Descriptive statistics (Part I)

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Lecture 2

• Descriptive statistics (Part I)

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Lecture 2 Descriptive statistics

• Data in raw form are usually not easy to use for
  decision making
• Some type of organization is needed
• Table
• Graph
• Techniques reviewed here
• Bar charts and pie charts
• Ordered array
• Stem-and-leaf display
• Frequency distributions, histograms
• Cumulative distributions
• Contingency tables

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Tabulating and Graphing Univariate Categorical
Data
Categorical Data
Graphing Data
Tabulating Data
Pie Charts
Summary Table
Bar Charts
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Summary Table(for an Investors Portfolio)
Investment Category Amount Percentage (in
thousands ) Stocks 46.5
42.27 Bonds 32 29.09 CD
15.5 14.09 Savings 16
14.55 Total 110 100
Variables are Categorical
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Bar Chart(for an Investors Portfolio)
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Pie Chart (for an Investors Portfolio)
Amount Invested in K
Savings 15
Stocks 42
CD 14
Percentages are rounded to the nearest percent
Bonds 29
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Organizing Numerical Data
Numerical Data
41, 24, 32, 26, 27, 27, 30, 24, 38, 21
Frequency Distributions Cumulative
Distributions
Ordered Array
Stem and Leaf Display
2 144677 3 028 4 1
21, 24, 24, 26, 27, 27, 30, 32, 38, 41
Histograms
Tables
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The Ordered Array

• Data in raw form (as collected)
• 24, 26, 24, 21, 27, 27, 30, 41, 32, 38
• Data in ordered array from smallest to largest
  21, 24, 24, 26, 27, 27, 30, 32, 38, 41
• Shows range (min to max)
• May help identify outliers (unusual
  observations)
• If the data set is large, the ordered array is
  less useful

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Stem-and-Leaf Display

• A simple way to see distribution details in a
  data set
• METHOD Separate the sorted data series
  into leading digits (the stem) and
  the trailing digits (the leaves)

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Example

• Data in Raw Form (as Collected) 24, 26, 24,
  21, 27, 27, 30, 41, 32, 38
• Data in Ordered Array from Smallest to Largest
  21, 24, 24, 26, 27, 27, 30, 32, 38, 41
• Stem-and-Leaf Display

2 1 4 4 6 7 7
3 0 2 8
4 1
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Tabulating Numerical Data Frequency Distributions

• What is a Frequency Distribution?
• A frequency distribution is a list or a table
• containing class groupings (ranges within which
  the data fall) ...
• and the corresponding frequencies with which data
  fall within each grouping or category
• It allows for a quick visual interpretation of
  the data

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Tabulating Numerical Data Frequency Distributions

• Example A manufacturer of insulation randomly
  selects 20 winter days and records the daily high
  temperature
• 24, 35, 17, 21, 24, 37, 26, 46, 58, 30,
• 32, 13, 12, 38, 41, 43, 44, 27, 53, 27

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• Sort Raw Data on days in Ascending Order12, 13,
  17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38,
  41, 43, 44, 46, 53, 58
• Find Range 58 - 12 46
• Select Number of Classes 5 (usually between 5
  and 15)
• Compute Class Interval (Width) 10 (46/5 then
  round up)
• Determine Class Boundaries (Limits)10, 20, 30,
  40, 50, 60
• Count Observations Assign to Classes

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Frequency Distributions, Relative Frequency
Distributions and Percentage Distributions
Data in Ordered Array 12, 13, 17, 21, 24, 24,
26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46,
53, 58
Relative Frequency
Percentage
Class Frequency
10, 20) 3
.15 15 20, 30) 6
.30 30 30,
40) 5 .25
25 40, 50)
4 .20
20 50, 60) 2 .10
10 Total 20
1 100
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Graphing Numerical Data The Histogram

• A graph of the data in a frequency distribution
  is called a histogram
• The class boundaries (or class midpoints) are
  shown on the horizontal axis
• the vertical axis is either frequency, relative
  frequency, or percentage
• Bars of the appropriate heights are used to
  represent the number of observations within each
  class

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Histogram Example
Class Midpoint
Class
Frequency
10, 20) 15
3 20, 30) 25
6 30, 40) 35
5 40, 50) 45
4 50, 60) 55
2
(No gaps between bars)
Class Midpoints
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Tabulating Numerical Data Cumulative Frequency
Data in Ordered Array 12, 13, 17, 21, 24, 24,
26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46,
53, 58
Upper Cumulative Cumulative Limit
Frequency Frequency 10
0
0 20 3
15 30 9
45 40 14
70 50
18 90 60
20 100
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Two categorical variables (contingency table)

• The following data represent the responses to a
  question asked in a survey of 20 college students
  majoring in business
• What is your gender? (Male M Female F)
• What is your major? (Accountancy A
  Information System I Market M)
• Gender M M M F M F F M F M F M M M M F F M
  F F
• Major A I I M A I A A I I A A A M I
  M A A A I

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Contingency table (contd)

• Raw data set
• Gender M M M F M F F M F M F M M M M
  F F M F F
• Major A I I M A I A A I I A A
  A M I M A A A I

A I M Total
Male 6 4 1 11
Female 4 3 2 9
Total 10 7 3 20
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Graphical methods are

• Good in presenting data
• Not easy for comparison
• Difficult to use for statistical inference

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Numerical description
Summary Measures
Variation
Central Tendency (location measures)
Quartiles
Range
Mean
Median
Mode
Variance
Interquartile range
Standard Deviation
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Mean

• Mean (Arithmetic Mean) of Data Values
• Sample mean
• Population mean

Sample Size
Population Size
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An example

• TV watching hours/week 5, 7, 3, 38, 7
• Mean (5 7 3 38 7)/5 60/5 12
• If the correct time for 4th subject is 8 (not 38)
• Mean (5 7 3 8 7)/5 30/5 6

3 5 6 7 8
3 5 7 12
38
Mean 6
Mean 12
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Mean (Contd)

• The Most Common Measure of Central Tendency
  especially when n is large due to its good
  theoretical properties
• Affected by Extreme Values (Outliers)

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Median

• Robust measure of central tendency
• Not affected by extreme values
• In an ordered array, the median is the middle
  number
• If n is odd, the median is the middle number
  (i.e,(n1)/2 th measurement)
• If n is even, the median is the average of the
  n/2 th and (n/2 1) th measurement

3 5 7 8
3 5 7
38
Median 7
Median 7
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Mode

• A Measure of Central Tendency
• Value that Occurs Most Often
• Not Affected by Extreme Values
• There May Not Be a Mode
• There May Be Several Modes
• Used for Either Numerical or Categorical Data

0 1 2 3 4 5 6
0 1 2 3 4 5 6 7 8 9 10 11
12 13 14
No Mode
Mode 9
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Quartiles

• Split ordered data into 4 quarters
• Position of i-th quartile
• (1st quartile) and (3rd quartile)
  are measures of Noncentral Location
• are called 25th, 50th, and
  75th percentile respectively. A pth percentile
  is the value of X such that p of the
  measurements are less than X and (100-p) are
  greater than X.

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25
25
25
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Quartiles (example)
Data in Ordered Array 3 6 6 12 12 12 15
15 18 21

• Position of first quartile is
• Position of third quartile is

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5-number summary

• Box-and-Whisker Plot
• Graphical display of data using 5-numbers

Data in Ordered Array 3 6 6 12 12 12 15
15 18 21
Median( )
X
X
largest
smallest
21
6
3
12
15.75