Standard RATIOS AND 6.G.1 Through composition into 6.EE.2a. Write expressions that rectangles or decomposition
More informationt dt Estimate the value of the integral with the trapezoidal rule. Use n = 4.
Trapezoidal Rule We have already found the value of an integral using rectangles in the first lesson of this module. In this section we will again be estimating the value of an integral using geometric
More informationStandards for Mathematical Practice. Ratio and Proportional Relationships
North Carolina Standard Course of Study Sixth Grade Mathematics 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique
More informationAlgebraic Expressions
ALGEBRAIC EXPRESSIONS 229 Algebraic Expressions Chapter 12 12.1 INTRODUCTION We have already come across simple algebraic expressions like x + 3, y 5, 4x + 5, 10y 5 and so on. In Class VI, we have seen
More informationFor those of you who are taking Calculus AB concurrently with AP Physics, I have developed a
AP Physics C: Mechanics Greetings, For those of you who are taking Calculus AB concurrently with AP Physics, I have developed a brief introduction to Calculus that gives you an operational knowledge of
More informationTOPIC: Descriptive Statistics Single Variable
TOPIC: Descriptive Statistics Single Variable I. Numerical data summary measurements A. Measures of Location. Measures of central tendency Mean; Median; Mode. Quantiles - measures of noncentral tendency
More informationAlgebra Readiness. Curriculum (445 topics additional topics)
Algebra Readiness This course covers the topics shown below; new topics have been highlighted. Students navigate learning paths based on their level of readiness. Institutional users may customize the
More informationGrade 6 - SBA Claim 1 Example Stems
Grade 6 - SBA Claim 1 Example Stems This document takes publicly available information about the Smarter Balanced Assessment (SBA) in Mathematics, namely the Claim 1 Item Specifications, and combines and
More informationMeasures of Central Tendency
Measures of Central Tendency Summary Measures Summary Measures Central Tendency Mean Median Mode Quartile Range Variance Variation Coefficient of Variation Standard Deviation Measures of Central Tendency
More informationMATHEMATICS Math I. Number and Quantity The Real Number System
MATHEMATICS Math I The high school mathematics curriculum is designed to develop deep understanding of foundational math ideas. In order to allow time for such understanding, each level focuses on concepts
More informationGanado Unified School District (Math/6 Grade)
5-4 5-5 5-6 6-1 6-2 6-3 6-4 7-1 7-3 7-4 7-5 7-6 8-1 8-2 8-3 8-4 9-1 9-2 9-3 9-4 6.NS.A.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions,
More informationMiddle School Math Solution: Course 3
Ohio 8.MP MATHEMATICAL PRACTICES The s for Mathematical Practice describe the skills that mathematics educators should seek to develop in their students. The descriptions of the mathematical practices
More informationMIA Textbook INTERMEDIATE 1 CHECKLIST
St Ninian s High School MIA Textbook INTERMEDIATE 1 CHECKLIST I understand this part of the course = I am unsure of this part of the course = I do not understand this part of the course = Name Class Teacher
More informationClinton Community School District K-8 Mathematics Scope and Sequence
6_RP_1 6_RP_2 6_RP_3 Domain: Ratios and Proportional Relationships Grade 6 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. Understand the
More informationLearning Expectations for Sample Middle School. Math - Grade 6
Ratio and Proportional Relationships (Reporting Category #1) Domain: Ratios & Proportional Relationships Cluster: Understand ratio concepts and use ratio reasoning to solve problems. MAFS.6.RP.1.1 Understand
More informationFORCE TABLE INTRODUCTION
FORCE TABLE INTRODUCTION All measurable quantities can be classified as either a scalar 1 or a vector 2. A scalar has only magnitude while a vector has both magnitude and direction. Examples of scalar
More informationThe Matrix Vector Product and the Matrix Product
The Matrix Vector Product and the Matrix Product As we have seen a matrix is just a rectangular array of scalars (real numbers) The size of a matrix indicates its number of rows and columns A matrix with
More information1 Measurement Uncertainties
1 Measurement Uncertainties (Adapted stolen, really from work by Amin Jaziri) 1.1 Introduction No measurement can be perfectly certain. No measuring device is infinitely sensitive or infinitely precise.
More informationOhio s Learning Standards-Extended. Mathematics. Ratio and Proportional Relationships Complexity a Complexity b Complexity c
Ohio s Learning Standards-Extended Mathematics Ratio and Proportional Relationships Complexity a Complexity b Complexity c Most Complex Least Complex Understand ratio concepts and use ratio reasoning to
More informationDestination Math. Scope & Sequence. Grades K 12 solutions
Destination Math Scope & Sequence Grades K 12 solutions Table of Contents Destination Math Mastering Skills & Concepts I: Pre-Primary Mathematics, Grades K-1... 3 Destination Math Mastering Skills & Concepts
More informationArithmetic with Whole Numbers and Money Variables and Evaluation (page 6)
LESSON Name 1 Arithmetic with Whole Numbers and Money Variables and Evaluation (page 6) Counting numbers or natural numbers are the numbers we use to count: {1, 2, 3, 4, 5, ) Whole numbers are the counting
More informationMapping Common Core State Standard Clusters and. Ohio Grade Level Indicator. Grade 6 Mathematics
Mapping Common Core State Clusters and Ohio s Grade Level Indicators: Grade 6 Mathematics Ratios and Proportional Relationships: Understand ratio concepts and use ratio reasoning to solve problems. 1.
More informationChapter 2. Motion in One Dimension. AIT AP Physics C
Chapter 2 Motion in One Dimension Kinematics Describes motion while ignoring the agents that caused the motion For now, will consider motion in one dimension Along a straight line Will use the particle
More informationMath 1 Variable Manipulation Part 5 Absolute Value & Inequalities
Math 1 Variable Manipulation Part 5 Absolute Value & Inequalities 1 ABSOLUTE VALUE REVIEW Absolute value is a measure of distance; how far a number is from zero: 6 is 6 away from zero, and " 6" is also
More information