Statistical Methods STAT 302 Chapter 2 Summarizing Data Copyright 2025 by Aburweis All rights reserved No part of this work may be reproduced in any form without written permission Chapter 2 Learning Objectives 1 Construct frequency tables 2 Create and interpret graphs for qualitative data 3 Calculate mean median and mode 4 Select the appropriate measure of center 5 Compute range variance and standard deviation 6 Determine quartiles and five number summary 7 Create histograms box plots and scatter plots 8 Recognize different distribution shapes 2 Qualitative Categorical Data Summarizing and Displaying 3 Summarizing and Displaying Data Numerical Summarization Methods Frequency table frequency distribution Relative frequency table Contingency table Graphical Display and Representation Methods Bar chart bar plot Mosaic plot Pareto chart Pie chart 4 Qualitative Categorical Data Summarizing 5 Frequency Table Frequency Distribution A frequency table organizes and displays the counts frequencies for each category of a categorical variable Example 1 Suppose we have a sample of 1 000 undergraduate students at Texas A M University and we record their academic classifications e g freshman sophomore junior senior Category Frequency Relative Frequency Cumulative Frequency Cumulative Relative Frequency Freshman Sophomore Junior Senior Total 193 286 267 254 1000 0 193 0 286 0 267 0 254 1 000 193 479 746 1000 19 3 47 9 74 6 100 6 Contingency Table A contingency table is used to display the relationship between two categorical variables Example 2 Consider a sample of 1 000 undergraduate students at Texas A M University We can use a contingency table to examine the association between class year freshman sophomore junior senior and shirt size small medium large r a e Y s s a C l Shirt Size Category Small Medium Large Total Freshman Sophomore Junior Senior Total 53 67 76 62 258 97 118 112 106 433 43 101 79 86 193 286 267 254 309 1000 7 Qualitative Categorical Data Displaying 8 Bar Chart Bar Graph Bar Plot A bar chart is an effective tool for summarizing and organizing categorical data It visually represents the frequency or relative frequency of each category In a bar chart the height or length of each bar is proportional to the frequency or relative frequency of the corresponding category Example 3 A student organization ordered T shirts for its members The table below shows the sizes of the T shirts ordered Construct a bar chart to visually represent this data Frequency Distribution of T Shirt Sizes T Shirt Size Extra Small Small Medium Large Extra Large 8 17 33 25 11 33 25 17 8 11 y c n e u q e r F 35 30 25 20 15 10 5 0 Extra Small Small Medium Large Extra Large T Shirt Size 9 Bar Chart with two variables Side by Side Bar Chart A side by side bar chart displays individual bars for each category or group positioned next to each other allowing easy comparison between groups Stacked Bar Chart A stacked bar chart represents individual data values as segments within a single bar Instead of placing bars side by side each bar combines all categories with each segment representing a sub category or component of the total Standardized Stacked Bar Chart A standardized stacked bar chart is similar to a stacked bar chart but each bar is scaled to the same total height usually 100 This allows for the comparison of relative proportions or percentages across groups rather than absolute values 10 Bar Chart with two variables Cont Example 4 Construct a side by side bar chart a stacked bar chart and a standardized stacked bar chart to visualize the following data which represents the distribution of housing types across two distinct student classes Student Housing Types in a Statistics Course Type of Housing Class A Class B Apartment Dorm House Fraternity House 53 67 76 62 97 118 112 106 11 Bar Chart with two variables Cont s t n e d u t S f o r e b m u N 140 120 100 80 60 40 20 0 Side by Side Bar Chart of Student Housing Types 97 53 118 67 112 76 106 62 Apartment Dorm House Fraternity House Type of Housing Class A Class B Stacked Bar Chart of Student Housing Types s t n e d u t S f o r e b m u N 200 150 100 50 0 97 53 118 67 112 76 Type of Housing Class A Class B 106 62 Apartment Dorm House Fraternity House Standardized Stacked Bar Chart of Student Housing Types e g a t n e c r e P 100 80 60 40 20 0 65 35 64 36 Dorm 60 40 63 37 Apartment House Fraternity House Type of Housing Class A Class B 12 Mosaic Plot A mosaic plot is a graphical tool used to represent the relationship between two or more categorical variables It displays the proportions of each category combination with the area of each rectangle proportional to the number of observations in that combination Mosaic plots are particularly useful for visualizing associations in contingency tables allowing for a visual assessment of patterns relationships and dependencies within the data Question 1 What are the differences between the two visualizations shown below Question 2 How does the survival rate differ between First Class and Third Class passengers 13 Pareto Chart Pareto Diagram A Pareto chart is a specialized type of bar chart in which the height of each bar represents the frequency or magnitude of an event The bars are arranged in descending order from left to right highlighting the most significant categories Example 5 The table below lists the leading causes of death in the United States in 2014 Source Health United States 2015 Table 19 Construct a Pareto chart to organize and visualize this data Cause of death Accidents Cancer Heart disease Chronic lower respiratory disease Stroke cerebrovascular diseases of death 136 053 591 699 147 101 614 348 133 103 14 Pie Chart A pie chart is used to visually represent a single qualitative categorical variable typically with a small number of categories The circle is divided into sectors slices with each slice proportional to the frequency or relative frequency of the corresponding category Pie charts like bar charts are suitable for summarizing categorical data Example 6 Six students were asked about their vegetable preferences peas carrots spinach as shown in the table below Construct a pie chart to visually represent this data Student Preference 1 2 3 4 5 6 Peas Carrots Carrots Spinach Carrots Peas Percent who prefer Peas Carrots Spinach 33 50 17 Vegetable Preferences Spinach 17 Peas 33 Carrots 50 Peas Carrots Spinach 15 Pie Chart Limitations Pie charts can be difficult to interpret particularly when
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