2 Central Tendency n A single summary score that best describes the central location of an entire distribution of scores. n Measures of Central Tendency: n Mean n The sum of all scores divided by the number of scores. n Median n The value that divides the distribution in half when observations are ordered. n Mode n The most frequent score.
3 Central Tendency Example: Mode n 52, 76, 100, 136, 186, 196, 205, 150, 257, 264, 264, 280, 282, 283, 303, 313, 317, 317, 325, 373, 384, 384, 400, 402, 417, 422, 472, 480, 643, 693, 732, 749, 750, 791, 891 n Mode: most frequent observation n Mode(s) for hotel rates: n 264, 317, 384
4 Pros and Cons of the Mode n Pros n Good for nominal data. n Good when there are two typical scores. n Easiest to compute and understand. n The score comes from the data set. n Cons n Ignores most of the information in a distribution. n Small samples may not have a mode.
5 Central Tendency Example: Median n 52, 76, 100, 136, 186, 196, 205, 150, 257, 264, 264, 280, 282, 283, 303, 313, 317, 317, 325, 373, 384, 384, 400, 402, 417, 422, 472, 480, 643, 693, 732, 749, 750, 791, 891 n The median is the middle value when observations are ordered. n To find the middle, count in (N+1)/2 scores when observations are ordered lowest to highest. n Median hotel rate: n (35+1)/2 = 18 n 317
6 Finding the median with an even number of scores. n 2, 2, 3, 5, 6, 7, 7, 7, 8, 9 n With an even number of scores, the median is the average of the middle two observations when observations are ordered. n Find the average of the N/2 and the (N/2)+1 score. n N/2 = 5 th score, (N/2)+1 = 6 th score n Add middle two observations and divide by two. n (6+7)/2 = 6.5 n Median is 6.5
7 Pros and Cons of Median n Pros n Not influenced by extreme scores or skewed distributions. n Good with ordinal data. n Easier to compute than the mean. n Cons n May not exist in the data. n Doesn t take actual values into account.
8 Mean: the average of all the scores mean= x N eg mean = = 4.1 *the most commonly used measure of central tendency *the balance point of a distribution (illustrated next slide)
9 Mean n Is the balance point of a distribution.
10 Mean n Population mu µ = ΣX N sigma, the sum of X, add up all scores N, the total number of scores in a population n Sample sigma, the sum of X, add up all scores X bar X = ΣX n n, the total number of scores in a sample
11 Central Tendency Example: Mean n 52, 76, 100, 136, 186, 196, 205, 150, 257, 264, 264, 280, 282, 283, 303, 313, 317, 317, 325, 373, 384, 384, 400, 402, 417, 422, 472, 480, 643, 693, 732, 749, 750, 791, 891 n Mean hotel rate: X X = ΣX n = = Mean hotel rate: $371.60
12 Pros and Cons of the Mean n Pros n Mathematical center of a distribution. n Just as far from scores above it as it is from scores below it. n Good for interval and ratio data. n Does not ignore any information. n Inferential statistics is based on mathematical properties of the mean. n Cons n Influenced by extreme scores and skewed distributions. n May not exist in the data.
13 The effect of skew on average. n In a normal distribution, the mean, median, and mode are the same. n In a skewed distribution, the mean is pulled toward the tail.
14 Which average? n Each measure contains a different kind of information. n For example, all three measures are useful for summarizing the distribution of American household incomes. n In 1998, the income common to the greatest number of households was $25,000. n Half the households earned less than $38,885. n The mean income was $50,600. n Reporting only one measure of central tendency might be misleading and perhaps reflect a bias.
15 Describing Data with Tables & Graphs
16 Descriptive Statistics n The goal of descriptive statistics is to summarize a collection of data in a clear and understandable way. n What is the pattern of scores over the range of possible values? n Where, on the scale of possible scores, is a point that best represents the set of scores? n Do the scores cluster about their central point or do they spread out around it?
17 What is the pattern of scores? n Graphs often make it easier to see certain characteristics and trends in a set of data. n Graphs for quantitative data. n n n Stem and Leaf Display Histogram Frequency Polygon n Graphs for qualitative data. n n Bar Chart Pie Chart number/count class/category
18 Histogram n Consists of a number of bars placed side by side. n The width of each bar indicates the interval size. n The height of each bar indicates the frequency of the interval. n There are no gaps between adjacent bars. n Continuous nature of quantitative data.
19 Tables: Frequency Distributions n (def) organizes raw data or observations that have been collected by showing how often an observation occurs in each class n For quantitative data n Ungrouped: list all possible scores that occur, and then indicate how often each score occurs n (Maximizes information about individual scores) n Grouped: combine all possible scores into classes and indicate how often each score occurs within a class n (Easier to see patterns in data, but lose information bout individual scores) n Do not use if you have more than 20 classes!
20 Organize these depression scores! (ranging from 1 to 10) Ungrouped 1. Each observation should be included in only one class 2. List all classes, even those with zero frequencies score f Total 32
21 Organize these depression scores! (ranging from 1 to 10) Grouped 1. Find the lowest and highest scores 2. Classes should have (roughly) equal intervals 3. Each observation should be included in only one class 4. List all classes, even those with zero frequencies score 1-3 (no depression) (moderate depression) 7-10 (high depression) 4 Total 32 f 8
22 Tables: Relative Frequency Distributions n (def) a frequency distribution that also includes the proportion of each group n Divide the frequency of each class by the total frequency 20/32 =.625 score f Relative f 1-3 (no depression) (moderate depression) (high depression) Total
23 Tables: Cumulative Frequency Distributions n (def) a frequency distribution that also includes the total number of observations in each class and all lower-ranked classes n Cumulative f: Add the frequency of each class to the sum of all classes ranked before it n Cumulative %: Divide the cumulative frequency by the total sample (i.e., percentile rank) score f Relative f Cumulative f Cumulative % 1-3 (no depression) % 4-6 (moderate depression) % 7-10 (high depression) % Total = 28 28/32 =.875
24 Graphs for quantitative data: Histogram n Consists of bars placed side by side (to represent continuity of data), the width of each bar indicates a single interval, the height indicates the frequency of the interval 10! 8! Frequency! 6! 4! 2! 0! 1! 2! 3! 4! 5! 6! 7! 8! 9! Level of Depression! 10!
25 Outliers n (def) a very extreme score n Is it accurate? n Should you segregate it from your summary? n Will it enhance your understanding?
26 Graphs: Shape
27 Review! Assume all scores go from low to high n Draw and describe the shape: n IQ scores for the general population n Attractiveness scores for a bunch of supermodels n 1 st grader s math scores on a college-level exam n Annual income level for a community consisting of equal numbers of relatively poor and relatively wealthy people
28 Graphs for quantitative data: Frequency Polygons n A line graph that emphasizes the continuity of continuous variables n Uses a single point rather than a bar Frequency! 10! 8! 6! 4! 2! 0! 3! 3! 2! 2! 1! 1! 4! 5! 6! 7! Level of Depression! 10! 10! 9! 9! 8! 8!
29 Graphs for quantitative data: Stem & Leaf n A display for sorting data on the basis of leading and trailing digits
30 Raw data Stem Leaf
31 Histogram Las Vegas Hotel R ates Frequency hotel rates Rates
32 Shapes of Histograms
33 Frequency Polygons 35 n Uses a single point rather than a bar Frequency Range
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
Northumberland Knowledge Know Guide How to Analyse Data - November 2012 - This page has been left blank 2 About this guide The Know Guides are a suite of documents that provide useful information about
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
DESCRIPTIVE STATISTICS AND EXPLORATORY DATA ANALYSIS SEEMA JAGGI Indian Agricultural Statistics Research Institute Library Avenue, New Delhi - 110 012 firstname.lastname@example.org 1. Descriptive Statistics Statistics
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
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
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
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
Statistics Revision Sheet Question 6 of Paper The Statistics question is concerned mainly with the following terms. The Mean and the Median and are two ways of measuring the average. sumof values no. of
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
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
BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential
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,
Data exploration with Microsoft Excel: univariate analysis Contents 1 Introduction... 1 2 Exploring a variable s frequency distribution... 2 3 Calculating measures of central tendency... 16 4 Calculating
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
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
Lecture 1: Review and Exploratory Data Analysis (EDA) Sandy Eckel email@example.com Department of Biostatistics, The Johns Hopkins University, Baltimore USA 21 April 2008 1 / 40 Course Information I Course
Chapter : Frequency Distributions and Graphs 1 Presenting frequency distributions as graphs In a statistical study, researchers gather data that describe the particular variable under study. To present
A Few Sources for Data Examples Used Introduction to Environmental Statistics Professor Jessica Utts University of California, Irvine firstname.lastname@example.org 1. Statistical Methods in Water Resources by D.R. Helsel
Chapter 1: Exploring Data Chapter 1 Review 1. As part of survey of college students a researcher is interested in the variable class standing. She records a 1 if the student is a freshman, a 2 if the student
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
CHAPTER8 Elementary Quantitative Data Analysis Why Do Statistics? Case Study: The Likelihood of Voting How to Prepare Data for Analysis What Are the Options for Displaying Distributions? Graphs Frequency
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
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,
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
Data used in this guide: studentp.sav (http://people.ysu.edu/~gchang/stat/studentp.sav) Organize and Display One Quantitative Variable (Descriptive Statistics, Boxplot & Histogram) 1. Move the mouse pointer
There are three kinds of people in the world those who are good at math and those who are not. PSY 511: Advanced Statistics for Psychological and Behavioral Research 1 Positive Views The record of a month
4 A Picture Really Is Worth a Thousand Words Difficulty Scale (pretty easy, but not a cinch) What you ll learn about in this chapter Why a picture is really worth a thousand words How to create a histogram
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
Essentials of Statistics for the Social and Behavioral Sciences Barry H. Cohen R. Brooke Lea John Wiley & Sons, Inc. Essentials of Statistics for the Social and Behavioral Sciences Essentials of Behavioral
Information Technology Services will be updating the mark sense test scoring hardware and software on Monday, May 18, 2015. We will continue to score all Spring term exams utilizing the current hardware
Wilder Research Analyzing and interpreting data Evaluation resources from Wilder Research Once data are collected, the next step is to analyze the data. A plan for analyzing your data should be developed
Iris Sample Data Set Basic Visualization Techniques: Charts, Graphs and Maps CS598 Information Visualization Spring 2010 Many of the exploratory data techniques are illustrated with the Iris Plant data
Exploratory Data Analysis Johannes Schauer email@example.com Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction
Appendix 2.1 Tabular and Graphical Methods Using Excel 1 Appendix 2.1 Tabular and Graphical Methods Using Excel The instructions in this section begin by describing the entry of data into an Excel spreadsheet.
Engineering Problem Solving and Excel EGN 1006 Introduction to Engineering Mathematical Solution Procedures Commonly Used in Engineering Analysis Data Analysis Techniques (Statistics) Curve Fitting techniques
Introduction to Statistics and Quantitative Research Methods Purpose of Presentation To aid in the understanding of basic statistics, including terminology, common terms, and common statistical methods.
Chapter 6 Data Analysis, Statistics, and Probability Content Strand Description Questions in this content strand assessed students skills in collecting, organizing, reading, representing, and interpreting
Common Tools for Displaying and Communicating Data for Process Improvement Packet includes: Tool Use Page # Box and Whisker Plot Check Sheet Control Chart Histogram Pareto Diagram Run Chart Scatter Plot
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
STATISTICAL ANALYSIS WITH EXCEL COURSE OUTLINE Perhaps Microsoft has taken pains to hide some of the most powerful tools in Excel. These add-ins tools work on top of Excel, extending its power and abilities
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
THE UNIVERSITY OF TEXAS AT TYLER COLLEGE OF NURSING 1 COURSE SYLLABUS NURS 5317 STATISTICS FOR HEALTH PROVIDERS Fall 2013 & Danice B. Greer, Ph.D., RN, BC firstname.lastname@example.org Office BRB 1115 (903) 565-5766
Contacts University of California Curriculum Integration (UCCI) Institute Sarah Fidelibus, UCCI Program Manager Street Address: 1111 Franklin Street Oakland, CA 94607 1. Program Information Mailing Address:
Prescription: 430 Statistics and Financial Mathematics for Business Elective prescription Level 4 Credit 20 Version 2 Aim Students will be able to summarise, analyse, interpret and present data, make predictions
Quantitative Methods for Finance Module 1: The Time Value of Money 1 Learning how to interpret interest rates as required rates of return, discount rates, or opportunity costs. 2 Learning how to explain
2. Filling Data Gaps, Data validation & Descriptive Statistics Dr. Prasad Modak Background Data collected from field may suffer from these problems Data may contain gaps ( = no readings during this period)
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
An Introduction to Statistics using Microsoft Excel BY Dan Remenyi George Onofrei Joe English Published by Academic Publishing Limited Copyright 2009 Academic Publishing Limited All rights reserved. No
Chapter 2 Describing Data: Frequency Distributions and Graphic Presentation GOALS When you have completed this chapter, you will be able to: Organize raw data into a frequency distribution Produce a histogram,
Activity 3.7 Statistical Analysis with Excel Introduction Engineers use various tools to make their jobs easier. Spreadsheets can greatly improve the accuracy and efficiency of repetitive and common calculations;
Improving the Performance of Data Mining Models with Data Preparation Using SAS Enterprise Miner Ricardo Galante, SAS Institute Brasil, São Paulo, SP ABSTRACT In data mining modelling, data preparation
When to Use a Particular Statistical Test Central Tendency Univariate Descriptive Mode the most commonly occurring value 6 people with ages 21, 22, 21, 23, 19, 21 - mode = 21 Median the center value the
You can analyze your data and display it in a histogram (a column chart that displays frequency data) by using the Histogram tool of the Analysis ToolPak. This data analysis add-in is available when you
SECTION 2-1: OVERVIEW Chapter 2 Describing, Exploring and Comparing Data 19 In this chapter, we will use the capabilities of Excel to help us look more carefully at sets of data. We can do this by re-organizing
Contents An introduction to using Microsoft Excel for quantitative data analysis 1 Introduction... 1 2 Why use Excel?... 2 3 Quantitative data analysis tools in Excel... 3 4 Entering your data... 6 5 Preparing
4. Give 5 different numbers such that their average is 21. The numbers are: I found these numbers by: 5. A median of a set of scores is the value in the middle when the scores are placed in order. In case
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
Data Preparation Part 1: Exploratory Data Analysis & Data Cleaning, Missing Data CAS Predictive Modeling Seminar Louise Francis Francis Analytics and Actuarial Data Mining, Inc. www.data-mines.com Louise.email@example.com
TIPS FOR DOING STATISTICS IN EXCEL Before you begin, make sure that you have the DATA ANALYSIS pack running on your machine. It comes with Excel. Here s how to check if you have it, and what to do if you
1. Data: exploratory data analysis Content 1.1 Introduction 1.2 Tables and diagrams 1.3 Describing univariate numerical data Ref: Pagano and Gauvreau, Chapters 2 and 3. Data! Data! Data! I can t make bricks
Running head: EVALUATING THE RESULTS OF A CAR CRASH STUDY USING SAS 1 Evaluating the results of a car crash study using Statistical Analysis System Kennesaw State University 2 Abstract Part 1. The study
Chapter 2 Overview Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Classify as categorical or qualitative data. 1) A survey of autos parked in
CONCEPT DEVELOPMENT Mathematics Assessment Project CLASSROOM CHALLENGES A Formative Assessment Lesson Representing Data with Graphs Mathematics Assessment Resource Service University of Nottingham & UC
Chapter 2 Data Exploration 2.1 Data Visualization and Summary Statistics After clearly defining the scientific question we try to answer, selecting a set of representative members from the population of
G3658-6 Program Development and Evaluation Analyzing Quantitative Data Ellen Taylor-Powell Statistical analysis can be quite involved. However, there are some common mathematical techniques that can make
MAS131: Introduction to Probability and Statistics Semester 1: Introduction to Probability Lecturer: Dr D J Wilkinson Statistics is concerned with making inferences about the way the world is, based upon
Name Score: Math 227 Review Exam 1 Chapter 2& Fall 2011 ********************************************************************************************************************** SHORT ANSWER. Show work on
My presentation is about data visualization. How to use visual graphs and charts in order to explore data, discover meaning and report findings. The goal is to show that visual displays can be very effective
1 Using SAS outside of ITCs Statistical Methods and Computing, 22S:30/105 Instructor: Cowles Lab 1 Jan 31, 2014 You can access SAS from off campus by using the ITC Virtual Desktop Go to https://virtualdesktopuiowaedu
Basic Probability and Statistics Review Six Sigma Black Belt Primer Pat Hammett, Ph.D. January 2003 Instructor Comments: This document contains a review of basic probability and statistics. It also includes
Goals: Recognize variables as: Qualitative or Quantitative Discrete Continuous Study Ch. 2.1, # 1 13 : Prof. G. Battaly, Westchester Community College, NY Study Ch. 2.1, # 1 13 Variable: characteristic
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
Introduction to Michael P. Wiper, Universidad Carlos III de Madrid Course objectives This course provides a brief introduction to statistics and probability. Firstly, we shall analyze the different methods
Data Mining: Exploring Data Lecture Notes for Chapter 3 Introduction to Data Mining by Tan, Steinbach, Kumar Tan,Steinbach, Kumar Introduction to Data Mining 8/05/2005 1 What is data exploration? A preliminary
INTRODUCTION The aim of this How To guide is to provide advice on how to analyse your data and how to present it. If you require any help with your data analysis please discuss with your divisional Clinical
Math 10 Name W C Exam 5: Chapter 13 Each problem is worth 10 points. You must show all work to receive full credit. 1. State whether each statement is true or false. a. A normal distribution has a bell
1 Statistical Data analysis With Excel For HSMG.632 students Dialog Boxes Descriptive Statistics with Excel To find a single descriptive value of a data set such as mean, median, mode or the standard deviation,
Bowerman, O'Connell, Aitken Schermer, & Adcock, Business Statistics in Practice, Canadian edition Online Learning Centre Technology Step-by-Step - Excel Microsoft Excel is a spreadsheet software application
Student Learning Development Using averages This guide explains the different types of average (mean, median and mode). It details their use, how to calculate them, and when they can be used most effectively.
Math Elementary Statistics: A Brief Version, 5/e Bluman Ch. 3. # 3, 4,, 30, 3, 3 Find (a) the mean, (b) the median, (c) the mode, and (d) the midrange. 3) High Temperatures The reported high temperatures
INTRODUCING THE NORMAL DISTRIBUTION IN A DATA ANALYSIS COURSE: SPECIFIC MEANING CONTRIBUTED BY THE USE OF COMPUTERS Liliana Tauber Universidad Nacional del Litoral Argentina Victoria Sánchez Universidad
Keep equipment too long and drive cost higher than the point of purchasing a new vehicle. Disposing of equipment too early and you lose some of the useful life further increasing cost. The general accepted