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Agresti/Franklin Statistics, 1 of 63 Chapter 2 Exploring Data with Graphs and Numerical Summaries Learn …. The Different Types of Data The Use of Graphs to Describe Data The Numerical Methods of Summarizing Data

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Agresti/Franklin Statistics, 2 of 63 Section 2.1 What are the Types of Data?

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Agresti/Franklin Statistics, 3 of 63 In Every Statistical Study: Questions are posed Characteristics are observed

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Agresti/Franklin Statistics, 4 of 63 Characteristics are Variables A Variable is any characteristic that is recorded for subjects in the study

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Agresti/Franklin Statistics, 5 of 63 Variation in Data The terminology variable highlights the fact that data values vary.

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Agresti/Franklin Statistics, 6 of 63 Example: Students in a Statistics Class Variables: Age GPA Major Smoking Status …

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Agresti/Franklin Statistics, 7 of 63 Data values are called observations Each observation can be: Quantitative Categorical

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Agresti/Franklin Statistics, 8 of 63 Categorical Variable Each observation belongs to one of a set of categories Examples: Gender (Male or Female) Religious Affiliation (Catholic, Jewish, …) Place of residence (Apt, Condo, …) Belief in Life After Death (Yes or No)

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Agresti/Franklin Statistics, 9 of 63 Quantitative Variable Observations take numerical values Examples: Age Number of siblings Annual Income Number of years of education completed

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Agresti/Franklin Statistics, 10 of 63 Graphs and Numerical Summaries Describe the main features of a variable For Quantitative variables: key features are center and spread For Categorical variables: key feature is the percentage in each of the categories

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Agresti/Franklin Statistics, 11 of 63 Quantitative Variables Discrete Quantitative Variables and Continuous Quantitative Variables

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Agresti/Franklin Statistics, 12 of 63 Discrete A quantitative variable is discrete if its possible values form a set of separate numbers such as 0, 1, 2, 3, …

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Agresti/Franklin Statistics, 13 of 63 Examples of discrete variables Number of pets in a household Number of children in a family Number of foreign languages spoken

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Agresti/Franklin Statistics, 14 of 63 Continuous A quantitative variable is continuous if its possible values form an interval

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Agresti/Franklin Statistics, 15 of 63 Examples of Continuous Variables Height Weight Age Amount of time it takes to complete an assignment

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Agresti/Franklin Statistics, 16 of 63 Frequency Table A method of organizing data Lists all possible values for a variable along with the number of observations for each value

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Agresti/Franklin Statistics, 17 of 63 Example: Shark Attacks

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Agresti/Franklin Statistics, 18 of 63 Example: Shark Attacks What is the variable? Is it categorical or quantitative? How is the proportion for Florida calculated? How is the % for Florida calculated? Example: Shark Attacks

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Agresti/Franklin Statistics, 19 of 63 Insights – what the data tells us about shark attacks Example: Shark Attacks

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Agresti/Franklin Statistics, 20 of 63 Identify the following variable as categorical or quantitative: Choice of diet (vegetarian or non-vegetarian): a. Categorical b. Quantitative

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Agresti/Franklin Statistics, 21 of 63 Number of people you have known who have been elected to political office: a. Categorical b. Quantitative Identify the following variable as categorical or quantitative:

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Agresti/Franklin Statistics, 22 of 63 Identify the following variable as discrete or continuous: The number of people in line at a box office to purchase theater tickets: a. Continuous b. Discrete

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Agresti/Franklin Statistics, 23 of 63 The weight of a dog: a. Continuous b. Discrete Identify the following variable as discrete or continuous:

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Agresti/Franklin Statistics, 24 of 63 Section 2.2 How Can We Describe Data Using Graphical Summaries?

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Agresti/Franklin Statistics, 25 of 63 Graphs for Categorical Data Pie Chart: A circle having a “slice of pie” for each category Bar Graph: A graph that displays a vertical bar for each category

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Agresti/Franklin Statistics, 26 of 63 Example: Sources of Electricity Use in the U.S. and Canada

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Agresti/Franklin Statistics, 27 of 63 Pie Chart

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Agresti/Franklin Statistics, 28 of 63 Bar Chart

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Agresti/Franklin Statistics, 29 of 63 Pie Chart vs. Bar Chart Which graph do you prefer? Why?

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Agresti/Franklin Statistics, 30 of 63 Graphs for Quantitative Data Dot Plot: shows a dot for each observation Stem-and-Leaf Plot: portrays the individual observations Histogram: uses bars to portray the data

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Agresti/Franklin Statistics, 31 of 63 Example: Sodium and Sugar Amounts in Cereals

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Agresti/Franklin Statistics, 32 of 63 Dotplot for Sodium in Cereals Sodium Data: 0 210 260 125 220 290 210 140 220 200 125 170 250 150 170 70 230 200 290 180

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Agresti/Franklin Statistics, 33 of 63 Stem-and-Leaf Plot for Sodium in Cereal Sodium Data: 0 210 260 125 220 290 210 140 220 200 125 170 250 150 170 70 230 200 290 180

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Agresti/Franklin Statistics, 34 of 63 Frequency Table Sodium Data: 0 210 260 125 220 290 210 140 220 200 125 170 250 150 170 70 230 200 290 180

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Agresti/Franklin Statistics, 35 of 63 Histogram for Sodium in Cereals

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Agresti/Franklin Statistics, 36 of 63 Which Graph? Dot-plot and stem-and-leaf plot: More useful for small data sets Data values are retained Histogram More useful for large data sets Most compact display More flexibility in defining intervals

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Agresti/Franklin Statistics, 37 of 63 Shape of a Distribution Overall pattern Clusters? Outliers? Symmetric? Skewed? Unimodal? Bimodal?

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Agresti/Franklin Statistics, 38 of 63 Symmetric or Skewed ?

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Agresti/Franklin Statistics, 39 of 63 Example: Hours of TV Watching

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Agresti/Franklin Statistics, 40 of 63 Identify the minimum and maximum sugar values: a. 2 and 14 b. 1 and 3 c. 1 and 15 d. 0 and 16

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Agresti/Franklin Statistics, 41 of 63 Consider a data set containing IQ scores for the general public: What shape would you expect a histogram of this data set to have? a. Symmetric b. Skewed to the left c. Skewed to the right d. Bimodal

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Agresti/Franklin Statistics, 42 of 63 Consider a data set of the scores of students on a very easy exam in which most score very well but a few score very poorly: What shape would you expect a histogram of this data set to have? a. Symmetric b. Skewed to the left c. Skewed to the right d. Bimodal

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Agresti/Franklin Statistics, 43 of 63 Section 2.3 How Can We describe the Center of Quantitative Data?

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Agresti/Franklin Statistics, 44 of 63 Mean The sum of the observations divided by the number of observations

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Agresti/Franklin Statistics, 45 of 63 Median The midpoint of the observations when they are ordered from the smallest to the largest (or from the largest to the smallest)

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Agresti/Franklin Statistics, 46 of 63 Find the mean and median CO 2 Pollution levels in 8 largest nations measured in metric tons per person: 2.3 1.1 19.7 9.8 1.8 1.2 0.7 0.2 a. Mean = 4.6 Median = 1.5 b. Mean = 4.6 Median = 5.8 c. Mean = 1.5 Median = 4.6

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Agresti/Franklin Statistics, 47 of 63 Outlier An observation that falls well above or below the overall set of data The mean can be highly influenced by an outlier The median is resistant: not affected by an outlier

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Agresti/Franklin Statistics, 48 of 63 Mode The value that occurs most frequently. The mode is most often used with categorical data

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Agresti/Franklin Statistics, 49 of 63 Section 2.4 How Can We Describe the Spread of Quantitative Data?

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Agresti/Franklin Statistics, 50 of 63 Measuring Spread: Range Range: difference between the largest and smallest observations

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Agresti/Franklin Statistics, 51 of 63 Measuring Spread: Standard Deviation Creates a measure of variation by summarizing the deviations of each observation from the mean and calculating an adjusted average of these deviations

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Agresti/Franklin Statistics, 52 of 63 Empirical Rule For bell-shaped data sets: Approximately 68% of the observations fall within 1 standard deviation of the mean Approximately 95% of the observations fall within 2 standard deviations of the mean Approximately 100% of the observations fall within 3 standard deviations of the mean

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Agresti/Franklin Statistics, 53 of 63 Parameter and Statistic A parameter is a numerical summary of the population A statistic is a numerical summary of a sample taken from a population

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Agresti/Franklin Statistics, 54 of 63 Section 2.5 How Can Measures of Position Describe Spread?

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Agresti/Franklin Statistics, 55 of 63 Quartiles Splits the data into four parts The median is the second quartile, Q 2 The first quartile, Q 1, is the median of the lower half of the observations The third quartile, Q 3, is the median of the upper half of the observations

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Agresti/Franklin Statistics, 56 of 63 Example: Find the first and third quartiles Prices per share of 10 most actively traded stocks on NYSE (rounded to nearest $) 2 4 11 12 13 15 31 31 37 47 a. Q 1 = 2 Q 3 = 47 b. Q 1 = 12 Q 3 = 31 c. Q 1 = 11 Q 3 = 31 d. Q 1 =11.5 Q 3 = 32

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Agresti/Franklin Statistics, 57 of 63 Measuring Spread: Interquartile Range The interquartile range is the distance between the third quartile and first quartile: IQR = Q3 – Q1

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Agresti/Franklin Statistics, 58 of 63 Detecting Potential Outliers An observation is a potential outlier if it falls more than 1.5 x IQR below the first quartile or more than 1.5 x IQR above the third quartile

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Agresti/Franklin Statistics, 59 of 63 The Five-Number Summary The five number summary of a dataset: Minimum value First Quartile Median Third Quartile Maximum value

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Agresti/Franklin Statistics, 60 of 63 Boxplot A box is constructed from Q 1 to Q 3 A line is drawn inside the box at the median A line extends outward from the lower end of the box to the smallest observation that is not a potential outlier A line extends outward from the upper end of the box to the largest observation that is not a potential outlier

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Agresti/Franklin Statistics, 61 of 63 Boxplot for Sodium Data Sodium Data: 0 200 Five Number Summary: 70 210 125 210 Min: 0 125 220 Q1: 145 140 220 Med: 200 150 230 Q3: 225 170 250 Max: 290 170 260 180 290 200 290

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Agresti/Franklin Statistics, 62 of 63 Boxplot for Sodium in Cereals Sodium Data: 0 210 260 125 220 290 210 140 220 200 125 170 250 150 170 70 230 200 290 180

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Agresti/Franklin Statistics, 63 of 63 Z-Score The z-score for an observation measures how far an observation is from the mean in standard deviation units An observation in a bell-shaped distribution is a potential outlier if its z-score +3

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