Chapter 2 Describing, Exploring, and Comparing Data Sections 2.12.4

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Chapter 2--Describing, Exploring, and Comparing
Data(Sections 2.1-2.4)

• MATH320

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Overview

• Descriptive Statisticssummarize or describe the
  important characteristics of a known set of
  population data
• Inferential Statisticsuse sample data to make
  inferences (or generalizations) about a
  population

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Important Characteristics of Data

• Center A representative or average value that
  indicates where the middle of the data set is
  located
• Variation A measure of the amount that the
  values vary among themselves
• Distribution The nature or shape of the
  distribution of data (such as bell-shaped,
  uniform, or skewed)
• Outliers Sample values that lie very far away
  from the vast majority of other sample values
• Time Changing characteristics of the data over
  time

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Section 2.2Frequency Distributions
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Frequency Distribution

• Lists data values (either individually or by
  groups of intervals), along with their
  corresponding frequencies or counts.

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Lower Class Limits

• The smallest numbers that can actually belong to
  different classes

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Upper Class Limits

• The largest numbers that can actually belong to
  different classes

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Class Boundaries

• Number separating classes

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Class Width

• The difference between two consecutive lower
  class limits or two consecutive lower class
  boundaries

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Reasons To Construct Frequency Distributions

• Large data sets can be summarized
• Can gain some insight into the nature of the data
• Have a basis for constructing graphs
• Great overview
• Ability to see any strange or unusual data
• Quickly identifies any outliners

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Constructing a Frequency Distribution

• Decide on the number of classes (between 5 and 20
  good start)
• Calculate widths
• Starting point Choose a lower limit of the first
  class
• Using the lower limit of the first class and
  class width, proceed to list the lower class
  limits
• List the lower class limits in a vertical column
  and proceed to enter the upper class limits
• Go through the data set putting a tally in the
  appropriate class for each data value

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Example Determine Class Width
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Example Set Up Frequency Distribution Table

• Class width 34
• Take lowest number and add 33
• You want 34 units in the frequency.
• Count first lowest class as first unit
• Build from there

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Relative Frequency

• Percent of the whole

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Cumulative Frequency

• Write out new classes using next class
• Add each to the previous class
• Should end up with total number of items at the
  end

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Frequency Tables
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Section 2.3Visualizing Data
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Visualizing Data
Depict the nature of shape or shape of the data
distribution
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Histogram
A bar graph in which the horizontal scale
represents the classes of data values and the
vertical scale represents the frequencies.
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Relative Frequency Histogram
Has the same shape and horizontal scale as a
histogram, but the vertical scale is marked with
relative frequencies.
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Histogram Relative Frequency Histogram
Figure 2-1
Figure 2-2
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Frequency Polygon
Uses line segments connected to points directly
above class midpoint values
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Ogive
A line graph that depicts cumulative frequencies
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Dot Plot
Consists of a graph in which each data value is
plotted as a point along a scale of values
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Stem-and Leaf Plot
Represents data by separating each value into two
parts the stem (such as the leftmost digit) and
the leaf (such as the rightmost digit)
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Pareto Chart
A bar graph for qualitative data, with the bars
arranged in order according to frequencies
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Pie Chart
A graph depicting qualitative data as slices of a
pie
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Scatter Diagram
A plot of paired (x,y) data with a horizontal
x-axis and a vertical y-axis
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Time-Series Graph
Data that have been collected at different points
in time
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Losses of Napoleon's Army
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Section 2.4Measures of Center
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Definitions

• Measure of Center The value at the center or
  middle of a data set
• Arithmetic Mean (Mean) The measure of center
  obtained by adding the values and dividing the
  total by the number of values.
• Median The middle value when the original data
  values are arranged in order of increasing (or
  decreasing) magnitude.
• Median is not affected by an extreme value.

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Notation (Mean)

• ? denotes the addition of a set of values
• x is the variable usually used to represent the
  individual data values
• n represents the number of values in a sample
• N represents the number of values in a population

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Notation (Mean)

• µ is pronounced mu and denotes the mean of all
  values in a population

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Finding the Median

• Odd Number of Values the median is the number
  located in the exact middle of the list.
• Even Number of Values the median is found by
  computing the mean of the two middle numbers.

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Odd Number of Values
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Even Number of Values
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Mode (M)

• Value that occurs most frequently
• Mode may be
• Mode (one value)
• Bimodal (two values)
• Multimodal (multiple values)
• No Mode (no valuesno repeats)

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Midrange

• The value midway between the highest and lowest
  values in the original data set
• To compute

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Round-off Rule for Measures of Center

• Carry one more decimal place than is present in
  the original set of values
• 10.2 12.3 9.5 5.2 6.5
• Median 8.74
• Rules of Rounding
• Between 0 4, round down (9.83 becomes 9.8)
• Between 5 9, round up (9.86 becomes 9.9)
• Both are rounded to the tenths place

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Mean From Frequency Distribution

• Assume that in each class, all sample values are
  equal to the class midpoint

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Best Measure of Center
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Definitions

• Symmetric Data is symmetric if the left half of
  its histogram is roughly a mirror image of its
  right half.
• Skewed Data is skewed if it is not symmetric
  and if it extends more to one side than the
  other.

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Skewness
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Recap
In this section we have discussed