Chapter 6. Descriptive Statistics

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Chapter 6. Descriptive Statistics

• 6.1 Experimentation
• 6.2 Data Presentation
• 6.3 Sample Statistics
• 6.4 Examples

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• Data a mixture of nature and noise.
• Is the noise manageable?
• The noise is desired to be represented by a
  probability distribution.
• Statistical inference
• The science of deducing properties of an
  underlying probability distribution from data
• Can we have information on the underlying
  probability distribution?
• The information is given in the form of
  (functions of) data.

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Figure 6.1 The relationship between probability
theory and statistical inference
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6.1 Experimentation6.1.1 Samples

• Population the set of all the possible
  observations available from a particular
  probability distribution.
• Sample a subset of a population.
• Random sample a sample where the elements are
  chosen at random from the population
• A sample is desired to be representative of the
  population.
• Types of observations numerical and categorical
  (nominal)
• - numerical observation either integers or
  real numbers
• - categorical observation a machine breakdown
  is classified as either mechanical, electrical or
  misuse.

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6.1.2 Examples

• Example 1 Machine breakdowns
• Suppose that an engineer in charge of the
  maintenance of a machine keeps records on the
  breakdown causes over a period of a year.
• Suppose that 46 breakdowns were observed by the
  engineer (see Figure 6.2).
• What is the population from which this sample is
  drawn?
• Factors to consider to check the representative
  of data
• Quality of operators
• Working load on the machine
• Particularity of data observation (e.g., more
  rainy days than other years)

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Figure 6.2 Data set of machine breakdowns
How representative is this years data set of
future years?
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Figure 6.4Data set of defective computer chips

• Example 2 Defective computer chips
• The chip boxes are selected at random.

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• Points to check on data
• What is the data type?
• Are the data representative?
• How is the randomness of the data realized?
• Statistical problem
• What is the population from which the data are
  sampled?

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6.2 Data presentation

• 6.2.1 Bar and Pareto charts
• 6.2.2 Pie charts
• 6.2.3 Histograms
• 6.2.4 Outliers

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Figure 6.7 Bar chart of machine breakdowns data
set

• Bar chart uses bars to represent the data.
• The length of a bar is proportional to the
  frequency

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Figure 6.9 Pareto chart of customer complaints
for Internet company

• Pareto chart is a bar chart where the categories
• are arranged in order of decreasing frequency.

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Figure 6.12Pie chart for machine breakdowns data
set

• Pie Charts emphasize the proportion of each
  category.

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Figure 6.14Histogram of computer chips data set

• Histograms are used to represent numerical data.

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Figure 6.16 Histograms of metal cylinder
diameter data set with different bandwidths
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Figure 6.18 A histogram with positive (or right)
skewness
The right-hand tail is longer and flatter than
the left-hand tail
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Figure 6.19 A histogram with negative (or left)
skewness
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Figure 6.20 A histogram for a bimodal
distribution
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Figure 6.21 Histogram of a data set with a
possible outlier
An outlier is an observation which is not from
the distribution from which the main body of the
sample is collected.
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6.3 Sample statistics

• 6.3.1 Sample mean of a data set
  
• 6.3.2 Sample median
• the value of the middle of the ordered data
  points
• ex) if n 2k1 (odd), the sample median
• if n 2k (even), the sample median
• 6.3.3 Sample trimmed mean
• A trimmed mean is obtained by deleting some of
  the largest and some of the smallest data
  observations.
• Usually a 10 trimmed mean is employed where the
  top 10 of the data points are removed together
  with the bottom 10 of the data points.

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Figure 6.22 Illustrative data set
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Figure 6.23Relationship between the samplemean,
median, and trimmed meanfor positively and
negativelyskewed data sets
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• 6.3.4 Sample mode
• For categorical or discrete data, the sample
  mode may be used to denote the category or data
  value that contains the largest number of data
  observations.
• 6.3.5 Sample variance (s2)

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• 6.3.6 The 100p Sample Percentile (pth Sample
  Quantile) and sample quartile
• The 100p sample percentile is the value such
  that at least 100p percent of the data are less
  than or equal to it and at least 100(1-p) percent
  are greater than or equal to it. If there are two
  values satisfying the condition, the 100p sample
  percentile is the arithmetic average of these two
  values.
• a data set 2,5,6,7,8
• - p 0.1, i.e., 10 percentile sample
  percentile is 2
• - p 0.2, i.e., 20 percentile sample
  percentile is (25)/2 3.5
• The 25 sample percentile the first quartile
• The 50 sample percentile the second quartile
  or the sample median
• The 75 sample percentile the third quartile

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• 6.3.7 Boxplots

Figure 6.24 Boxplot of a data set
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• 6..8. Coefficient of variation (CV)
• the spread of the data relative to the middle
  value

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• Recall the Chebyshevs inequality
• Let
• Then,
• In general,
• Theorem the weak law of large numbers
• Let be a sequence of i.i.d.
  random variables, each having mean and
  variance
• Then, for any

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• (proof)

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• Homework 6
• Read Chapter 6.
• Review Chapter 1 Chapter 5.
• Midterm Exam
• Date 10.24 (Wed)
• Time 900-1100