Descriptive%20Statistics

1
Descriptive Statistics

• Lecture 02
• Tabular and Graphical Presentation of Data and
  Measures of Locations

2
Presentation of Qualitative Variables

• The simplest way of presenting/summarizing a
  qualitative variable is by using a frequency
  table, which shows the frequency of occurrence of
  each of the different categories.
• Such a table could also include the relative
  frequency, which indicates the proportion or
  percentage of occurrence of each of the
  categories.
• The frequency table could then be pictorially
  represented by a bar graph or a pie diagram.

3
An Example

• A manufacturer of jeans has plants in California
  (CA), Arizona (AZ), and Texas (TX). A sample of
  25 pairs of jeans was randomly selected from a
  computerized database, and the state in which
  each was produced was recorded. The data are as
  follows
• CA AZ AZ TX CA CA CA TX TX TX AZ AZ CA AZ TX CA
  AZ TX TX TX CA AZ AZ CA CA
• Quite uninformative at this stage!
• Need to summarize to reveal information.

4
The Frequency Table
5
The Bar Chart
Frequency
10
5
0
TX
CA
AZ
6
Example continued

• By looking at this frequency table and bar graph,
  one is able to obtain the information that there
  seems to be equal proportions of pairs of jeans
  being manufactured in the three states.
• Frequency table and bar graph certainly more
  informative than the raw presentation of the
  sample data.
• Another method of pictorial presentation of
  qualitative data is by using the pie diagram. In
  this case a pie is divided into the categories
  with a given categorys angle being equal to 360
  degrees times the relative frequency of
  occurrence of that category.

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Pie Diagram
Angles (in degrees) CA(360)(.36)129.6 AZ(360)(
.32)115.2 TX(360)(.32)115.2
129.6o
115.2o
115.2o
8
Pie Chart from Minitab
9
Presentation of Quantitative Variables

• When the quantitative variable is discrete (such
  as counts), a frequency table and a bar graph
  could also be used for summarizing it.
• Only difference is that the values of the
  variables could not be reshuffled in the graph,
  in contrast to when the variable is categorical
  or qualitative.
• For example suppose that we asked a sample 20
  students about the number of siblings in their
  family. The sample data might be
• 4, 1, 6, 2, 2, 3, 4, 1, 2, 2, 3, 7, 2, 1, 1, 5,
  3, 4, 6, 3

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Its Bar Graph is
11
An Example of a Real Data Set Poverty versus
PACT in SC
74 48 54 77 43 55 94 41 62 88 49 62 78 50 59 79 46
58 61 41 47 45 26 34 87 49 62 68 36 52 76 45 56 3
2 22 31 63 39 53 33 20 26 64 44 53 39 20 22 37 21
27 47 23 30 40 29 41 43 25 27 37 24 31 64 37 43 59
36 45 70 32 41 55 37 46 90 38 47 45 32 35 31 25 2
4 35 29 32 15 14 18
73 30 41 31 24 30 75 45 57 57 29 40 80 51 63 54 30
44 67 28 33 76 45 50 87 61 61 54 27 33 60 32 41 3
5 26 35 51 29 36 50 35 42 43 23 26 66 32 44 86 63
75 54 25 33 87 60 69 49 29 37 46 38 43 50 38 44 57
40 50 90 60 75 26 17 20 47 23 27 53 37 39 58 34 4
3 16 13 15
Lunch ActualLang ActualMath 59 32 38 46 26 30 90 6
3 67 29 17 24 41 24 26 51 30 41 41 25 30 43 32 36
70 33 36 93 50 66 84 50 66 64 27 32 52 36 43 50 31
43 53 28 35 78 36 41 57 31 42 51 39 42 55 41 53 6
0 37 45 96 46 66 75 34 45 60 29 36 71 43 53 68 42
51 76 47 52 82 49 55
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Frequency Tables and Histograms
Consider the variable Lunch, which represents
the percentage of students in the school district
whose lunches are not free. The higher the value
of this variable, the richer the district. n
Number of Observations 86 LV Lowest Value
15 HV Highest Value 96 Let us construct a
frequency table with classes 10,20), 20,30),
30,40), , 90,100)
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Frequency Table for Variable Lunch
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Frequency Histogram
15
Stem-and-Leaf Plots

• An important tool for presenting quantitative
  data when the sample size is not too large is via
  a stem-and-leaf plot.
• By using this method, there is usually no loss
  of information in that the exact values of the
  observations could be recovered (in contrast to a
  frequency table for continuous data).
• Basic idea To divide each observation into a
  stem and a leaf.
• The stems will serve as the body of the plant
  while the leaves will serve as the branches or
  leaves of the plant.
• An illustration makes the idea transparent.

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An Example

• A random sample of 30 subjects from the 1910
  subjects in the blood pressure data set was
  selected. We present here the systolic blood
  pressures of these 30 subjects.
• 30 Systolic Blood Pressures 122 135 110 126 100
  110 110 126 94 124 108 110 92 98 118 110 102 108
  126 104 110 120 110 118 100 110 120 100 120 92
• Lowest Value 92, Highest Value 135
• Stems 9,10, 11, 12, 13
• Leaves Ones Digit

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Stem-and-Leaf Plot

• 9 224
• 9 8
• 10 00024
• 10 88
• 11 00000000
• 11 88
• 12 00024
• 12 666
• 13
• 13 5

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Stem-and-Leaf continued

• In this stem-and-leaf plot, because there will
  only be 5 stems if we use 9, 10, 11, 12, 13, we
  decided to subdivide each stem into two parts
  corresponding to leaf values lt 4, and those gt
  5.
• Such a procedure usually produces better looking
  distributions.
• Looking at this stem-and-leaf plot, notice that
  many of the observations are in the range of
  100-126.
• The exact values could be recovered from this
  plot.
• By arranging the leaves in ascending order, the
  plot also becomes more informative.

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Comparative Stem-and-Leaf Plots

• When comparing the distributions of two groups
  (e.g., when classified according to GENDER),
  side-by-side stem-and-leaf plots (also
  side-by-side histograms) could be used.
• To illustrate, consider 30 observations from the
  blood pressure data set with Gender and Systolic
  Blood Pressure being the observed variables.
• For the males (Sex 0) 122, 120, 130, 110, 134,
  136, 142, 100, 120, 162, 126, 132, 124, 130
• For the females (Sex 1) 132, 94, 104, 100,
  130, 110, 102, 110, 130, 92, 125, 108, 100, 130,
  100, 100

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Comparing Male/Female Systolic Blood Pressures
21
Scatterplots Studying Relationship Between
Poverty and Math
Question What kind of relationship is there
between Lunch and PACT Math Scores?
22
Numerical Summary Measures

• Overview
• Why do we need numerical summary measures?
• Measures of Location
• Measures of Variation
• Measures of Position
• Box Plots

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Why we Need Summary Measures?

• A picture is worth a thousand words, but beauty
  is always in the eyes of the beholder!
• Graphs or pictures sometimes unwieldy
• Usually wants a small set of numbers that could
  provide the important features of the data set
• When making decisions, objectivity is enhanced
  when they are based on numbers!
• Numerical summaries and tabular/graphical
  presentations complement each other

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The Setting

• In defining and illustrating our summary
  measures, assume that we have sample data
• Sample Data X1, X2, X3, , Xn
• Sample Size n
• These summary measures are thus (sample)
  statistics.
• If instead they are based on the population
  values, they will be (population) parameters.

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Measures of Location or Center

• These are summary measures that provide
  information on the center of the data set
• Usually, these measures of location are where the
  observations cluster, but not always
• In laymans terms, these measures are what we
  associate with averages
• Will discuss two measures sample mean and sample
  median

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Sample Mean or Arithmetic Average

• The sample mean equals the sum of the
  observations divided by the number of
  observations.
• It is defined symbolically via

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Properties of the Sample Mean

• Center of Gravity
• Sum of the deviations of the observations from
  the mean is always zero (barring rounding errors)
• Sample mean could however be affected drastically
  by extreme or outliers
• The sample mean is very conducive to mathematical
  analysis compared to other measures of location

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Illustration

• Consider the systolic blood pressure data set
  considered in Lecture 01
• Sample Size n 30
• Data 122, 135, 110, 126, 100, 110, 110, 126, 94,
  124, 108, 110, 92, 98, 118, 110, 102, 108, 126,
  104, 110, 120, 110, 118, 100, 110, 120, 100, 120,
  92

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Sample Mean Computation

• This value of 111.1 could be interpreted as the
  balancing point of the 30 systolic blood pressure
  observations.
• Locating this in the histogram we have

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Sample Mean in Histogram
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Sample Median

• Sample median (M) value that divides the
  arranged/ordered data set into two equal parts.
• At least 50 are lt M and at least 50 are gt M
• Not sensitive to outliers but harder to deal with
  mathematically
• Appropriate when histogram is left or
  right-skewed
• Better to present both mean and median in practice

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Illustration of Computation of Median

• Consider again the blood pressure data earlier.
• n30 an even number.
• Median will be the average of the 15th and 16th
  observations in arranged data.
• Arranged data 92, 92, 94, 98, 100, 100, 100,
  102, 104, 108, 108, 110, 110, 110, 110, 110, 110,
  110, 110, 118, 118, 120, 120, 120, 122, 124, 126,
  126, 126, 135

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Continued ...

• The sample median is the average of 110 and 110,
  which are the 15th and 16th observations in the
  arranged data.
• The median equals 110.
• Note that it is very close to the sample mean
  value of 111.1
• This closeness is because of the near symmetry of
  the distribution

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Relative Positions of Mean and Median

• For symmetric distributions, the mean and the
  median coincide.
• For right-skewed distributions, the mean tends to
  be larger than the median (mean pulled up by the
  large extreme values)
• For left-skewed distributions, the mean tends to
  be smaller than the median (mean pulled down by
  the small extreme values)