Lecture 4
DESCRIPTIVE STATISTICS:
Numerical summaries

BUSINESS STATISTICS
Advanced Educational Program

Reading materials:
Chap 4 (Keller)

1

Outline

2

Measure of center and spread

¥  Measures of center:
-  Mean, median, mode
-  Selection of measures of location

¥  Measures of dispersion (spread):
-  Range, quartile range, quartile deviation,
variance, standard deviation

¥  Empirical rule (general case: ChebyshevÕs
law)
¥  Coefficient of variation
¥  Coefficient of skewness
3

Measures of center

4

Measures of center
¥  A measure of center or location shows
where the center of the data is
Ơ Three most useful measures of location:
Đ Arithmetic mean/average
§  Median
§  Mode

5

6


Arithmetic mean from frequency table

Arithmetic mean from raw data
N

¥  Arithmetic mean from population:

X
à=

i


Ơ Apply this formula for the sample:

i =1

N

k

x f

n

¥  Arithmetic mean from sample:

i i

∑x

i

x=

x=

i =1

i =1
k

∑f


n

i

i =1

Where:

Xi, xi - the value of each item
N, n - total number of items

Where: xi - the value of class i
fi Ð frequency of class i

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8

Mean is sensitive to outliers

Advantages and disadvantages of arithmetic mean
¥  Advantages:
Ð  Easy to understand and calculate
Ð  Values of every items are included => representative for
the whole set of data

¥  Disadvantages
Ð  Sensitive to outliers:
Sample: (43; 38; 37; : : : ; 27; 34): =>

x = 33.5
Contaminated sample
(43; 38; 37; : : : ; 27; 1934): =>
x = 71.5

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10

Median

Calculate median from raw data
¥ 

—  Median is the value of the observation which is
located in the middle of the data set
— 

If the data has an odd number of observations:
(n + 1)th
Ð  Middle observation:
2

Steps to find median:

Median = x ( n +1)th

1.  Arrange the observations in order of size (normally
ascending order)


2

¥ 

2.  Find the number of observations and hence the middle
observation

If the data has an even number of observations:
Ð 

There are two observations located in the middle and

Median = ( x

3.  The median is the value of the middle observation

th

⎛n⎞
⎜ ⎟
⎝2⎠

11

+x

⎛n ⎞
⎜ +1⎟
⎝2 ⎠


th

)/2

12


Example

Advantages and disadvantages of median
¥  Advantages:

¥ 

E.g1. Raw data: 11, 11, 13, 14, 17 => find median

¥ 

E.g 2. Raw data: 11, 11, 13, 14, 16, 17 => find
median

Ð  Easy to understand and calculate
Ð  Not affected by outlying values => thus can be used
when the mean would be misleading

¥  Disadvantages
Ð  Value of one observation => fails to reflect the whole
data set
Ð  Not easy to use in other analysis
13


14

Mode
Example to calculate mode
¥ 
¥ 

Mode is the value which occurs most
frequently in the data set
Steps to find mode
1.  Draw a frequency table for the data
2.  Identify the mode as the most frequent value

15

Frequency

8

3

12

7

16

12


17

8

19

5

16

Mean, median and mode in normal and skewed
distributions

Bimodal and multimodal data

Bimodal (two modes)

X

Multimodal (several modes)
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18


Which measure of centre is best?

Measures of dispersion (variability)

¥  Mean generally most commonly used

¥  Sensitive to extreme values
¥  If data skewed/extreme values present, median better, e.g.
real estate prices
¥  Mode generally best for categorical data Ð e.g. restaurant
service quality (below): mode is very good. (ordinal)
Rating

# customers

Excellent

20

Very good

50

Good

30

Satisfactory

12

Poor

10

Very Poor


6

¥ 

Measures of dispersion tell you how spread
out all other values of the distribution from
the central tendency
Measures of dispersion

¥ 
¥ 

The range, quartile range, and quartile deviation

¥ 

Variance and standard deviation

19

Why do we need measures of dispersion?

20

Why measures of dispersion? (1)

¥  Two data sets of midterm marks of 5 students:
Ð  First set: 100, 40, 40, 35, 35 => Mean: 50
Ð  Second set: 70, 55, 50, 40, 35 => Mean: 50

Ø Which mean (first or second) is more reliable?

¥  Need to know the spread of other values around the
central tendency, especially important in analysing
stock market.

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Range

Why measures of dispersion? (2)

¥  Range is the difference between the largest and
smallest value => Sort data before computing range
¥  Formula: Range = maximum value - minimum
value
¥  Advantages of Range: easy to calculate for
ungrouped data.
¥  Disadvantages:
Ð  Take into account only two values
Ð  Affected by one or two extreme values
Ð  More difficult to calculate for grouped data
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Quartiles


Quartile range and quartile deviation

¥  Quartiles: are defined as values of observations
which are a quarter of the way through data

¥  Quartile range = Q3 Ð Q1

Ð  Q1 - the first quartile: the value of the
observation of which 25% of observations fall
below

¥  Quartile deviation =

Ð  Q2 - the second quartile: the median (50% of the
observations fall below)

¥  Advantages of quartile deviation (semi-interquartile range):
less affected by extreme value

Q3 − Q1
2

¥  Disadvantages: take into account only 50% of the data

Ð  Q3 - the third quartile: the value of the
observation of which 75% of observations fall
below
25


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Variance
¥  Variance from population:
¥  Variance from sample

Standard deviation (σ )

2 =

s2 =

( X i à )2

Ơ Standard deviation (S.D) is the square root of variance
¥  S.D from population:

N

∑ ( x − x)

2

σ = σ2

n −1
¥  S.D from sample:

¥  Advantages:
¥  Take into account all values

¥  Easy to interpret the result.

s = s2

¥  Advantages:
¥  Overcome the disadvantage of meaningless unit of
variance
¥  The most widely used measure of dispersion (the bigger
its value => the more spread out are the data)

¥  Disadvantages: the unit of variance has no meaning

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Application of this in finance
¥  Variance (or S.D) of an investment, can be used
as a measure of risk e.g. on profits/return.
¥  Larger variance è larger risk
¥  Usually, higher rate of return, higher risk

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Example Ð 2 funds over 10 years (1)
¥  Rates of return
A

8.3 -6.2 20.9 -2.7 33.6 42.9 24.4 5.2

3.1 30.5


B 12.1 -2.8 6.4 12.2 27.8 25.3 18.2 10.7 -1.3 11.4

x A = 16%

xB = 12%

s A2 = 280.34(%) 2

s A2 = 99.37(%) 2

¥  Which fund will you invest?


Empirical rules or the law of 3 σ

Example Ð 2 funds over 10 years (2)

¥  For a normal or symmetrical distribution:
l 

Ð  68.26% of all obs fall within 1 standard deviation of the
mean, i.e. in the range:

Depending on how Risk-averse you are:
Fund A: higher risk, but also higher average rate
of return.

( x − 1s) ↔ ( x + 1s)
Ð  95.45% of all obs fall within 2 standard deviation of the
mean, i.e. in the range:


( x − 2s) ↔ ( x + 2s)
Ð  99.73% of all obs fall within 3 standard deviation of the
mean, i.e. in the range:

( x − 3s ) ↔ ( x + 3s )
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Meaning of the law of 3σ

Boxplot

¥  Convert z-score to probability (next lecture)
Here is the Boxplot of height of international students
studying at UNSW

¥  Identify outliers

Boxplot of Height
200

whisker
190

upper quartile

Height

180


170

median

box

lower quartile

160

whisker
150

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Boxplots

Shapes of Boxplots

¥  Need MEDIAN and QUARTILES to create a boxplot
¥  MEDIAN = middle of observations, i.e. ! way through
observations
¥  QUARTILES = mark quarter points of observations, i.e. "
(Q1) and # (Q3) of the way through data [(n+1)/4; 3(n+1)/
4]
¥  INTERQUARTILE RANGE = Q3-Q1
¥  Whiskers: max length is 1.5*IQR; stretch from box to
furthest data point (within this range)

¥  Points further out from box marked with stars; called
outliers

Boxplot of Symmetric, Positive skew, Negative skew, Bimodal
5.0

¥  Skewness/
symmetry
¥  Modality
¥  Range

Data

2.5

0.0

-2.5

-5.0
Symmetric

35

Positive skew

Negative skew

Bimodal


36


Coefficient of skewness (C of S)

Activity 1
¥  Summary statistics of two data sets are as follows

¥  This measures the shape of distribution
¥  There are some measures of skewness.
¥  Below is a common one: PearsonÕs coefficient of skewness.
Coefficient of skewness = 3 x (mean-median)/standard
deviation
¥  If C of S is nearly +1 or -1, the distribution is highly skewed
¥  If C of S is positive => distribution is skewed to the right
(positive skew)

n 

¥  If C of S is negative => distribution is skewed to the left
(negative skew)

Set 1:
Ages of students
studying at UNSW

Set 2:
Wages of staffs
294.3


Mean

22.4839

Median

21

292.5

Standard deviation

6.3756

125.93

Compute the PearsonÕs coefficient of skewness of these data
sets and describe their shapes of distribution

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Investigating the relationship between variables

Distribution shapes

¥  Methods:

6


Frequency

4

100

o Multiple bar chart
o Scatterplot (mentioned in lecture 8)

2

50

0

0

Frequency

150

8

200

10

Ð  Table: Cross-table
Ð  Charts:


20

40

60
age

Skewed to the right

80

100

200

300

400

500

600

wages

Nearly normal
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Cross-table


Cross-table

¥  Cross-table is used to investigate the relationship
b/w two categorical vars or discrete variables with
few values.

¥  EX: use gss.sav data file to explore the
relationship b/w internet use and degree

¥  Note:
Ð  Need to identify dependent and independent variables.
Ð  Know how to calculate row and column percentages
Ð  Rule of thumb: independent var in row and dependent
var in column

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Multiple bar chart

Multiple bar chat
Here you are

¥  We can use multiple bar chart to explore the
relationship b/w variables.
¥  The skill is to know how to draw chart
¥  EX: use gss.sav data file to explore the

relationship b/w internet use, age, and degree

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