Chapter 4. Displaying and Summarizing. Quantitative Data
|
|
- Kathryn Griffin
- 5 years ago
- Views:
Transcription
1 STAT 141 Introduction to Statistics Chapter 4 Displaying and Summarizing Quantitative Data Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
2 4.1 Histograms 1 We divide the range of the data into classes of equal width. 2 Count the number of observations in each class. 3 Draw the histogram. Each bar has equal width and the height of each bar is the class count. Relative frequency histograms Replace the counts on the vertical axis with the percentage or proportion of the total number of cases falling in each class. (Step 2) Question? Compare histograms with bar charts. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
3 Example 4.1 Three steps to create a histogram for quantitative data. Figure: Quantitative data Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
4 Example 4.1 (cont d) 1: Find the range of the data: [ , ]. Consider a expended range: [1980,2024]. Use class width 4 to divide the range into 11 classes. Division is NOT unique. We normally choose integer width. 2: Count the number of observations in each class : Draw the histogram. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
5 Example 4.1 (cont d) Figure: Histogram of Example 4.1 Bin Zou STAT 141 University of Alberta Winter / 31
6 4.2 Stem-and-Leaf Displays 1: Divide each observed value into two parts: - The leading digit(s) of a number is called stem. - The rest of the digit(s) of a number is called leaf. Example: for a two-digit number 19, 1 is the stem and 9 is the leaf. Note: use only one digit for each leaf, either round or truncate the data values to one decimal place after the stem. 2: List the stems in a column (with the smallest at the bottom), and place a vertical line to the right of this column. 3: For each measurement, record the leaf portion in the same row as its corresponding stem. 4: Order the leaves from lowest to highest in each stem. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
7 Example 4.2 Construct a stem-and-leaf plot for the prices of walking shoes. Prices of walking shoes in $: First, we order the data from smallest to largest: The first digit of each observation is the stem, while the second digit is the leaf. The stems of the data are: 4, 6, 7, 8 and 9. Place them in a column. Put the leaves in each column with the ascending order. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
8 Stem-and-leaf Plot for Example Table: Stem-and-leaf Plot for Example 4.2 Remark: count all repeating leaves in each stem. order is important for both stems and leaves. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
9 More Questions about Stem-and-leaf Question 1 What if the data were 520, 541, 542,...? Or even in decimals, like 12.2, 14.8, 18.9,...? Use the first two digits as the stem, and the last one as the leaf, e.g., 52 0, Question 2 What if there are too many leaves for a stem? For instance, We can further divide the stem 5 into 5a (containing all observations from 50 to 54) and 5b (containing all observations from 55 to 59). Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
10 4.3 Dotplots A dotplot places a dot along an axis for each case in the data. Example 4.3 Assume the data are given by: Figure: Dotplot of Example 4.3 Note: Dotplots can be drawn vertically as well, meaning switch the axes in the figure above. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
11 4.4 The Shape of a Distribution Modes Modes are peaks (or humps) of a histogram. Equivalently, the mode is the value that occurs with the highest frequency in a data set. A histogram with one mode is called unimodal. Histograms with two peaks are bimodal. Those with three or more are called multimodal. A histogram with no mode (namely, all categories have approximately the same counts) is called uniform. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
12 Examples Figure: Histograms of different types Bin Zou STAT 141 University of Alberta Winter / 31
13 Symmetric Histograms When folded along a vertical line through the middle, a symmetric histogram should have the edges match pretty closely. Figure: Example of a symmetric histogram Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
14 Skewed Histograms If one tail stretches out farther than the other, the histogram is said to be skewed to the side of the longer tail. Figure: Example of skewed histograms In the above example, the histogram on the left (colored in blue) is skewed to the left (also called negatively skewed), and the one on the right is skewed to the right (positively skewed). Bin Zou STAT 141 University of Alberta Winter / 31
15 Outliers Outliers are extremely large or small observations, which are located away from the main body of the distribution. Apparently, the three observations in the leftmost bar are outliers. Bin Zou STAT 141 University of Alberta Winter / 31
16 Review of the Shape of A Distribution Example 4.4 Recall that there are three aspects regarding the shape of a distributions: number of modes, symmetry, and outliers. Discuss the shape of the distribution described in the picture below. Figure: The shape of a distribution Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
17 4.5 The Centre of a Distribution Measuring the Center: Mean Suppose a sample consists of n observations: x 1,x 2,...,x n. The mean (or average) of these values is given by x := n x i i=1 n = x 1 + x x n. n x is pronounced x-bar. The notation reads as sigma (Greek letter S ), meaning the summation over the index. Example 4.5 Given x 1 = 14.1, x 2 = 3.2, x 3 = 25.3, x 4 = 2.8,x 5 = 17.5,x 6 = 13.9, x 7 = Calculate the mean. x = x 1 + x x 7 7 = = Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
18 More about Mean Mean is a good measure of the centre when the data are (approximately) symmetric. If a distribution is skewed or has outliers, then mean is NOT a reliable measure of the centre. To see the impact of a single observation on the mean, we change x 7 in Example 4.5 from 45.8 to The new mean is calculated as follows x = = Compared with the mean calculated in Example 4.5, the new mean is more than 10 times bigger. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
19 Measuring the Centre: Median For a skewed distribution, the median is a better measure of the centre comparing with the mean. For a set of data in ascending (or descending) order, the median is the value that divide the data in half. Denote n the total number of observations. Assume that the data has been sorted in ascending order. If n is odd, then the median is the observation in the n+1 2 position. If n is even, then the median is the average of the two values in positions n 2 and n Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
20 Examples Example 4.6 The data are the same as in Example 4.5: x 1 = 14.1, x 2 = 3.2, x 3 = 25.3, x 4 = 2.8,x 5 = 17.5,x 6 = 13.9, x 7 = Rearrange the data in ascending order as follows: 17.5, 2.8, 3.2, 13.9, 14.1, 25.3, The total number of observations is 7, an odd number. Hence, the median is in the = 4th position. Namely, 13.9 is the median. Note: since the median is only related to the middle value(s), it is NOT affected by outliers. In the above example, if we change x 7 from 45.8 to 1000, the median stays the same. What happened to the mean example? Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
21 Examples Example 4.7 The previous example consists of odd number observations. Now, we add another observation with the value 35.7 to the previous data, and recalculate the median. The ordered data in Example 4.7 are listed as: 17.5, 2.8, 3.2, 13.9, 14.1, 25.3, 35.7, With n = 8, the median is the average of the values in the 4th and 5th positions. Hence, the median is given by = Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
22 Mean VS Median In a symmetric distribution, mean = median. In a positively (right) skewed distribution, mean > median. In a negatively (left) skewed distribution, mean < median. Example 4.8 Given the data: -2, -1, 0, 1, 2. The data set is symmetric about 0, and mean = median = 0. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
23 4.6 The Spread of a Distribution The more the data vary, the less a measure of centre can tell us. Why? Let s have a look up an extreme example. Assume you know the mean of the given data is 10, but the data are unknown to you. Both the sets (-10000, 30, 10000) and (10, 10, 10) have the same mean 10. However, these two sets of data are significantly different. We need to know the spread of a distribution as well. Range Range = max - min. Note: range defined here is a singe number, not an interval as in common sense. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
24 The Interquartile Range A percentile is a measure that indicates the value below which a given percentage of observations in a group of observations fall. The first quartile (also called lower quartile), Q1, is the percentile of 25%, meaning 25% percentage of observations are below Q1. The median (second quartile) is the percentile of 50%. Sometimes, we use Q2 to denote the median. Similarly, we define the third quartile (also called upper quartile), Q3, as the percentile of 75%. Interquartile range (IQR) = Q3 Q1. Note: IQR is a proper measure of spread when a distribution is skewed. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
25 Examples Example 4.6 revisited Data are ordered as: 17.5, 2.8, 3.2, 13.9, 14.1, 25.3, We have calculated Q2 = 13.9 in Example 4.6. Split the data into two equal parts: 17.5,2.8,3.2,13.9 and 13.9, 14.1, 25.3, Note: when n is odd (7 in this example), the median is included in both parts. Q1 is then the median of the first part of the data, calculated as Q1 = = 3.0. Q3 is the median of the second part of the data, calculated as Q3 = = Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
26 Examples Example 4.7 revisited Data are ordered as: 17.5, 2.8, 3.2, 13.9, 14.1, 25.3, 35.7, Separate the data into two parts with equal number. The set of the data has 8 observations, so each new part has 4 observations: 17.5, 2.8, 3.2, 13.9 and 14.1, 25.3, 35.7, Same as the example in the previous slide, we calculate the first quartile Q1 = = 3.0, 2 and the third quartile Q3 = = Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
27 The Standard Deviation If a distribution is skewed, use IQR. What if a distribution is (approximately) symmetric? The answer is: the standard deviation. Assume a data set of n observations: y 1,y 2,...,y n. The mean is denoted as ȳ. The formula to calculate the standard deviation is n (y i ȳ) 2 n y i=1 i 2 nȳ 2 i=1 s = =. n 1 n 1 s 2 is called the variance. Note: the standard deviation defined here is actually the standard deviation of a sample, NOT a population. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
28 Examples Figure: Finding the standard deviation Bin Zou STAT 141 University of Alberta Winter / 31
29 Empirical Rules There are three empirical rules about the standard deviation. (1) About 68% of the data will lie within 1 standard deviation of the mean. (2) Nearly all, about 95%, of the data will lie within 2 standard deviations of the mean. (3) Virtually all, about %, of the data will lie within 3 standard deviations of the mean. You will know the reason behind these rules after we have covered the Normal distribution. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
30 5-Number Summary The 5-number summary for a set of data reports: Minimum, Lower quartile (Q1), Median (Q2), Upper quartile (Q3), Maximum. The 5-number summary provides measures for centre, spread, and skewness. For a right skewed distribution, Q2 Q1 < Q3 Q2. For a left skewed distribution, Q2 Q1 > Q3 Q2. Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
31 Summarizations of Chapter 4 1 Displaying quantitative data: histogram, stem-and-leaf plot, dotplot. 2 Shape: unimodal/bimodal/multimodal/uniform, symmetric/skewed, outlier. 3 Symmetric distributions: mean + standard deviation. 4 Skewed distribution: median + IQR (5-number summary). Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter / 31
Chapter 4.notebook. August 30, 2017
Sep 1 7:53 AM Sep 1 8:21 AM Sep 1 8:21 AM 1 Sep 1 8:23 AM Sep 1 8:23 AM Sep 1 8:23 AM SOCS When describing a distribution, make sure to always tell about three things: shape, outliers, center, and spread
More informationChapter 5. Understanding and Comparing. Distributions
STAT 141 Introduction to Statistics Chapter 5 Understanding and Comparing Distributions Bin Zou (bzou@ualberta.ca) STAT 141 University of Alberta Winter 2015 1 / 27 Boxplots How to create a boxplot? Assume
More information1-1. Chapter 1. Sampling and Descriptive Statistics by The McGraw-Hill Companies, Inc. All rights reserved.
1-1 Chapter 1 Sampling and Descriptive Statistics 1-2 Why Statistics? Deal with uncertainty in repeated scientific measurements Draw conclusions from data Design valid experiments and draw reliable conclusions
More informationSTP 420 INTRODUCTION TO APPLIED STATISTICS NOTES
INTRODUCTION TO APPLIED STATISTICS NOTES PART - DATA CHAPTER LOOKING AT DATA - DISTRIBUTIONS Individuals objects described by a set of data (people, animals, things) - all the data for one individual make
More informationHistograms allow a visual interpretation
Chapter 4: Displaying and Summarizing i Quantitative Data s allow a visual interpretation of quantitative (numerical) data by indicating the number of data points that lie within a range of values, called
More informationQUANTITATIVE DATA. UNIVARIATE DATA data for one variable
QUANTITATIVE DATA Recall that quantitative (numeric) data values are numbers where data take numerical values for which it is sensible to find averages, such as height, hourly pay, and pulse rates. UNIVARIATE
More informationSTAT 200 Chapter 1 Looking at Data - Distributions
STAT 200 Chapter 1 Looking at Data - Distributions What is Statistics? Statistics is a science that involves the design of studies, data collection, summarizing and analyzing the data, interpreting the
More informationLecture 6: Chapter 4, Section 2 Quantitative Variables (Displays, Begin Summaries)
Lecture 6: Chapter 4, Section 2 Quantitative Variables (Displays, Begin Summaries) Summarize with Shape, Center, Spread Displays: Stemplots, Histograms Five Number Summary, Outliers, Boxplots Cengage Learning
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 informationFurther Mathematics 2018 CORE: Data analysis Chapter 2 Summarising numerical data
Chapter 2: Summarising numerical data Further Mathematics 2018 CORE: Data analysis Chapter 2 Summarising numerical data Extract from Study Design Key knowledge Types of data: categorical (nominal and ordinal)
More informationStat 20: Intro to Probability and Statistics
Stat 20: Intro to Probability and Statistics Lecture 5: Summary Statistics Tessa L. Childers-Day UC Berkeley 30 June 2014 By the end of this lecture... You will be able to: Describe a data set by its:
More informationLecture 2. Quantitative variables. There are three main graphical methods for describing, summarizing, and detecting patterns in quantitative data:
Lecture 2 Quantitative variables There are three main graphical methods for describing, summarizing, and detecting patterns in quantitative data: Stemplot (stem-and-leaf plot) Histogram Dot plot Stemplots
More informationChapter 3. Data Description
Chapter 3. Data Description Graphical Methods Pie chart It is used to display the percentage of the total number of measurements falling into each of the categories of the variable by partition a circle.
More informationLecture 3B: Chapter 4, Section 2 Quantitative Variables (Displays, Begin Summaries)
Lecture 3B: Chapter 4, Section 2 Quantitative Variables (Displays, Begin Summaries) Summarize with Shape, Center, Spread Displays: Stemplots, Histograms Five Number Summary, Outliers, Boxplots Mean vs.
More informationChapter 1: Exploring Data
Chapter 1: Exploring Data Section 1.3 with Numbers The Practice of Statistics, 4 th edition - For AP* STARNES, YATES, MOORE Chapter 1 Exploring Data Introduction: Data Analysis: Making Sense of Data 1.1
More informationUnits. Exploratory Data Analysis. Variables. Student Data
Units Exploratory Data Analysis Bret Larget Departments of Botany and of Statistics University of Wisconsin Madison Statistics 371 13th September 2005 A unit is an object that can be measured, such as
More informationDescriptive Statistics
Descriptive Statistics CHAPTER OUTLINE 6-1 Numerical Summaries of Data 6- Stem-and-Leaf Diagrams 6-3 Frequency Distributions and Histograms 6-4 Box Plots 6-5 Time Sequence Plots 6-6 Probability Plots Chapter
More informationIntroduction to Statistics
Introduction to Statistics Data and Statistics Data consists of information coming from observations, counts, measurements, or responses. Statistics is the science of collecting, organizing, analyzing,
More informationDescribing distributions with numbers
Describing distributions with numbers A large number or numerical methods are available for describing quantitative data sets. Most of these methods measure one of two data characteristics: The central
More informationSTT 315 This lecture is based on Chapter 2 of the textbook.
STT 315 This lecture is based on Chapter 2 of the textbook. Acknowledgement: Author is thankful to Dr. Ashok Sinha, Dr. Jennifer Kaplan and Dr. Parthanil Roy for allowing him to use/edit some of their
More information3.1 Measure of Center
3.1 Measure of Center Calculate the mean for a given data set Find the median, and describe why the median is sometimes preferable to the mean Find the mode of a data set Describe how skewness affects
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 informationStatistics and parameters
Statistics and parameters Tables, histograms and other charts are used to summarize large amounts of data. Often, an even more extreme summary is desirable. Statistics and parameters are numbers that characterize
More informationSummarizing and Displaying Measurement Data/Understanding and Comparing Distributions
Summarizing and Displaying Measurement Data/Understanding and Comparing Distributions Histograms, Mean, Median, Five-Number Summary and Boxplots, Standard Deviation Thought Questions 1. If you were to
More information3.1 Measures of Central Tendency: Mode, Median and Mean. Average a single number that is used to describe the entire sample or population
. Measures of Central Tendency: Mode, Median and Mean Average a single number that is used to describe the entire sample or population. Mode a. Easiest to compute, but not too stable i. Changing just one
More informationDescribing distributions with numbers
Describing distributions with numbers A large number or numerical methods are available for describing quantitative data sets. Most of these methods measure one of two data characteristics: The central
More informationCHAPTER 5: EXPLORING DATA DISTRIBUTIONS. Individuals are the objects described by a set of data. These individuals may be people, animals or things.
(c) Epstein 2013 Chapter 5: Exploring Data Distributions Page 1 CHAPTER 5: EXPLORING DATA DISTRIBUTIONS 5.1 Creating Histograms Individuals are the objects described by a set of data. These individuals
More informationDetermining the Spread of a Distribution
Determining the Spread of a Distribution 1.3-1.5 Cathy Poliak, Ph.D. cathy@math.uh.edu Department of Mathematics University of Houston Lecture 3-2311 Lecture 3-2311 1 / 58 Outline 1 Describing Quantitative
More informationDetermining the Spread of a Distribution
Determining the Spread of a Distribution 1.3-1.5 Cathy Poliak, Ph.D. cathy@math.uh.edu Department of Mathematics University of Houston Lecture 3-2311 Lecture 3-2311 1 / 58 Outline 1 Describing Quantitative
More informationare the objects described by a set of data. They may be people, animals or things.
( c ) E p s t e i n, C a r t e r a n d B o l l i n g e r 2016 C h a p t e r 5 : E x p l o r i n g D a t a : D i s t r i b u t i o n s P a g e 1 CHAPTER 5: EXPLORING DATA DISTRIBUTIONS 5.1 Creating Histograms
More information1.3: Describing Quantitative Data with Numbers
1.3: Describing Quantitative Data with Numbers Section 1.3 Describing Quantitative Data with Numbers After this section, you should be able to MEASURE center with the mean and median MEASURE spread with
More informationTopic 3: Introduction to Statistics. Algebra 1. Collecting Data. Table of Contents. Categorical or Quantitative? What is the Study of Statistics?!
Topic 3: Introduction to Statistics Collecting Data We collect data through observation, surveys and experiments. We can collect two different types of data: Categorical Quantitative Algebra 1 Table of
More informationStat 101 Exam 1 Important Formulas and Concepts 1
1 Chapter 1 1.1 Definitions Stat 101 Exam 1 Important Formulas and Concepts 1 1. Data Any collection of numbers, characters, images, or other items that provide information about something. 2. Categorical/Qualitative
More informationUnit Two Descriptive Biostatistics. Dr Mahmoud Alhussami
Unit Two Descriptive Biostatistics Dr Mahmoud Alhussami Descriptive Biostatistics The best way to work with data is to summarize and organize them. Numbers that have not been summarized and organized are
More informationMATH 1150 Chapter 2 Notation and Terminology
MATH 1150 Chapter 2 Notation and Terminology Categorical Data The following is a dataset for 30 randomly selected adults in the U.S., showing the values of two categorical variables: whether or not the
More informationChapter 4: Displaying and Summarizing Quantitative Data
Chapter 4: Displaying and Summarizing Quantitative Data This chapter discusses methods of displaying quantitative data. The objective is describe the distribution of the data. The figure below shows three
More informationCHAPTER 2: Describing Distributions with Numbers
CHAPTER 2: Describing Distributions with Numbers The Basic Practice of Statistics 6 th Edition Moore / Notz / Fligner Lecture PowerPoint Slides Chapter 2 Concepts 2 Measuring Center: Mean and Median Measuring
More informationAP Final Review II Exploring Data (20% 30%)
AP Final Review II Exploring Data (20% 30%) Quantitative vs Categorical Variables Quantitative variables are numerical values for which arithmetic operations such as means make sense. It is usually a measure
More information1. Exploratory Data Analysis
1. Exploratory Data Analysis 1.1 Methods of Displaying Data A visual display aids understanding and can highlight features which may be worth exploring more formally. Displays should have impact and be
More informationCHAPTER 1. Introduction
CHAPTER 1 Introduction Engineers and scientists are constantly exposed to collections of facts, or data. The discipline of statistics provides methods for organizing and summarizing data, and for drawing
More informationMATH 117 Statistical Methods for Management I Chapter Three
Jubail University College MATH 117 Statistical Methods for Management I Chapter Three This chapter covers the following topics: I. Measures of Center Tendency. 1. Mean for Ungrouped Data (Raw Data) 2.
More informationElementary Statistics
Elementary Statistics Q: What is data? Q: What does the data look like? Q: What conclusions can we draw from the data? Q: Where is the middle of the data? Q: Why is the spread of the data important? Q:
More informationChapter 3. Measuring data
Chapter 3 Measuring data 1 Measuring data versus presenting data We present data to help us draw meaning from it But pictures of data are subjective They re also not susceptible to rigorous inference Measuring
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series. Slide 1
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Overview 3-2 Measures
More informationMath 140 Introductory Statistics
Math 140 Introductory Statistics Professor Silvia Fernández Chapter 2 Based on the book Statistics in Action by A. Watkins, R. Scheaffer, and G. Cobb. Visualizing Distributions Recall the definition: The
More informationMath 140 Introductory Statistics
Visualizing Distributions Math 140 Introductory Statistics Professor Silvia Fernández Chapter Based on the book Statistics in Action by A. Watkins, R. Scheaffer, and G. Cobb. Recall the definition: The
More informationDescribing Distributions
Describing Distributions With Numbers April 18, 2012 Summary Statistics. Measures of Center. Percentiles. Measures of Spread. A Summary Statement. Choosing Numerical Summaries. 1.0 What Are Summary Statistics?
More informationChapter2 Description of samples and populations. 2.1 Introduction.
Chapter2 Description of samples and populations. 2.1 Introduction. Statistics=science of analyzing data. Information collected (data) is gathered in terms of variables (characteristics of a subject that
More informationStatistics I Chapter 2: Univariate data analysis
Statistics I Chapter 2: Univariate data analysis Chapter 2: Univariate data analysis Contents Graphical displays for categorical data (barchart, piechart) Graphical displays for numerical data data (histogram,
More informationChapter 1. Looking at Data
Chapter 1 Looking at Data Types of variables Looking at Data Be sure that each variable really does measure what you want it to. A poor choice of variables can lead to misleading conclusions!! For example,
More information3 GRAPHICAL DISPLAYS OF DATA
some without indicating nonnormality. If a sample of 30 observations contains 4 outliers, two of which are extreme, would it be reasonable to assume the population from which the data were collected has
More informationDescriptive Statistics-I. Dr Mahmoud Alhussami
Descriptive Statistics-I Dr Mahmoud Alhussami Biostatistics What is the biostatistics? A branch of applied math. that deals with collecting, organizing and interpreting data using well-defined procedures.
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 2 Methods for Describing Sets of Data Summary of Central Tendency Measures Measure Formula Description Mean x i / n Balance Point Median ( n +1) Middle Value
More informationDescriptive Univariate Statistics and Bivariate Correlation
ESC 100 Exploring Engineering Descriptive Univariate Statistics and Bivariate Correlation Instructor: Sudhir Khetan, Ph.D. Wednesday/Friday, October 17/19, 2012 The Central Dogma of Statistics used to
More informationStatistics I Chapter 2: Univariate data analysis
Statistics I Chapter 2: Univariate data analysis Chapter 2: Univariate data analysis Contents Graphical displays for categorical data (barchart, piechart) Graphical displays for numerical data data (histogram,
More informationExample 2. Given the data below, complete the chart:
Statistics 2035 Quiz 1 Solutions Example 1. 2 64 150 150 2 128 150 2 256 150 8 8 Example 2. Given the data below, complete the chart: 52.4, 68.1, 66.5, 75.0, 60.5, 78.8, 63.5, 48.9, 81.3 n=9 The data is
More informationMeasures of. U4 C 1.2 Dot plot and Histogram 2 January 15 16, 2015
U4 C 1. Dot plot and Histogram January 15 16, 015 U 4 : C 1.1 CCSS. 9 1.S ID.1 Dot Plots and Histograms Objective: We will be able to represent data with plots on the real number line, using: Dot Plots
More informationLecture 1: Descriptive Statistics
Lecture 1: Descriptive Statistics MSU-STT-351-Sum 15 (P. Vellaisamy: MSU-STT-351-Sum 15) Probability & Statistics for Engineers 1 / 56 Contents 1 Introduction 2 Branches of Statistics Descriptive Statistics
More informationNumerical Measures of Central Tendency
ҧ Numerical Measures of Central Tendency The central tendency of the set of measurements that is, the tendency of the data to cluster, or center, about certain numerical values; usually the Mean, Median
More informationMath 120 Introduction to Statistics Mr. Toner s Lecture Notes 3.1 Measures of Central Tendency
Math 1 Introduction to Statistics Mr. Toner s Lecture Notes 3.1 Measures of Central Tendency The word average: is very ambiguous and can actually refer to the mean, median, mode or midrange. Notation:
More informationTypes of Information. Topic 2 - Descriptive Statistics. Examples. Sample and Sample Size. Background Reading. Variables classified as STAT 511
Topic 2 - Descriptive Statistics STAT 511 Professor Bruce Craig Types of Information Variables classified as Categorical (qualitative) - variable classifies individual into one of several groups or categories
More informationA is one of the categories into which qualitative data can be classified.
Chapter 2 Methods for Describing Sets of Data 2.1 Describing qualitative data Recall qualitative data: non-numerical or categorical data Basic definitions: A is one of the categories into which qualitative
More informationStatistics for Managers using Microsoft Excel 6 th Edition
Statistics for Managers using Microsoft Excel 6 th Edition Chapter 3 Numerical Descriptive Measures 3-1 Learning Objectives In this chapter, you learn: To describe the properties of central tendency, variation,
More informationLecture 2 and Lecture 3
Lecture 2 and Lecture 3 1 Lecture 2 and Lecture 3 We can describe distributions using 3 characteristics: shape, center and spread. These characteristics have been discussed since the foundation of statistics.
More informationHomework Example Chapter 1 Similar to Problem #14
Chapter 1 Similar to Problem #14 Given a sample of n = 129 observations of shower-flow-rate, do this: a.) Construct a stem-and-leaf display of the data. b.) What is a typical, or representative flow rate?
More informationWeek 1: Intro to R and EDA
Statistical Methods APPM 4570/5570, STAT 4000/5000 Populations and Samples 1 Week 1: Intro to R and EDA Introduction to EDA Objective: study of a characteristic (measurable quantity, random variable) for
More informationSection 3. Measures of Variation
Section 3 Measures of Variation Range Range = (maximum value) (minimum value) It is very sensitive to extreme values; therefore not as useful as other measures of variation. Sample Standard Deviation The
More informationChapter 1 Descriptive Statistics
MICHIGAN STATE UNIVERSITY STT 351 SECTION 2 FALL 2008 LECTURE NOTES Chapter 1 Descriptive Statistics Nao Mimoto Contents 1 Overview 2 2 Pictorial Methods in Descriptive Statistics 3 2.1 Different Kinds
More informationExploring Data. How to Explore Data
Exploring Data Statistics is the art and science of learning from data. This may include: Designing appropriate tools to collect data. Organizing data in a meaningful way. Displaying data with appropriate
More informationChapter 6 Group Activity - SOLUTIONS
Chapter 6 Group Activity - SOLUTIONS Group Activity Summarizing a Distribution 1. The following data are the number of credit hours taken by Math 105 students during a summer term. You will be analyzing
More informationChapter 2: Descriptive Analysis and Presentation of Single- Variable Data
Chapter 2: Descriptive Analysis and Presentation of Single- Variable Data Mean 26.86667 Standard Error 2.816392 Median 25 Mode 20 Standard Deviation 10.90784 Sample Variance 118.981 Kurtosis -0.61717 Skewness
More informationSections 2.3 and 2.4
1 / 24 Sections 2.3 and 2.4 Note made by: Dr. Timothy Hanson Instructor: Peijie Hou Department of Statistics, University of South Carolina Stat 205: Elementary Statistics for the Biological and Life Sciences
More informationADMS2320.com. We Make Stats Easy. Chapter 4. ADMS2320.com Tutorials Past Tests. Tutorial Length 1 Hour 45 Minutes
We Make Stats Easy. Chapter 4 Tutorial Length 1 Hour 45 Minutes Tutorials Past Tests Chapter 4 Page 1 Chapter 4 Note The following topics will be covered in this chapter: Measures of central location Measures
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 informationMeasures of center. The mean The mean of a distribution is the arithmetic average of the observations:
Measures of center The mean The mean of a distribution is the arithmetic average of the observations: x = x 1 + + x n n n = 1 x i n i=1 The median The median is the midpoint of a distribution: the number
More informationChapter 3: Displaying and summarizing quantitative data p52 The pattern of variation of a variable is called its distribution.
Chapter 3: Displaying and summarizing quantitative data p52 The pattern of variation of a variable is called its distribution. 1 Histograms p53 Spoiled ballots are a real threat to democracy. Below are
More informationAlgebra 2. Outliers. Measures of Central Tendency (Mean, Median, Mode) Standard Deviation Normal Distribution (Bell Curves)
Algebra 2 Outliers Measures of Central Tendency (Mean, Median, Mode) Standard Deviation Normal Distribution (Bell Curves) Algebra 2 Notes #1 Chp 12 Outliers In a set of numbers, sometimes there will be
More informationChapter 6 The Standard Deviation as a Ruler and the Normal Model
Chapter 6 The Standard Deviation as a Ruler and the Normal Model Overview Key Concepts Understand how adding (subtracting) a constant or multiplying (dividing) by a constant changes the center and/or spread
More informationChapter 2 Descriptive Statistics
Chapter 2 Descriptive Statistics Lecture 1: Measures of Central Tendency and Dispersion Donald E. Mercante, PhD Biostatistics May 2010 Biostatistics (LSUHSC) Chapter 2 05/10 1 / 34 Lecture 1: Descriptive
More informationChapter 2 Solutions Page 15 of 28
Chapter Solutions Page 15 of 8.50 a. The median is 55. The mean is about 105. b. The median is a more representative average" than the median here. Notice in the stem-and-leaf plot on p.3 of the text that
More informationDescribing Distributions with Numbers
Describing Distributions with Numbers Using graphs, we could determine the center, spread, and shape of the distribution of a quantitative variable. We can also use numbers (called summary statistics)
More informationLecture 1: Description of Data. Readings: Sections 1.2,
Lecture 1: Description of Data Readings: Sections 1.,.1-.3 1 Variable Example 1 a. Write two complete and grammatically correct sentences, explaining your primary reason for taking this course and then
More informationObjective A: Mean, Median and Mode Three measures of central of tendency: the mean, the median, and the mode.
Chapter 3 Numerically Summarizing Data Chapter 3.1 Measures of Central Tendency Objective A: Mean, Median and Mode Three measures of central of tendency: the mean, the median, and the mode. A1. Mean The
More informationBNG 495 Capstone Design. Descriptive Statistics
BNG 495 Capstone Design Descriptive Statistics Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential statistical methods, with a focus
More informationSets and Set notation. Algebra 2 Unit 8 Notes
Sets and Set notation Section 11-2 Probability Experimental Probability experimental probability of an event: Theoretical Probability number of time the event occurs P(event) = number of trials Sample
More information1.3.1 Measuring Center: The Mean
1.3.1 Measuring Center: The Mean Mean - The arithmetic average. To find the mean (pronounced x bar) of a set of observations, add their values and divide by the number of observations. If the n observations
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 informationUnit 2. Describing Data: Numerical
Unit 2 Describing Data: Numerical Describing Data Numerically Describing Data Numerically Central Tendency Arithmetic Mean Median Mode Variation Range Interquartile Range Variance Standard Deviation Coefficient
More informationSTATISTICS 1 REVISION NOTES
STATISTICS 1 REVISION NOTES Statistical Model Representing and summarising Sample Data Key words: Quantitative Data This is data in NUMERICAL FORM such as shoe size, height etc. Qualitative Data This is
More informationPerformance of fourth-grade students on an agility test
Starter Ch. 5 2005 #1a CW Ch. 4: Regression L1 L2 87 88 84 86 83 73 81 67 78 83 65 80 50 78 78? 93? 86? Create a scatterplot Find the equation of the regression line Predict the scores Chapter 5: Understanding
More informationDescribing Distributions With Numbers
Describing Distributions With Numbers October 24, 2012 What Do We Usually Summarize? Measures of Center. Percentiles. Measures of Spread. A Summary Statement. Choosing Numerical Summaries. 1.0 What Do
More informationLast Lecture. Distinguish Populations from Samples. Knowing different Sampling Techniques. Distinguish Parameters from Statistics
Last Lecture Distinguish Populations from Samples Importance of identifying a population and well chosen sample Knowing different Sampling Techniques Distinguish Parameters from Statistics Knowing different
More informationDescriptive Data Summarization
Descriptive Data Summarization Descriptive data summarization gives the general characteristics of the data and identify the presence of noise or outliers, which is useful for successful data cleaning
More information2.1 Measures of Location (P.9-11)
MATH1015 Biostatistics Week.1 Measures of Location (P.9-11).1.1 Summation Notation Suppose that we observe n values from an experiment. This collection (or set) of n values is called a sample. Let x 1
More informationChapter 1: Introduction. Material from Devore s book (Ed 8), and Cengagebrain.com
1 Chapter 1: Introduction Material from Devore s book (Ed 8), and Cengagebrain.com Populations and Samples An investigation of some characteristic of a population of interest. Example: Say you want to
More informationLecture Slides. Elementary Statistics Twelfth Edition. by Mario F. Triola. and the Triola Statistics Series. Section 3.1- #
Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola Chapter 3 Statistics for Describing, Exploring, and Comparing Data 3-1 Review and Preview 3-2 Measures
More information2/2/2015 GEOGRAPHY 204: STATISTICAL PROBLEM SOLVING IN GEOGRAPHY MEASURES OF CENTRAL TENDENCY CHAPTER 3: DESCRIPTIVE STATISTICS AND GRAPHICS
Spring 2015: Lembo GEOGRAPHY 204: STATISTICAL PROBLEM SOLVING IN GEOGRAPHY CHAPTER 3: DESCRIPTIVE STATISTICS AND GRAPHICS Descriptive statistics concise and easily understood summary of data set characteristics
More informationReview for Exam #1. Chapter 1. The Nature of Data. Definitions. Population. Sample. Quantitative data. Qualitative (attribute) data
Review for Exam #1 1 Chapter 1 Population the complete collection of elements (scores, people, measurements, etc.) to be studied Sample a subcollection of elements drawn from a population 11 The Nature
More informationIntroduction to Statistics for Traffic Crash Reconstruction
Introduction to Statistics for Traffic Crash Reconstruction Jeremy Daily Jackson Hole Scientific Investigations, Inc. c 2003 www.jhscientific.com Why Use and Learn Statistics? 1. We already do when ranging
More informationWhat is statistics? Statistics is the science of: Collecting information. Organizing and summarizing the information collected
What is statistics? Statistics is the science of: Collecting information Organizing and summarizing the information collected Analyzing the information collected in order to draw conclusions Two types
More information