Chapter 3 Descriptive Statistics

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Chapter 3 - Descriptive Statistics

• Given precise enough measurement, even supposedly
  constant process conditions produce differing
  responses.
• For this reason we are not as interested in
  individual data values as we are in the pattern
  or distribution of the data as a whole.
• Well use graphs to display the distribution and
  statistics to describe the distribution.

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3.1 Graphing Quantitative Data

• Dot diagram
• Each observation is a dot placed at a position
  corresponding to its numerical value.
• Stem-and-leaf plot
• Made by using the last few digits of each data
  point to indicate where it falls
• Both displays require that each data point covers
  the same amount of space.

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Example 3.1

• The government requires manufacturers to monitor
  the amount of radiation emitted through the
  closed door of a microwave. The following are
  radiation amounts emitted by 24 microwaves
  measured by one manufacturer.

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Example 3.1

• Its easiest to first order your data.

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Dot Diagram
.00
.30
.20
.10
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Stem-and-Leaf Plot

• Use the first digit after the decimal place as
  the stem and the second as the leaf

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Example 3.2

• Other examples of splitting data into stem and
  leaf

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Stem-and-leaf Plot Vocabulary

• Back-to-back
• Example Mens vs. Womens heights
• Recorded in inches
• Men Women 630 0064000
  65000000006600

• Split Stems
• There are multiple rows for each stem.
• Bins are divided at between digits.
• Womens heights639641647865011655

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Graphing Quantitative Data

• Frequency tables
• Break data into intervals of equal length and
  then tally the data points in each interval
• The number of intervals we use varies
• Every data point is included in exactly one
  interval.
• Histograms
• Break data into intervals of equal length and
  then create a connected bar chart
• Begin vertical axis at zero
• Draw bars with equal width

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Frequency Table

• Data from example 3.1

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Histogram

• Data from example 3.1

6
4
2
.03
.21
.15
.09
.27
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Histogram (different intervals)
8
6
4
2
.05
.25
.15
13
BAD Histogram
8
6
4
2
.05
.25
.15
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Distributional Shapes
Bell-shaped
Right-skewed
Left-skewed
Bimodal (Multimodal)
Uniform (Unimodal)
Truncated
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Bivariate Quantitative Data

• Scatterplot
• Plot each data point as a dot where its response
  variable and supervised variable
• Look for patterns
• Run Chart
• Plot data points in order determined by the time
  of observation
• Look for patterns

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Example 3.3

• Scatterplot for ACT score and Highschool GPA for
  12 students

The scatterplot of GPA against ACT score shows a
fairly strong, positive, linear relationship.
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Example 3.4
strong, positive, linear relationship
strong, negative, linear relationship
strong curved relationship
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Example 3.4 (continued)
weak, positive, linear relationship
weak, negative, linear relationship
no relationship
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Example 3.5

• Run Chart - Suppose that we plot the number of
  typing errors made per minute by a typist against
  time.

Pattern steady increase until we hit minute 10,
then there is a sharp drop followed by another
steady increase. It was discovered that the
typist was given a five minute break after minute
10.
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3.2 Quantiles

• The p quantile, denoted as Q(p), is the number
  such that p is the percentage of the distribution
  that lies to the left of Q(p), and 1-p is the
  percentage of the distribution that lies to the
  right of Q(p).

1/2
Relative Frequency
Q(1/3) 2/3
2/3 4/3 2
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Quantiles (Book)

• For empirical distributions, the above definition
  translates to
• For an ordered data set x1 x2 xn
• 1. For i 1,2,,n the p quantile of
  the data set is
• the ith smallest data point, xi. That is

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Quantiles (Book)

• 2. For any p not equal to for some integer i
  n such
• that , the p quantile is obtained by
  linear
• interpolation between the values of
• with corresponding that bracket p

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Finding Quantiles

• General procedure for finding the p quantile of
  an empirical distribution
• Order data values x(1) x(2) x(n)
• Set i np0.5
• If i 1, 2, , n then
• otherwise,

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Example 3.5

• Ten batteries were tested to determine how long
  the batteries would last (hrs) under normal
  conditions. Below are the ten values that were
  obtained
• 100, 120, 80, 90, 95, 115, 120, 110, 105, 95
• Give values for Q(.35) and Q(.42)
• Give the values for Q(.68) and Q(.90)
• Give the values for Q(.25), Q(.50) and Q(.75)
• Step1 Order the data
• 80,90,95,95,100,105,110,115,120,120

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Example 3.5a

• Give the values for Q(.35) and Q(.42)

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Example 3.5b

• Give the values for Q(0.68) and Q(0.90)

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Example 3.5c

• Give the values for Q(0.25),Q(.50) and Q(0.75)

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Quantile Terminology

• Special quantiles
• Q(.25) Q1, 1st quartile, lower quartile
• Q(.5) Q2, 2nd quartile, median
• Q(.75) Q3, 3rd quartile, upper quartile
• Special values associated with quartiles
• Inter-quartile range (IQR) Q3 Q1
• Upper fence Q3 1.5IQR
• Lower fence Q1 1.5IQR

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Boxplots

• Another tool used to illustrate the distribution
• Steps for making a boxplot
• Order the observed data values
• Find Q1, Q2, Q3, IQR, UF and LF
• Draw a box that spans the IQR
• Divide the box at the median (Q2)
• Draw asterisks (or dots) for any data values less
  than the lower fence and any values greater than
  the upper fence
• Draw a line from the sides of the box to the
  smallest value greater than the LF and the
  largest value smaller than the UF

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Example 3.6

• Draw a boxplot based on the 15 ordered values
  below
• 75, 80, 80, 85, 90, 95, 95, 100, 105, 110, 110,
  115, 120, 120, 125
• Find the necessary values

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Example 3.6

• Q186.25, Q2100, Q3113.75, so
• IQR 113.75 86.25 41.25
• UF Q(.75)1.5(IQR) 155
• LF Q(.25) 1.5(IQR) 45
• Identify all values outside the upper and lower
  fences none

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Example 3.6
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Q-Q Plots

• Q-Q plot Quantile-Quantile Plot
• Used for two data sets
• We plot Q(p) for data set 1 (denoted Q1(p))
  versus Q(p) for data set 2
• We will only deal with two data sets of the same
  size
• Straight line indicates that the two data sets
  have the same distributional shape.

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Example 3.7

• Data Set 1 1, 2, 3, 4, 5
• Data Set 2 6, 7, 8, 9, 10

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Example 3.8

• Data set 1 1, 5, 7, 8, 9, 10
• Data set 2 -10, -9, -8, -7, -5, -1

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Normal Probability Plot

• A normal probability plot is a type of Q-Q plot
  that allows us to determine if the distribution
  of our data is bell-shaped (normal).
• A straight line is indication that our data is
  normal/bell-shaped.
• An S-shaped line indicates that our data is
  skewed.

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Table 3.10

• Table 3.10 (page 89) in the book gives some
  quantiles for a distribution that is known to be
  bell-shaped.
• The numbers in the body of the table give Q(p)
  for p given by the margins of the table.
• Q(.23) -.74
• Row .2 and Column .03 give the p0.23 quantile as
  -.74
• Q(.79) .81
• Row .7 and Column .09 give the p0.79 quantile as
  .81

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Example 3.9

• Annual incomes (in thousands of dollars) for 8
  families (in a common geographical location) are
  given below 23, 31, 43, 47, 51, 58, 67, 83
• Does this data appear to be from a bell-shaped
  distribution?
• Remember p

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Example 3.9
Bell-shaped distribution?
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3.3 Numerical Measures

• Measures of location
• Median
• Same as Q(.5)
• Not affected by skew or outliers (extreme
  observations)
• Sample Mean
• For data , the mean is
  given as
• Strongly affected by skew or outliers

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Example 3.10

• Data set 1 2, 3, 5, 8, 12
• Median 5
• Mean (235812)/5 6
• Data set 2 2, 3, 5, 8, 102
• Median 5
• Mean 24

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Measures of Spread

• IQR
• Measures the spread of the middle half of the
  data
• Not sensitive to skew or outliers
• Range
• Highly sensitive to outliers

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Measures of Spread

• Sample Variance
• How much the data is spread from the sample mean,
  .
• Sensitive to outliers or skew
• Sample Standard Deviation
• Sensitive to outliers or skew

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Example 3.11

• Same data as example 3.9 23, 31, 43, 47,
  51, 58, 67, 83
• Find the range
• Find the mean
• Find the variance
• Find the standard deviation

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Understanding Standard Deviation

• Chebyschevs Theorem
• For any data set and any number k larger than 1,
  a fraction of at least of the
  data are within ks of .
• 3/4 of the data will be within 2 standard
  deviations of the mean, 8/9 of the data will be
  within 3 standard deviations of the mean, etc.
• Standard deviation acts as a ruler

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Statistics vs. Parameters

• Numerical summaries of sample data are called
  statistics.
• Numerical summaries of population data are called
  parameters.
• Often represented by Greek letters

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Plots of Summary Statistics

• Example 9 from the book
• Three different glues are tested with three
  different types of wood (3x3 factorial study) and
  the mean strength is calculated (based on 3
  observations) for each combination.

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Example 3.12

• A plot of the means categorized by glue and wood
  shows that pine and fir have similar gluing
  properties with pine being stronger. The gluing
  properties of Oak are much different (opposite
  trend).

250
oak
200
pine
150
fir
100
white
cascamite
carpenter
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3.4 Statistics for Qualitative Data

• The fraction of items in the sample with a
  particular characteristic is
• The sample mean occurrences per unit of item is
• is closer in meaning to than to .

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Example 3.13

• A random sample of students from ISU is taken in
  which there ends up being 210 freshmen, 171
  sophomores, 182 juniors, and 115 seniors.
• Find for each of the classifications.

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Example 3.14

• When studying the number of towns reporting power
  outages caused by thunderstorms, it is found that
  there were 8 storms in which no outages were
  reported, 2 storms in which 3 outages were
  reported, and 5 storms in which 1 outage was
  reported. Find .

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Plotting Qualitative Data

• Bar Charts same as histograms, but without
  intervals
• Segmented Bar Charts each bar is a divided
  between different levels of an additional
  variable
• Run Chart taken on categorical times
• Example daily
• Read more in section 3.4.2