Excursions in Modern Mathematics Sixth Edition

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Excursions in Modern MathematicsSixth Edition

• Peter Tannenbaum

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Chapter 14Descriptive Statistics

• Graphing and Summarizing Data

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Descriptive StatisticsOutline/learning Objectives

• To interpret and produce an effective graphical
  summary of a data set.
• To identify various types of numerical variables.
• To interpret and produce numerical summaries of
  data including percentiles and five-number
  summaries.

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Descriptive StatisticsOutline/learning Objectives

• To describe the spread of a data set using range,
  interquartile range, and standard deviation.

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Descriptive Statistics

• 14.1 Graphical Descriptions of Data

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Descriptive Statistics

• Data set
• A collection of data values denoted by N.
• Data points
• Individual data values in a data set.

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Descriptive Statistics
Stat 101 Test Scores Part 1 Professor Blackbeard
has posted the results in the hallway outside his
office. The data set consists of N 75 data
points (the number of students that took the
test). Each data point is a raw score on the
midterm between 0 and 25. Each student has one
question on their mind How did I do? Its the
next question that is statistically more
interesting How did the class as a whole do?
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Descriptive Statistics
Stat 101 Test Scores Part 2 The first step in
summarizing the information is to organize the
scores in a frequency table. In this table, the
number below each score gives the frequency of
the score that is, the number of students
getting that particular score.
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Descriptive Statistics
Stat 101 Test Scores Part 2 The figure below
shows the information in a more visual way called
a bar graph. With a bar graph, it is easy to
detect outliers -- extreme data points that do
not fit into the overall pattern of the data (the
score of 1 and 24).
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Descriptive Statistics
Stat 101 Test Scores Part 2 Sometimes it is more
convenient to express the bar graph in a term of
relative frequencies that is, the frequencies
given in terms of percentages of the total
population.
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Descriptive Statistics
Stat 101 Test Scores Part 2 Frequency charts
that use icons or pictures instead of bars to
show the frequencies are commonly referred to as
pictograms.
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Descriptive Statistics

• 14.2 Variables

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Descriptive Statistics

• Variable
• Any characteristic that varies with the members
  of a population.
• Numerical (Quantitative) Variable
• A variable that represents a measurable quantity.

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Descriptive Statistics

• Continuous
• When the difference between the values of a
  numerical variable can be arbitrarily small.
• Discrete
• When possible values of the numerical variable
  change by minimum increments.

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Descriptive Statistics

• Categorical (Qualitative) Variables
• Variables can also describe characteristics that
  cannot be measured numerically.
• Pie Chart
• When the number of categories is small, another
  commonly used way to describe the relative
  frequencies of the categories.

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Descriptive Statistics

• Stat 101 Test Scores Part 3
• The process of converting test scores (a
  numerical variable) into grades ( a categorical
  variable) requires setting up class intervals for
  the various letter grades.
• The grade distribution in the Stat 101 midterm
  can now be seen by means of a bar graph.

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Descriptive Statistics

• Histograms
• When a numerical variable is continuous, its
  possible values can vary by infinitesimally small
  increments. As a consequence, there are no gaps
  between the class intervals.

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Descriptive Statistics

• 14.3 Numerical Summaries of Data

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Descriptive Statistics

• Measures of Location
• The mean (or average), the median, and the
  quartiles are numbers that provide information
  about the values of the data.
• Measures of Spread
• The range, the interquartile range, and the
  standard deviation are numbers that provide
  information about the spread within the data set.

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Descriptive Statistics

• Stat 101 Test Scores Part 4
• The average of a set of N numbers is found by
  adding the numbers and dividing the total by N.
• Step 1. Find the sum Sum d1 f1 d2 f2
  dk fk
• (1 1) (6 1) (24 1) 814
• Step 2. Find N N f1 f2 fk 75
• Step 3. Find A A Sum/N 814/75 ? 10.85

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Descriptive Statistics

• Percentile
• The pth percentile of a data set is a value such
  that p percent of the numbers fall at or below
  this value and the rest fall at or above it.
• Locator
• Computed by the pth percent of N and is denoted
  by L. L (p/100) N

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Descriptive Statistics

• Finding the pth Percentile of a Data Set
• Step 0. Sort the data set. Let d1, d2, d3,
  , dN represent the sorted data set.
• Step 1. Find the locator L (p/100) N
• Step 2. Find the pth percentile If L is a
  whole number, the pth percentile is given by d
  L.5. If L is not a whole number, the pth
  percentile is given by dL (L is L rounded up).

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Descriptive Statistics

• The 50th percentile of a data set is known as
  the median and denoted by M.
• Finding the Median of a Data Set
• Sort the data set. Let d1, d2, d3, , dN
  represent the sorted data set.
• If N is odd, the median is d (N1)/2 . If N
  is even, the median is the average of d N/2 and d
  (N/2)1 .

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Descriptive Statistics

After the median, the next most commonly used
set of percentiles are the first and third
quartiles. The first quartile (denoted by Q1) is
the 25th percentile, and the third quartile
(denoted by Q3) is the 75th percentile.
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Descriptive Statistics

• Stat 101 Test Scores Part 5
• We will now find the median and quartile scored
  for Stat 101.
• Here N 75 (odd), the median is d (751)/2 d
  38 . We conclude that the 38th test score is 11.
  Thus, M 11.
• The locator for the first quartile is L (0.25)
  X 75 18.75. We tally from left to right. Thus
  Q1 d 19 9 .
• Since the first and third quartiles are at equal
  distance, a quick way to locate the third
  quartile is to count from right to left. Thus,
  Q3 12.

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Descriptive Statistics

• A common way to summarize a large data set is by
  means of its five-number summary. The
  five-number summary is given by the smallest
  value in the data set (called the Min), the first
  quartile (Q1), the median (M), the third quartile
  (Q3), and the largest value in the data set
  (called the Max). These five numbers together
  often tells us a great deal about the data.

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Descriptive Statistics

• Stat 101 Test Scores Part 6
• For the Stat 101 data set, the five-number
  summary is Min 1, Q1 9, M 11, Q3 12 and
  Max 24.
• What useful information can we get out of this?
• The big picture we get from the five-number
  summary is that there were a lot of bunching up
  in a narrow band of scores.
• In general, this type of lumpy distribution of
  test scores is indicative of a test with an
  uneven level of difficulty.

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Descriptive Statistics

• 14.4 Measures of Spread

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Descriptive Statistics

• Range
• The difference between the highest and lowest
  values of the data and is denoted by R. Thus, R
  Max - Min.
• Interquartile Range
• The difference between the third quartile and the
  first quartile (IQR Q3 Q1), and it tells us
  how spread out the middle 50 of the data values
  are.

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Descriptive Statistics

• The Standard Deviation of a Data Set
• Let A denote the mean of the data set. For each
  number x in the data set, compute its deviation
  from the mean (x A), and square each of these
  numbers. These are called the squared
  deviations.
• Find the average of the squared deviations.
  This number is called the variance V.
• The standard deviation is the square root of
  the variance ( ).

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Descriptive Statistics Conclusion

• Basic concepts in statistics
• Graphical summaries
• Numerical summaries