Agresti/Franklin Statistics, 1 of 63 Chapter 2 Exploring Data with Graphs and Numerical Summaries...
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Transcript of Agresti/Franklin Statistics, 1 of 63 Chapter 2 Exploring Data with Graphs and Numerical Summaries...
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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
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Section 2.1
What are the Types of Data?
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In Every Statistical Study:
Questions are posed
Characteristics are observed
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Characteristics are Variables
A Variable is any characteristic that is recorded for subjects in the study
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Variation in Data
The terminology variable highlights the fact that data values vary.
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Example: Students in a Statistics Class
Variables:• Age
• GPA
• Major
• Smoking Status
• …
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Data values are called observations
Each observation can be:
• Quantitative
• Categorical
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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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Quantitative Variable
Observations take numerical values
Examples:• Age
• Number of siblings
• Annual Income
• Number of years of education completed
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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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Quantitative Variables
Discrete Quantitative Variables
and
Continuous Quantitative Variables
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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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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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Continuous
A quantitative variable is continuous if its possible values form an interval
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Examples of Continuous Variables
Height Weight Age Amount of time it takes to complete
an assignment
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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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Example: Shark Attacks
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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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Insights – what the data tells us about shark attacks
Example: Shark Attacks
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Identify the following variable as categorical or quantitative:
Choice of diet (vegetarian or non-vegetarian):
a. Categorical
b. Quantitative
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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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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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The weight of a dog:
a. Continuous
b. Discrete
Identify the following variable as discrete or continuous:
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Section 2.2
How Can We Describe Data Using Graphical Summaries?
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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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Example: Sources of Electricity Use in the U.S. and Canada
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Pie Chart
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Bar Chart
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Pie Chart vs. Bar Chart
Which graph do you prefer? Why?
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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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Example: Sodium and Sugar Amounts in Cereals
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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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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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Frequency Table
Sodium Data: 0 210
260 125220 290210 140220 200125 170250 150170 70230 200290 180
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Histogram for Sodium in Cereals
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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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Shape of a Distribution
Overall pattern• Clusters?
• Outliers?
• Symmetric?
• Skewed?
• Unimodal?
• Bimodal?
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Symmetric or Skewed ?
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Example: Hours of TV Watching
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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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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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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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Section 2.3
How Can We describe the Center of Quantitative Data?
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Mean
The sum of the observations divided by the number of observations
n
xx
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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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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
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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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Mode
The value that occurs most frequently.
The mode is most often used with categorical data
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Section 2.4
How Can We Describe the Spread of Quantitative Data?
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Measuring Spread: Range
Range: difference between the largest and smallest observations
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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
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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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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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Section 2.5
How Can Measures of Position Describe Spread?
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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
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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
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Measuring Spread: Interquartile Range
The interquartile range is the distance between the third quartile and first quartile:
IQR = Q3 – Q1
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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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The Five-Number Summary
The five number summary of a dataset:
• Minimum value
• First Quartile
• Median
• Third Quartile
• Maximum value
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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
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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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Boxplot for Sodium in Cereals
Sodium Data: 0 210
260 125220 290210 140220 200125 170250 150170 70230 200290 180
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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