Elementary Statistics Thirteenth Edition Chapter 2 Summarizing and Graphing Data Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Summarizing and Graphing Data 2-1 Frequency Distributions for Organizing and Summarizing Data 2-2 Histograms 2-3 Graphs that Enlighten and Graphs that Deceive 2-4 Scatterplots, Correlation, and Regression Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Key Concept While a frequency distribution is a useful tool for summarizing data and investigating the distribution of data, an even better tool is a histogram, which is a graph that is easier to interpret than a table of numbers. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Histogram • Histogram – A graph consisting of bars of equal width drawn adjacent to each other (unless there are gaps in the data) The horizontal scale represents classes of quantitative data values, and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Important Uses of a Histogram • Visually displays the shape of the distribution of the data • Shows the location of the center of the data • Shows the spread of the data • Identifies outliers Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Relative Frequency Histogram • Relative Frequency Histogram – It has the same shape and horizontal scale as a histogram, but the vertical scale is marked with relative frequencies instead of actual frequencies. Time (seconds) Frequency 75-124 11 125-174 24 175-224 10 225-274 3 275-324 2 Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Critical Thinking Interpreting Histograms Explore the data by analyzing the histogram to see what can be learned about “CVDOT”: • the Center of the data, • the Variation, • the shape of the Distribution, • whether there are any Outliers, • and Time. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Common Distribution Shapes Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Normal Distribution Because this histogram is roughly bell-shaped, we say that the data have a normal distribution. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Skewness (1 of 3) • Skewness – A distribution of data is skewed if it is not symmetric and extends more to one side than to the other. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Skewness (2 of 3) Data skewed to the right (positively skewed) have a longer right tail. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Skewness (3 of 3) Data skewed to the left (negative skewed) have a longer left tail. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Assessing Normality with Normal Quantile Plots (1 of 5) Criteria for Assessing Normality with a Normal Quantile Plot • Normal Distribution: The pattern of the points in the normal quantile plot is reasonably close to a straight line, and the points do not show some systematic pattern that is not a straight-line pattern. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Assessing Normality with Normal Quantile Plots (2 of 5) Criteria for Assessing Normality with a Normal Quantile Plot • Not a Normal Distribution: The population distribution is not normal if the normal quantile plot has either or both of these two conditions: – The points do not lie reasonably close to a straight-line pattern. – The points show some systematic pattern that is not a straight-line pattern. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Assessing Normality with Normal Quantile Plots (3 of 5) Normal Distribution: The points are reasonably close to a straight-line pattern, and there is no other systematic pattern that is not a straight-line pattern. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Assessing Normality with Normal Quantile Plots (4 of 5) Not a Normal Distribution: The points do not lie reasonably close to a straight line. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Assessing Normality with Normal Quantile Plots (5 of 5) Not a Normal Distribution: The points show a systematic pattern that is not a straight-line pattern. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Elementary Statistics Thirteenth Edition Chapter 2 Summarizing and Graphing Data Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Summarizing and Graphing Data 2-1 Frequency Distributions for Organizing and Summarizing Data 2-2 Histograms 2-3 Graphs that Enlighten and Graphs that Deceive 2-4 Scatterplots, Correlation, and Regression Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Key Concept Introduce other common graphs that foster understanding of data. Discuss some graphs that are deceptive because they create impressions about data that are somehow misleading or wrong. Technology now provides us with powerful tools for generating a wide variety of graphs. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Graphs that Enlighten: Dotplots (1 of 2) • Dotplots – A graph of quantitative data in which each data value is plotted as a point (or dot) above a horizontal scale of values. Dots representing equal values are stacked. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Graphs that Enlighten: Dotplots (2 of 2) • Dotplots – Features of a Dotplot ▪ Displays the shape of distribution of data. ▪ It is usually possible to recreate the original list of data values. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Stemplots (1 of 2) • Stemplots (or stem-and-leaf plot) – Represents quantitative data by separating each value into two parts: the stem (such as the leftmost digit) and the leaf (such as the rightmost digit). Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Stemplots (2 of 2) • Stemplots (or stem-and-leaf plot) – Features of a Stemplot ▪ Shows the shape of the distribution of the data. ▪ Retains the original data values. ▪ The sample data are sorted (arranged in order). Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Time-Series Graph (1 of 2) • Time-Series Graph – A graph of time-series data, which are quantitative data that have been collected at different points in time, such as monthly or yearly Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Time-Series Graph (2 of 2) • Time-Series Graph – Feature of a Time-Series Graph ▪ Reveals information about trends over time. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Bar Graph (1 of 2) • Bar Graphs – A graph of bars of equal width to show frequencies of categories of categorical (or qualitative) data. The bars may or may not be separated by small gaps. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Bar Graph (2 of 2) • Bar Graphs – Feature of a Bar Graph ▪ Shows the relative distribution of categorical data so that it is easier to compare the different categories. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Pareto Chart (1 of 3) • Pareto Charts – A Pareto chart is a bar graph for categorical data, with the added stipulation that the bars are arranged in descending order according to frequencies, so the bars decrease in height from left to right. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Pareto Chart (2 of 3) • Pareto Charts – Features of a Pareto Chart ▪ Shows the relative distribution of categorical data so that it is easier to compare the different categories. ▪ Draws attention to the more important categories. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Pareto Chart (3 of 3) Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Pie Chart (1 of 3) • Pie Charts – A very common graph that depicts categorical data as slices of a circle, in which the size of each slice is proportional to the frequency count for the category Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Pie Chart (2 of 3) • Pie Charts – Feature of a Pie Chart ▪ Shows the distribution of categorical data in a commonly used format. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Pie Chart (3 of 3) Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Frequency Polygon (1 of 3) • Frequency Polygon – A graph using line segments connected to points located directly above class midpoint values – A frequency polygon is very similar to a histogram, but a frequency polygon uses line segments instead of bars. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Frequency Polygon (2 of 3) • Frequency Polygon – A variation of the basic frequency polygon is the relative frequency polygon, which uses relative frequencies (proportions or percentages) for the vertical scale. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Frequency Polygon (3 of 3) Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Relative Frequency Polygon Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Graphs That Deceive (1 of 4) • Nonzero Vertical Axis – A common deceptive graph involves using a vertical scale that starts at some value greater than zero to exaggerate differences between groups. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Graphs That Deceive (2 of 4) • Nonzero Vertical Axis Always examine a graph carefully to see whether a vertical axis begins at some point other than zero so that differences are exaggerated. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Graphs That Deceive (3 of 4) • Pictographs – Drawings of objects, called pictographs, are often misleading. Data that are one-dimensional in nature (such as budget amounts) are often depicted with two-dimensional objects (such as dollar bills) or threedimensional objects (such as stacks of coins, homes, or barrels). Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Graphs That Deceive (4 of 4) • Pictographs – By using pictographs, artists can create false impressions that grossly distort differences by using these simple principles of basic geometry: ▪ When you double each side of a square, its area doesn’t merely double; it increases by a factor of four. ▪ When you double each side of a cube, its volume doesn’t merely double; it increases by a factor of eight. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Pictographs Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Concluding Thoughts (1 of 2) In addition to the graphs we have discussed in this section, there are many other useful graphs - some of which have not yet been created. The world needs more people who can create original graphs that enlighten us about the nature of data. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Concluding Thoughts (2 of 2) In The Visual Display of Quantitative Information, Edward Tufte offers these principles: • For small data sets of 20 values or fewer, use a table instead of a graph. • A graph of data should make us focus on the true nature of the data, not on other elements, such as eye-catching but distracting design features. • Do not distort data; construct a graph to reveal the true nature of the data. • Almost all of the ink in a graph should be used for the data, not for other design elements. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Elementary Statistics Thirteenth Edition Chapter 2 Summarizing and Graphing Data Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Summarizing and Graphing Data 2-1 Frequency Distributions for Organizing and Summarizing Data 2-2 Histograms 2-3 Graphs that Enlighten and Graphs that Deceive 2-4 Scatterplots, Correlation, and Regression Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Key Concept Introduce the analysis of paired sample data. Discuss correlation and the role of a graph called a scatterplot, and provide an introduction to the use of the linear correlation coefficient. Provide a very brief discussion of linear regression, which involves the equation and graph of the straight line that best fits the sample paired data. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Scatterplot and Correlation (1 of 2) • Correlation – A correlation exists between two variables when the values of one variable are somehow associated with the values of the other variable. • Linear Correlation – A linear correlation exists between two variables when there is a correlation and the plotted points of paired data result in a pattern that can be approximated by a straight line. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Scatterplot and Correlation (2 of 2) • Scatterplot (or Scatter Diagram) – A scatterplot (or scatter diagram) is a plot of paired (x, y) quantitative data with a horizontal x-axis and a vertical y-axis. The horizontal axis is used for the first variable (x), and the vertical axis is used for the second variable (y). Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Example: Waist and Arm Correlation (1 of 2) • Correlation: The distinct pattern of the plotted points suggests that there is a correlation between waist circumferences and arm circumferences. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Example: Waist and Arm Correlation (2 of 2) • No Correlation: The plotted points do not show a distinct pattern, so it appears that there is no correlation between weights and pulse rates. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Linear Correlation Coefficient r • Linear Correlation Coefficient r – The linear correlation coefficient is denoted by r, and it measures the strength of the linear association between two variables. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Using r for Determining Correlation The computed value of the linear correlation coefficient, r, is always between −1 and 1. • If r is close to −1 or close to 1, there appears to be a correlation. • If r is close to 0, there does not appear to be a linear correlation. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Example: Correlation between Shoe Print Lengths and Heights? (1 of 2) Shoe Print Length (cm) Height (cm) 29.7 29.7 31.4 31.8 27.6 175.3 177.8 185.4 175.3 172.7 Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Example: Correlation between Shoe Print Lengths and Heights? (2 of 2) It isn’t very clear whether there is a linear correlation. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved P-Value • P-Value – If there really is no linear correlation between two variables, the P-value is the probability of getting paired sample data with a linear correlation coefficient r that is at least as extreme as the one obtained from the paired sample data. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Interpreting a P-Value from the Previous Example The P-value of 0.294 is high. It shows there is a high chance of getting a linear correlation coefficient of r = 0.591 (or more extreme) by chance when there is no linear correlation between the two variables. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Interpreting a P-Value from the Example Where n = 5 Because the likelihood of getting r = 0.591 or a more extreme value is so high (29.4% chance), we conclude there is not sufficient evidence to conclude there is a linear correlation between shoe print lengths and heights. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Interpreting a P-Value Only a small P-value, such as 0.05 or less (or a 5% chance or less), suggests that the sample results are not likely to occur by chance when there is no linear correlation, so a small P-value supports a conclusion that there is a linear correlation between the two variables. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Example: Correlation between Shoe Print Lengths and Heights (n = 40) Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Example: Correlation between Shoe Print Lengths and Heights The scatterplot shows a distinct pattern. The value of the linear correlation coefficient is r = 0.813, and the P-value is 0.000. Because the P-value of 0.000 is small, we have sufficient evidence to conclude there is a linear correlation between shoe print lengths and heights. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Regression • Regression – Given a collection of paired sample data, the regression line (or line of best fit, or least-squares line) is the straight line that “best” fits the scatterplot of the data. Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Example: Regression Line (1 of 2) Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Example: Regression Line (2 of 2) The general form of the regression equation has a y-intercept of b0 = 80.9 and slope b1 = 3.22. Using variable names, the equation is: Height = 80.9 + 3.22 (Shoe Print Length) Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved
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