Data Description

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CHAPTER 3
Data Description
2
Objectives

• Summarize data using measures of central
  tendency, such as the mean, median, mode, and
  midrange.
• Describe data using the measures of variation,
  such as the range, variance, and standard
  deviation.
• Identify the position of a data value in a data
  set using various measures of position, such as
  percentiles, deciles, and quartiles.

3
Objectives (contd.)

• Use the techniques of exploratory data analysis,
  including boxplots and five-number summaries to
  discover various aspects of data.

4
Introduction

• Statistical methods can be used to summarize
  data.
• Measures of average are also called measures of
  central tendency and include the mean, median,
  mode, and midrange.
• Measures that determine the spread of data values
  are called measures of variation or measures of
  dispersion and include the range, variance, and
  standard deviation.

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Introduction (contd.)

• Measures of position tell where a specific data
  value falls within the data set or its relative
  position in comparison with other data values.
• The most common measures of position are
  percentiles, deciles, and quartiles.

6
Introduction (contd.)

• The measures of central tendency, variation, and
  position are part of what is called traditional
  statistics. This type of data is typically used
  to confirm conjectures about the data.

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Introduction (contd.)

• Another type of statistics is called exploratory
  data analysis (EDA). These techniques include the
  the box plot and the five-number summary. They
  can be used to explore data to see what they show.

8
Basic Vocabulary

• 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 for a
  specific population.
• When the data in a data set is ordered it is
  called a data array.

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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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(3.2) The Arithmetic Average

• The mean is the sum of the values divided by the
  total number of values.
• Rounding rule the mean should be rounded to one
  more decimal place than occurs in the raw data.
• The type of mean that considers an additional
  factor is called the weighted mean.

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The Arithmetic Average

• The Greek letter ? (mu) is used to represent the
  population mean.
• The symbol (x-bar) represents the sample
  mean.
• Assume that data are obtained from a sample
  unless otherwise specified.

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Median and Mode

• The median is the halfway point in a data set.
  The symbol for the median is MD.
• The median is found by arranging the data in
  order and selecting the middle point.
• The value that occurs most often in a data set is
  called the mode.
• The mode for grouped data, or the class with the
  highest frequency, is the modal class.

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Midrange

• The midrange is defined as the sum of the lowest
  and highest values in the data set divided by 2.
• The symbol for midrange is MR.

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Central Tendency The Mean

• One computes the mean by using all the values of
  the data.
• The mean varies less than the median or mode when
  samples are taken from the same population and
  all three measures are computed for these
  samples.
• The mean is used in computing other statistics,
  such as variance.

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Central Tendency The Mean (contd.)

• The mean for the data set is unique, and not
  necessarily one of the data values.
• The mean cannot be computed for an open-ended
  frequency distribution.
• The mean is affected by extremely high or low
  values and may not be the appropriate average to
  use in these situations.

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Central Tendency The Median

• The median is used when one must find the center
  or middle value of a data set.
• The median is used when one must determine
  whether the data values fall into the upper half
  or lower half of the distribution.
• The median is used to find the average of an
  open-ended distribution.
• The median is affected less than the mean by
  extremely high or extremely low values.

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Central Tendency The Mode

• The mode is used when the most typical case is
  desired.
• The mode is the easiest average to compute.
• The mode can be used when the data are nominal,
  such as religious preference, gender, or
  political affiliation.
• The mode is not always unique. A data set can
  have more than one mode, or the mode may not
  exist for a data set.

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Central Tendency The Midrange

• The midrange is easy to compute.
• The midrange gives the midpoint.
• The midrange is affected by extremely high or low
  values in a data set.

19
Distribution Shapes

• In a positively skewed or right skewed
  distribution, the majority of the data values
  fall to the left of the mean and cluster at the
  lower end of the distribution.

20
Distribution Shapes (contd.)

• In a symmetrical distribution, the data values
  are evenly distributed on both sides of the mean.

21
Distribution Shapes (contd.)

• When the majority of the data values fall to the
  right of the mean and cluster at the upper end of
  the distribution, with the tail to the left, the
  distribution is said to be negatively skewed or
  left skewed.

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The Range

• The range is the highest value minus the lowest
  value in a data set.
• The symbol R is used for the range.

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(3.3) Variance and Standard Deviation

• The variance is the average of the squares of the
  distance each value is from the mean. The symbol
  for the population variance is ?2.

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Variance and Standard Deviation

• The standard deviation is the square root of the
  variance. The symbol for the population standard
  deviation is ?. Rounding rule The final answer
  should be rounded to one more decimal place than
  the original data.

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Coefficient of Variation

• The coefficient of variation is the standard
  deviation divided by the mean. The result is
  expressed as a percentage.
• The coefficient of variation is used to compare
  standard deviations when the units are different
  for the two variables being compared.

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Variance and Standard Deviation

• Variances and standard deviations can be used to
  determine the spread of the data. If the variance
  or standard deviation is large, the data are more
  dispersed. The information is useful in comparing
  two or more data sets to determine which is more
  variable.
• The measures of variance and standard deviation
  are used to determine the consistency of a
  variable.

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Variance and Standard Deviation (contd.)

• The variance and standard deviation are used to
  determine the number of data values that fall
  within a specified interval in a distribution.
• The variance and standard deviation are used
  quite often in inferential statistics.

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Chebyshevs Theorem

• The proportion of values from a data set that
  will fall within k standard deviations of the
  mean will be at least 1 1/k2 where k is a
  number greater than 1.
• This theorem applies to any distribution
  regardless of its shape.

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Empirical Rule for Normal Distributions

• The following apply to a bell-shaped
  distribution.
• Approximately 68 of the data values fall within
  one standard deviation of the mean.
• Approximately 95 of the data values fall within
  two standard deviations of the mean.
• Approximately 99.75 of the data values fall
  within three standard deviations of the mean.

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Standard Scores

• A standard score or z score is used when direct
  comparison of raw scores is impossible.
• A standard score or z score for a value is
  obtained by subtracting the mean from the value
  and dividing the result by the standard
  deviation.

31
Percentiles

• Percentiles are position measures used in
  educational and health-related fields to indicate
  the position of an individual in a group.
• A percentile, P, is an integer between 1 and 99
  such that the Pth percentile is a value where P
  of the data values are less than or equal to the
  value and 100 P of the data values are
  greater than or equal to the value.

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Quartiles and Deciles

• Quartiles divide the distribution into four
  groups, denoted by Q1, Q2, Q3. Note that Q1 is
  the same as the 25th percentile Q2 is the same
  as the 50th percentile or the median and Q3
  corresponds to the 75th percentile.
• Deciles divide the distribution into 10 groups.
  They are denoted by D1, D2, , D10.

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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.
• Outliers can be the result of measurement or
  observational error.
• When a distribution is normal or bell-shaped,
  data values that are beyond three standard
  deviations of the mean can be considered
  suspected outliers.

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Exploratory Data Analysis

• The purpose of exploratory data analysis is to
  examine data in order to find out what
  information can be discovered. For example
• Are there any gaps in the data?
• Can any patterns be discerned?

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Boxplots and Five-Number Summaries

• Boxplots are graphical representations of a
  five-number summary of a data set. The five
  specific values that make up a five-number
  summary are
• The lowest value of data set (minimum)
• Q1 (or 25th percentile)
• The median (or 50th percentile)
• Q3 (or 75th percentile)
• The highest value of data set (maximum)