Objectives

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Objectives

• Summarize data using measures of central
• tendency, such as the mean, median, mode,
• and midrange.
• Describe data using measures of variation,
• such as the range, variance, and standard
• deviation.

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Objectives

• Identify the position of a data value in a data
  set, using various measures of position, such as
  percentiles, deciles, and quartiles.
• Use the techniques of exploratory data analysis,
  including boxplots and five-number summaries, to
  discover various aspects of data.

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Introduction

• Chapter 2 showed how one can gain useful
  information from raw data by organizing them into
  a frequency distribution and then presenting the
  data by using various graphs.
• This chapter shows the statistical methods that
  can be used to summarize data. The most familiar
  of these methods is the finding of averages.

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Outline of Chapter 3

• 3-2 Measures of Central Tendency
• 3-3 Measures of Variation
• 3-4 Measures of Position
• 3-5 Exploratory Data Analysis
• 3-6 Summary

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3-2 Measures of Central Tendency

• A statistic is a characteristic or measure
  obtained by using the data values from a sample.
• A parameter is a characteristic or measure
  obtained by using all the data values from a
• specific population.
• Example Population mean (parameter) vs. sample
  mean (statistic)

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General Rounding Rule

• In statistics the basic rounding rule is that
  when computations are done in the calculation,
  rounding should not be done until the final
  answer is calculated.

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The Mean
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Example 3-3 (p99)
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The Median

• The median is the midpoint of the data array.
  The symbol for the median is MD.
• Steps for finding MD
• Arrange the data in order
• Select the middle point
• Examples 3-4 3-8 (p101)

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The Mode

• The value that occurs most often in a data set
  is called the mode.
• Unimodal, bimodal, multimodal
• Examples 3-9 3-14 (p103)

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The Midrange
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The Weighted Mean
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Properties and uses of central tendency (p108)
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Distribution Shapes
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3-3 Measures of Variation

• In statistics, to describe the data set
  accurately, statisticians must know more than the
  measures of central tendency.
• Example 3-18 (p115) mean is not enough to

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Range
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Population Variance and Standard Deviation

• Before the variance and standard deviation are
  defined formally, the computational procedure
  will be shown, since the definition is derived
  from the procedure.
• Example 3-21 (p117)

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Rounding Rule for the Standard Deviation

• The rounding rule for the standard deviation is
  the same as that for the mean. The final answer
  should be rounded to one more decimal place than
  that of the original data.

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Sample Variance and Standard Deviation
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Variance and Standard Deviation for Grouped Data

• The procedure for finding the variance and
  standard deviation for grouped data is similar to
  that for finding the mean for grouped data, and
  it uses the midpoints of each class.
• See Procedure Table on p123

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Q How to interpret the values of range,
variance, and standard deviation? A Uses of the
variance and standard deviation (p123)
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Coefficient of Variation (????)
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Range Rule of Thumb

• The range can be used to approximate the
  standard deviation. The approximation is called
  the range rule of thumb.

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Chebyshevs Theorem

• Notes
• Chebyshevs Theorem is valid for any shape of
  the population distribution (symmetric or
  asymmetric).
• Fig 3-3 for demonstrating Chebyshevs Theorem
• Practice Examples 3-27 3-28

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Empirical (Normal) Rule

• When a distribution is bell-shaped (or what is
  called normal), the following statements, which
  make up the empirical rule, are true.
• Approximately 68 of the data values will fall
  within 1 standard deviation of the men.
• Approximately 95 of the data values will fall
  within 2 standard deviations of the men.
• Approximately 99.7 of the data values will fall
  within 3 standard deviation of the men.
• Fig 3-4 for demonstrating the Empirical Rule
• Application case- Blood pressure (p128)
• Chebyshevs rule is more conservative than
  Empirical rule

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3-4 Measures of Position

• In addition to measures of central tendency and
  measures of variation, there are measures of
  position or location.
• These measurements are used to locate the
  relative position of a data value in the data
  set.

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Standard Scores
Q How to interpret the z-score in
words? Practice Examples 3-29 3-30
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Percentiles

• Percentiles divide the data set into 100 equal
  groups.
• Interesting facts
• The highest recorded temperature on Earth was
  136 (F) in Libya (???, ?????)
• The lowest recorded temperature on Earth was
  -129 (F) in Antarctica (???)
• Table 3-3

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Practice Examples 3-33 3-34
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Outliers

• An outlier is an extremely high or an extremely
  low data value when compared with the rest of the
  data values.

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Practice Application case 3-4 (p144)
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3-5 Exploratory Data Analysis

• In traditional statistics, data are organized by
  using a frequency distribution. From this
  distribution various graphs such as the
  histogram, frequency polygon, and ogive can be
  constructed to determine the shape or nature of
  the distribution. In addition, various statistics
  such as the mean and standard deviation can be
  computed to summarize the data.

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The Five-Number Summary and Boxplots

• A boxplot is a graph of a data set obtained by
  drawing a horizontal line from the minimum data
  value to Q1, drawing a horizontal line from Q3 to
  the maximum data value,and drawing a box whose
  vertical sides pass through Q1 and Q3 with a
  vertical line inside the box passing through the
  median or Q2.

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