S'A'B' Help Desk

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S.A.B. Help Desk

• City West, Yungondi Building, level 3, room 73
• Mondays 500pm 630pm
• Wednesdays 1230pm 230pm
• Thursdays 200pm 400pm
• Fridays 200pm 400pm
• Mawson Lakes, Maths Help Centre, OC1-60
• SAB tutor on Wednesdays 12noon 200pm

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Week 3 Objectives

• 1. Measures of location and dispersion
• 2. Attributes of distribution shape
• 3. Boxplots
• 4. Assessing skewness and bi-modality
• 5. Weighted averages
• 6. Index numbers

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Measures of location and dispersion
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Measures of location and dispersion revision

• Location mean and median (which is robust
  against outliers?)
• Dispersion standard deviation and IQR (which is
  robust against outliers?)
• What is the definition of the coefficient of
  variation (CV)?
• If data units are in minutes, what are the units
  of
• The median?
• The CV?
• The variance?

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Distribution shapes (a) symmetric or skewed?
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Distribution shapes (b) uni- or bi-modal?
Note for discrete observations, the mode is the
most frequent value. What does modality mean for
histograms?

• Uni-modal
• one peak only, so the mode is at the position of
  the peak, and is the most frequently occurring
  interval
• Bi-modal
• two peaks. Is this caused by genuine bimodality,
  i.e. two component distributions, or is it
  because of random fluctuations?

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Can uni-modality be distinguished from
bi-modality?

• Be cautious
• Multiple peaks may be caused by random
  fluctuations in the data
• Very large sample sizes are needed to accurately
  assess modality
• But bi-modality is of interest, where it
  indicates two underlying components

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This histogram is of 100 values sampled from a
uni-modal symmetric distribution
Is the multi-modality real?
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Lecture exercise 1
Questions of distribution shape

• (i) Skew or symmetric?
• (ii) Modality?

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3. Box-and-whisker plot

• Box-and-whisker plot is constructed based on the
  5-number summary, min, Q1, median, Q3, max.
• Two Box plots for (2.10.1) of the textbook with
  and without the outlier.

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Uses of box-plots

• Signal possible outliers
• Good visual presentation of data set
• Vertically stacked box-plots (on the same scale)
  enable easy comparison of several data sets
• Easy in Minitab Graph gt Boxplot or as option
  under Graphs after Stat gt Basic Statistics gt
  Display Descriptive Statistics

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Indicators of skewness

• Position of the median relative to Q1, Q3 on a
  boxplot, i.e. two unequal halves of the box
• Median closer to Q1 for right skewed
• Median closer to Q3 for left skewed
• Different lengths of the whiskers
• Longer right (left) whisker for right (left)
  skewed
• Different values for mean and median
• Mean gt Median for right-skewed
• Mean lt Median for left-skewed

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Lecture example 1
From the boxplot Median is closer to Q3 (bigger
left half of the box) Left whisker is
longer From the summary measures Mean lt
Median Conclusion Distribution of weight gain is
skewed-to-the-left
Descriptive Statistics Variable N Mean
Median StDev Q1 Q3 rat weight
17 0.055 0.26 0.9478 -0.689 0.609
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Can skewness be assessed from the boxplot and the
summary measures?
Lecture exercise 2
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Can skewness be assessed from the boxplot and the
summary measures?
Lecture exercise 3
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Histogram of the Depth data in Lecture Exercise 3
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Which location measure to use?
It depends on the context.

• For example, if the purpose of gathering the data
  is to find out about an average (say, average
  consumption of a product), then use the mean.
• Otherwise, if a general measure of centrality
  is needed, the mean could be used for nearly
  symmetric data sets without outliers, and the
  median could be used for skew distributions.

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Lecture exercise 4

• Describe the following distribution in terms of
  shape (skewness and modality), location and
  dispersion.
• Which location and dispersion measures should you
  use?

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Solution to Lecture exercise 4
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5. Weighted averages
Every Sunday a social club has a pizza night,
choosing a different pizza supplier each time.
Costs and quantities for 3 weeks are as follows.
What is average unit cost?
Which is the more appropriate, and why?
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A small school clothing business trades Monday to
Friday
Textbook example (3.7.1)
What is its average daily profit over the first
four months?
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Weighted or unweighted average?
Does it make a difference?
Which is the more appropriate?
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Weighted averages general formula
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Index numbers

• A series of index numbers is a sequence of
  figures, which keep track of the relative
  percentage value of some quantity of interest
  (say price)
• Familiar indexes are CPI, All-ords, Dow Jones,
  etc.

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Number of motor vehicles (millions) in the USA
An example of Index number
(Textbook example (3.9.1)
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Classification of Index Numbers
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Fixed and Chain based indexes

• A Fixed base index is calculated as
• current value base value 100
• A Chain-based index is calculated as
• current value previous value 100

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Number of motor vehicles (millions) in the USA
Textbook example (3.9.1)
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Simple and Composite indexes

• Simple index a series of percentages relating
  to a single commodity
• Composite index based upon a collection of
  commodities, called a basket of commodities.
  Examples CPI, all-ordinaries (a composite
  index from ASX stocks)

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How to make a Composite Index?

• Unweighted when weights are unavailable. Based
  on simple averaging of Prices or of Price
  relatives. The types are aggregative and
  relative see the text for details.
• What is the major deficiency of unweighted
  indexes?
• Why are the most commonly used composite indexes
  of weighted type?

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Weighted, composite indexes
Usually weights quantities
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Laspeyres, Paasche formulas
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Find Laspeyeres index for 1998 relative to base
year 1994
Data from Textbook example (3.9.2)
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Solution
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Find Paasches index for 1998 relative to base
year 1994
Data from Textbook example (3.9.2)
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Solution Lecture exercise 5
What interpretation can be given to the fact that
the Paasche index is higher than the Laspeyeres
index?
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Solution to Lecture exercise 5
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Find Laspeyeres and Paasche indexes for 1997
relative to base year 1995
Lecture exercise 6
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Solution to Lecture Exercise 6