Descriptive Analysis and Presentation of SingleVariable Data

1
Descriptive Analysis and Presentation of
Single-Variable Data

• Descriptive Statistics for Short
• References Julio Rivera

2
Circle graph

• Good for summarizing attribute (nominal) data

3
Bar Graphs

• These are more traditional
• Provide basic, quick information

4
Stem and Leaf Plots

• Useful for sorting data
• Useful for visualizing the spread of the data
• Stem is the line and the leaves are the values
  extended from the stem
• Sometimes shows things unexpected

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Frequency Distribution

• Suppose you have the values 3, 2, 2, 3, 2, 4,
  4, 1, 2, 2, 4, 3, 2, 0, 2, 2, 1, 3, 3, 1
• A list like this does not provide a clear picture
  to the reader
• Stem and Leaf or other graphs are not always the
  most effective way to display the data

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Ungrouped Frequency Distribution

• Used to represent the set of data in summary form
• This an ungrouped frequency distribution because
  each value of x stands alone

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Grouped Frequency Distribution

• Classes are formed
• Each class here has the same width (not always
  the case)
• Each class should not overlap with other classes

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Assume a data set

• 27 68 79 91 107 43 71 80 91 108 43 71 81 93 108 4
  4 71 82 94 116 47 73 82 94 12049 73 84 94 120 50 7
  4 84 96 122 54 75 86 97 123 58 76 88 103 127 65 77
  88 106 128

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Classify it

• No overlaps
• Upper class limit
• largest values fitting into each class
• Lower Class limit
• smallest piece of data that could go in each class

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Class Boundaries

• numbers that do not occur in the sample data, but
  are halfway between the upper limit of one class
  and the lower limit of the next
• In this data set the boundaries are 21.5, 32.5,
  43.5, 54.5, etc.

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Histogram

• A type of bar graph representing and entire set
  of data
• A histogram will contain
• title
• vertical scale (frequency)
• horizontal scale representing the variable

Exam Scores
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Relative Frequency Histogram

• A proportional measure of the frequency of
  occurrence (a percentage)
• SYSTAT puts it on the left of the graph

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Shapes of Histograms

• Described as Symmetrical, Normal or Triangular
• Both sides of the distribution are identical
• The normal distribution is your friend

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Shapes of Histograms

• Uniform or Rectangular
• Every value appears with equal frequency
• Notice there are no peaks in this
  distribution--we will talk about Kurtosis later.

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Shapes of Histograms

• Skewed
• One tail is stretched out longer than the other
• The direction of skewness is on the side of the
  longer tail
• This is skewed to the right (positive skew)

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Shapes of Histograms

• This is skewed to the left
• negative skew
• We will talk about the mathematical formula for
  skewness later

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Shapes of Histograms

• J-shaped
• There is no tail on the side of the class with
  the highest frequency

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Shapes of Histograms

• Bi-Modal or Multi-modal
• The most populous classes are separated by one or
  more classes.
• This situation often implies that two populations
  are being sampled

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Cumulative Frequency
Yuck--sorry about this definition!

• Using the frequencies to make subtotals of the
  accumulated frequencies of the classes and those
  classes which are less than the current class
• The relative cumulative frequency is a percentage
  of the total

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Classifying the Data

• Notice the classification of data
• Equal groups of eleven
• Only one of a number of ways of classifying data
• Remember thatbut first we have to talk about
  some characteristics of data

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Tangibility--the physical reality of the data

• Tangible
• Actually existing, capable of measurement
  (snowfall, mountain height, number of beagles,
  etc.)
• Abstract
• Computed or derived from other sources (percent
  of unemployment, median school years completed)

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Temporality--the situation of the data in time

• Status
• Showing things as they are at a particular time.
  (Land value in 1995, air traffic in December,
  Mean temperature 1950-80, etc.)
• Trend
• Showing how things change through time (urban
  growth since 1950, change in the extent of the
  rain forest since 1960)

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Measurement Levels--a quick review

• Nominal Data
• Ordinal Data
• Interval Data
• Ratio Data

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Classifying and Characterizing Data

• Classifications of data are useful because they
  allow us to describe data.
• The better we can describe the data, the more
  skillful we are at representing it.
• We are going to discuss some of these
  classifications in detail

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Exogenous Classifications

• Category boundaries are established based on
  threshold values set by criteria from another
  source
• Source is usually a government agency or
  professional organization
• Low Risk 30 to 49 PPM
• Medium Risk 50 to 99 PPM
• High Risk 100 to 1000 PPM

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Ideographic

• Category boundaries are established based on
  details within the data values
• Quantile
• Contains an equal number of pieces of data in
  each category
• 50 states, 5 categories, 10 states in each
  category
• 87 counties, 4 categories, 22 in three
  categories, 21 in one category

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• Multi-modal
• Uses natural breaks in the data to determine
  category boundaries
• Be careful--natural breaks--not blips
• Multi-step
• Breaks in the slope of a cumulative frequency
  curve

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Serial

• Category boundaries are established that have a
  mathematical relationship to one another
• Equal Interval
• divides the data into equally ranged classes
  based on the minimum and maximum data values

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• Arithmetic Progression
• Category ranges differ based on a rate of
  increase of a constant (Adding)
• 2, 4, 6, 8, 10, 12, 14, 16
• Geometric Progression
• Category ranges differ based on a rate of
  increase that takes in account the prior class
  range (Multiplying)
• 2, 4, 8, 16, 32, 64, 128, 256

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• Standard Deviation Units
• Centered on the mean of the data with class
  boundaries established by the standard deviations
  plus the mean
• Highlighted
• based on a decision to highlight particular
  sectors of the data set in greater or lesser
  detail

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Writing it out in a legend

• Categories designations should not overlap
• 0-10, 10-20, 20-30, 30-40, 40-50
• Designations should reflect the actual data that
  is in the category
• 0-10, 11-20, 21-30, 31-40, 41-50
• 0-9, 10-19, 20-29, 30-39, 41-50
• 1-10, 11-20, 21-30, 31-40, 41-50

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Classifying and Characterizing Data

• No where are these classifications more dramatic
  than on choropleth maps (and other thematic maps
  as well)
• The following seven maps were made with the same
  piece of data
• The only difference was how the data was
  classified
• All the classifications are standard, accepted
  ways of classifying data

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Theres more ways to describe data

• Shape and frequency are one way to describe data
  sets
• Then we can classify them
• We can describe the data relative to the middle
  of the data

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Measures of Central Tendency

• Different ways of describing the middle of the
  data set
• Some averages are more average than others
• Its mean to drive on the median, you might get
  mode down
• A really bad play on words--Im truly sorry
• Julio

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Mean

• Arithmetic Mean Sometimes called the
  average.
• This means add all the variables of x and then
  divide the number of values which is represented
  by n

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Median

• The middle value of the sample when the data are
  ranked in order according to size
• The median is the ith value in the ranked data
  set
• Median is between 25 and 26 (use both values)

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Other Averages

• Mode
• The value of x which occurs the most frequently
• Excellent for nominal data
• Midrange
• The midrange is the data value which is the
  average of the low and high values

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Weighted Mean

• Takes each class midpoint and multiples it by the
  class frequency.
• These are summed and divided by n

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Which tendency is most central?

• All these measures of central tendency describe
  the middle of the data.
• McGrew provides a good example with income data
• Mode 21,000
• Median 26,000
• Mean 71,428

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Using Central Tendency

• Mode, Median and Mean are the most common
• Means can be greatly influenced by outlying
  values
• Medians are always the middle value
• Modes are the most common value
• These can yield different results using the same
  data and have different bases in logic

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Measures of dispersion

• If you have a middle, how is the data arranged
  around the middle?
• Examine it graphically and statistically
• Graphically? did I hear John Tukey?

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Box and Whisker Plots

• Line in the center of the box is the median
• Box represents the Inter-quartile Range (25 to
  75 levels)
• Whiskers represent 25--sometimes outliers are
  marked

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Deviation from the Mean

• How much do the values vary from the mean?
• If the mean is 6 and your value is 3 what is the
  variation?
• If the mean is 6 and your value is 7 what is the
  variation?
• What is the absolute value of those two
  variations
• x reads the absolute value of x (always
  positive)
• x-y the absolute value of x-y (always positive)

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Mean absolute deviation

• The mean of the deviations
• Tells us the average distance that a piece of
  data is from the mean

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Standard Deviation and Variance

• These are your friends
• Think of them as Bert and Ernie
• These two statistics are two of the most commonly
  referred to in statistical work

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Standard Deviation

• First formula is for the sample, and the second
  is for the population
• Standard Deviation is a measure of fluctuation of
  the data (a yardstick if you will)

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Variance

• It is the standard deviation squared
• Like standard deviation it is a measurement of
  fluctuation of the data.
• Variances are used in other statistical
  procedures (later in the semester)

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Coefficient of Variation

• Comparison of data between two data sets
• Takes account the problems of different sample
  sizes
• A ratio of the standard deviation divided by the
  mean

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Coefficient of Variation

• Annual precipitation data for 40 years (n40)
• Although St. Louis has the larger SD, the CV for
  San Diego is larger.
• Why? and what does this mean?

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Measures of Shape and Relative Position

• Other measures of the frequency Distribution
• Skewness and Kurtosis

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Skewness

• Measures the degree of symmetry in the
  distribution
• the extent to which the bulk of values in a
  distribution are concentrated on one or the other
  side of the mean
• Remember shapes of histograms that were skewed to
  the right and skewed to the left (positive and
  negative)

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Skewness

• Skewness is important because many Geographic
  data sets are skewed
• Helps understand the mean
• Little skew, the mean is probably representative
• Lots--maybe not

s standard deviation
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Kurtosis

• Measures the flatness or the peakedness of the
  data
• Is the distribution full of peaks and valleys or
  is it flatter?
• Best compared to a normal curve (kurtosis3)

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Kurtosis

• If value is greater than 3 it is letptokurtic
  (peaky)
• If value is less than 3 it is platykurtic (flat)
• Note that some programs will use formula 2 so
  then zero is the dividing line for kurotsis

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Quartiles

• Data is divided into four equal parts (quarters)
• Inter-quartile range is between 25 and 50 marks
• Do not confuse quartiles with quantiles
• quartile is a type of quantile

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Z-score (Standard Score)

• The position of a value (x) away from the mean
  measured in standard deviations
• More on this later

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Chebyshevs Theorem

• A useful tidbit of knowledge
• The proportion of any distribution that lies
  within k standard deviations of the mean is at
  least 1-(1/k2) where k is any positive number
  larger than one
• This applies to any distribution of data

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Application of Theorem

• This means that for any distribution of data
• 75 of the data is within 2 standard deviations
  from the mean

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Empirical Rule

• For Normal Distributions
• 68 of the data is within 1 Standard Deviation of
  the Mean
• 95 is within 2 SDs
• 99.7 is within 3 SDs
• Really important to know (film at 11)