Visualizing Data: Freq. Tables, Histograms
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1 Visualizing Data: Freq. Tables, Histograms Engineering Statistics Section 1.2 Josh Engwer TTU 25 January 2016 Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
2 Descriptive Statistics Recall that Statistics consists of two broad branches: Descriptive Statistics Statistical Inference The remainder of this chapter focuses squarely on Descriptive Statistics: Definition Descriptive Statistics is the organization, summary, visualization and presentation of data that conveys useful information about the data. Descriptive Statistics involves: Data Visualization (this section) Numerical Summaries (Sections 1.3 & 1.4) Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
3 Data Visualizations The following data visualizations will never be considered in this course: Stem-and-Leaf Displays Bar Charts Pie Charts Control Charts Scatter Plots Dot Plots Line Plots Contour Plots Radar Plots Violin Plots Heatmaps So which data visualizations will be considered in this course?? Frequency Tables (this section) Histograms (this section) Boxplots (Section 1.4) Frequency Polygons (Chapter 3) Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
4 Frequency Tables Given a sample of eye colors: H, Br, Br, Br, S, A, H, H, G, A, Bl, Bl, Br, Bl, A, Br, H, G, A, A, Br, Bl, G, Bl, Bl Then the resulting frequency table is: EYE COLOR FREQUENCY RELATIVE FREQUENCY Amber (A) Blue (Bl) Brown (Br) Green (G) Hazel (H) Silver (S) TOTAL: Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
5 Frequency Tables Given a sample of eye colors: H, Br, Br, Br, S, A, H, H, G, A, Bl, Bl, Br, Bl, A, Br, H, G, A, A, Br, Bl, G, Bl, Bl Then the resulting frequency table is: EYE COLOR FREQUENCY RELATIVE FREQUENCY Amber (A) 5 Blue (Bl) 6 Brown (Br) 5 Green (G) 3 Hazel (H) 4 Silver (S) 1 TOTAL: 24 The frequency entails from counting the # data points of a given category. Then compute the total frequency. Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
6 Frequency Tables Given a sample of eye colors: H, Br, Br, Br, S, A, H, H, G, A, Bl, Bl, Br, Bl, A, Br, H, G, A, A, Br, Bl, G, Bl, Bl Then the resulting frequency table is: EYE COLOR FREQUENCY RELATIVE FREQUENCY Amber (A) 5 5/ Blue (Bl) 6 6/24 = Brown (Br) 5 5/ Green (G) 3 3/24 = Hazel (H) 4 4/ Silver (S) 1 1/ TOTAL: Each category s relative frequency is its frequency divided by the total freq. Round decimals to three decimal places as necessary. The total relative frequency should be very close to one (btw & 1.002) Frequency tables can also be made for numerical data (see the 1.2 Outline). Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
7 Histograms for Categorical Data Given a sample of eye colors: Br, G, Br, Br, S, A, H, H, G, A, Bl, Bl, Br, Bl, A, Br, H, G, A, A, Br, Bl, G, Bl, Bl Then the resulting histogram is: Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
8 Histograms for Categorical Data Given a sample of eye colors: Br, G, Br, Br, S, A, H, H, G, A, Bl, Bl, Br, Bl, A, Br, H, G, A, A, Br, Bl, G, Bl, Bl Then the resulting histogram is: Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
9 Histograms for Categorical Data The vertical axis of a histogram is always one of the following: Frequency (count of each category/bin) Relative Frequency = (Frequency)/(Total Frequency) Percent (%) = Relative Frequency 100% Density Categorical Data: Numerical Data: Density = Relative Frequency Density = (Relative Frequency)/(Bin Width) Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
10 Histograms for Discrete Data (Equal Bin Widths) Pick a bin width that avoids gaps (right figure) and overlumping (left figure). For this course, bin widths will be chosen a priori. Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23 Given a sample: 4.9, 4.9, 5.0, 5.7, 6.2, 5.3, 5.2, 5.5, 5.6, 5.7, 5.7, 4.1, 6.8 Here are three histograms using equal bin widths:
11 Histograms for Discrete Data (Unequal Bin Widths) Unequal bin widths are useful when there are some isolated data points. For this course, bin widths will be chosen a priori. Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23 Given a sample: 4.9, 4.9, 5.0, 5.7, 6.2, 5.3, 5.2, 5.5, 5.6, 5.7, 5.7, 4.1, 6.8 Here are three histograms using unequal bin widths:
12 Histograms for Continuous Numerical Data Continuous numerical data usually refer to infinite populations. A histogram for an infinite population looks like a smooth curve: Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
13 Modality of Discrete & Categorical Data Definition A dataset is unimodal if its histogram has exactly one peak. A dataset is bimodal if its histogram has exactly two peaks. A dataset is multimodal if its histogram has many peaks. See page 22 of the textbook for an example of multimodal discrete data. Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
14 Modality of Continuous Data Definition A dataset is unimodal if its histogram has exactly one peak. A dataset is bimodal if its histogram has exactly two peaks. A dataset is multimodal if its histogram has many peaks. See page 23 of the textbook for an example of multimodal continuous data. Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
15 Skewness of Discrete & Categorical Data Definition A dataset is positively skewed if its histogram has a long upper tail. A dataset is negatively skewed if its histogram has a long lower tail. Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
16 Skewness of Discrete & Categorical Data Definition A dataset is symmetric if its histogram s left half and right half are mirror images of each other. Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
17 Skewness of Continuous Data Definition A dataset is positively skewed if its histogram has a long upper tail. A dataset is negatively skewed if its histogram has a long lower tail. Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
18 Skewness of Continuous Data Definition A dataset is symmetric if its histogram s left half and right half are mirror images of each other. Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
19 Outlier(s) in Discrete & Categorical Data Definition A data point in a dataset is an outlier if it is far away from most of the data. Consider the dataset: 1,5,2,2,1,4,1,3,20,5,16,16 The left histogram (with equal bin widths) suggest that 16 & 20 are outliers. But identifying outliers is unclear with the right histogram (unequal bin widths). Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
20 Outlier(s) in Discrete & Categorical Data Definition A data point in a dataset is an outlier if it is far away from most of the data. Outliers are essentially extreme values of a dataset or sample. Outliers often occur due to catastrophic measurement errors: Instrumentation terribly mis-calibrated Instrumentation malfunctions during measurement Person deliberately lying in a survey Person deliberately exagerating measurements or counts However, not all outliers are due to errors: House prices Exam scores Histograms are not always effective in revealing outliers. Better visual and numerical methods for identifying outliers in Section 1.4 Outliers are never really considered for continuous data. Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
21 Textbook Logistics for Section 1.2 Difference(s) in Terminology: TEXTBOOK TERMINOLOGY Frequency Distribution Rectangle Height Class(es) Class Interval(s) Class Width SLIDES/OUTLINE TERMINOLOGY Frequency Table Bin Height Bin(s) Bin(s) Bin Width Difference(s) in Notation: CONCEPT TEXTBOOK NOTATION SLIDES/OUTLINE NOTATION Bin Widths 3.0 < 3.5 [3.0, 3.5) Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
22 Textbook Logistics for Section 1.2 Ignore Stem-and-Leaf Displays section (pg 13-15) Stem-and-leaf displays were popular prior to the 1980 s. Stem-and-leaf displays were useful when computers were text-only. Stem-and-leaf displays will never be used in this course. Ignore Dotplots section (pg 15-16) Dotplots are effectively histograms but with stacked dots instead of bars. Unfortunately, there s no freedom in choosing appropriate bin widths. Dotplots are only useful for small samples. Dotplots will never be used in this course. Ignore Multivariate Data section (pg 24) Multivariate Data can be hard or impossible to visualize Multivariate data is never considered in this course. Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
23 Fin Fin. Josh Engwer (TTU) Visualizing Data: Freq. Tables, Histograms 25 January / 23
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