Agresti/Franklin Statistics, 1 of 63 Chapter 2 Exploring Data with Graphs and Numerical Summaries...

Agresti/Franklin Statistics, 1 of 63

Chapter 2Exploring 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


Section 2.1

What are the Types of Data?


In Every Statistical Study:

Questions are posed

Characteristics are observed


Characteristics are Variables

A Variable is any characteristic that is recorded for subjects in the study


Variation in Data

The terminology variable highlights the fact that data values vary.


Example: Students in a Statistics Class

Variables:• Age

• GPA

• Major

• Smoking Status

• …


Data values are called observations

Each observation can be:

• Quantitative

• Categorical


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)


Quantitative Variable

Observations take numerical values

Examples:• Age

• Number of siblings

• Annual Income

• Number of years of education completed


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


Quantitative Variables

Discrete Quantitative Variables

and

Continuous Quantitative Variables


Discrete

A quantitative variable is discrete if its possible values form a set of separate numbers such as 0, 1, 2, 3, …


Examples of discrete variables

Number of pets in a household Number of children in a family Number of foreign languages spoken


Continuous

A quantitative variable is continuous if its possible values form an interval


Examples of Continuous Variables

Height Weight Age Amount of time it takes to complete

an assignment


Frequency Table

A method of organizing data

Lists all possible values for a variable along with the number of observations for each value


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?



Insights – what the data tells us about shark attacks



Identify the following variable as categorical or quantitative:

Choice of diet (vegetarian or non-vegetarian):

a. Categorical

b. Quantitative


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:


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


The weight of a dog:

a. Continuous

b. Discrete

Identify the following variable as discrete or continuous:


Section 2.2

How Can We Describe Data Using Graphical Summaries?


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


Example: Sources of Electricity Use in the U.S. and Canada


Pie Chart


Bar Chart


Pie Chart vs. Bar Chart

Which graph do you prefer? Why?


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


Example: Sodium and Sugar Amounts in Cereals


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


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


Frequency Table

Sodium Data: 0 210

260 125220 290210 140220 200125 170250 150170 70230 200290 180


Histogram for Sodium in Cereals


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


Shape of a Distribution

Overall pattern• Clusters?

• Outliers?

• Symmetric?

• Skewed?

• Unimodal?

• Bimodal?


Symmetric or Skewed ?


Example: Hours of TV Watching


Identify the minimum and maximum sugar values:

a. 2 and 14 b. 1 and 3

c. 1 and 15 d. 0 and 16


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


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


Section 2.3

How Can We describe the Center of Quantitative Data?


Mean

The sum of the observations divided by the number of observations

n

xx


Median

The midpoint of the observations when they are ordered from the smallest to the largest (or from the largest to the smallest)


Find the mean and median

CO2 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


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


Mode

The value that occurs most frequently.

The mode is most often used with categorical data


Section 2.4

How Can We Describe the Spread of Quantitative Data?


Measuring Spread: Range

Range: difference between the largest and smallest observations


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

1

)( 2

n

xxs


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


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


Section 2.5

How Can Measures of Position Describe Spread?


Quartiles

Splits the data into four parts The median is the second quartile, Q2

The first quartile, Q1, is the median of the lower half of the observations

The third quartile, Q3, is the median of the upper half of the observations


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. Q1 = 2 Q3 = 47

b. Q1 = 12 Q3 = 31

c. Q1 = 11 Q3 = 31

d. Q1 =11.5 Q3 = 32


Measuring Spread: Interquartile Range

The interquartile range is the distance between the third quartile and first quartile:

IQR = Q3 – Q1


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


The Five-Number Summary

The five number summary of a dataset:

• Minimum value

• First Quartile

• Median

• Third Quartile

• Maximum value


Boxplot

A box is constructed from Q1 to Q3

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


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


Boxplot for Sodium in Cereals

Sodium Data: 0 210

260 125220 290210 140220 200125 170250 150170 70230 200290 180


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 or > +3

deviation standard

mean -n observatio z

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Agresti/Franklin Statistics, 1 of 63 Chapter 2 Exploring Data with Graphs and Numerical Summaries...

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