Descriptive Statistics

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Descriptive Statistics
Spatial Statistics (SGG 2413)

• Assoc. Prof. Dr. Abdul Hamid b. Hj. Mar Iman
• Director
• Centre for Real Estate Studies
• Faculty of Engineering and Geoinformation Science
• Universiti Tekbnologi Malaysia
• Skudai, Johor

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Learning Objectives

• Overall To give students a basic understanding
  of descriptive statistics
• Specific Students will be able to
• understand the basic concept of descriptive
  
• statistics
• understand the concept of distribution
• can calculate measures of central tendency
• dispersion
• can calculate measures of kurtosis and
  skewness

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Contents

• What is descriptive statistics
• Central tendency, dispersion, kurtosis, skewness
• Distribution

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Descriptive Statistics

• Use sample information to explain/make
  abstraction of population phenomena.
• Common phenomena
• Association (e.g. s1,2.3 0.75)
• Tendency (left-skew, right-skew)
• Trend, pattern, location, dispersion, range
• Causal relationship (e.g. if X then Y)
• Emphasis on meaningful characterisation of data
  (e.g. central tendency, variability), graphics,
  and description
• Use non-parametric analysis (e.g. ?2, t-test,
  2-way anova)

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E.g. of Abstraction of phenomena
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Inferential Statistics

• Using sample statistics to infer some phenomena
  of population parameters
• Common phenomena cause-and-effect
• One-way r/ship
• Feedback r/ship
• Recursive
• Use parametric analysis (e.g. a and ?) through
  regression analysis
• Emphasis on hypothesis testing

Y f(X)
Y1 f(Y2, X, e1) Y2 f(Y1, Z, e2)
Y1 f(X, e1) Y2 f(Y1, Z, e2)
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Parametric statistics

• Statistical analysis that attempts to explain the
  population parameter using a sample
• E.g. of statistical parameters mean, variance,
  std. dev., R2, t-value, F-ratio, ?xy, etc.
• It assumes that the distributions of the
  variables being assessed belong to known
  parameterised families of probability
  distributions

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Examples of parametric relationship
Dep9t 215.8
Dep7t 192.6
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Non-parametric statistics

• First used by Wolfowitz (1942)
• Statistical analysis that attempts to explain the
  population parameter using a sample without
  making assumption about the frequency
  distribution of the assessed variable
• In other words, the variable being assessed is
  distribution-free
• E.g. of non-parametric statistics histogram,
  stochastic kernel, non-parametric regression

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Descriptive Inferential Statistics (DS IS)

• DS gather information about a population
  characteristic (e.g. income) and describe it with
  a parameter of interest (e.g. mean)
• IS uses the parameter to test a hypothesis
  pertaining to that characteristic. E.g.
• Ho mean income RM 4,000
• H1 mean income lt RM 4,000)
• The result for hypothesis testing is used to make
  inference about the characteristic of interest
  (e.g. Malaysian ? upper middle income)

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Sample Statistics Central Tendency
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Central Tendency Mean

• For individual observations, . E.g.
• X 3,5,7,7,8,8,8,9,9,10,10,12
• 96 n 12
• Thus, 96/12 8
• The above observations can be organised into a
  frequency table and mean calculated on the basis
  of frequencies
• 
  96 12
• Thus, 96/12 8

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Central Tendency - Mean and Mid-point

• Let say we have data like this

Price (RM 000/unit) of Shop Houses in Skudai
Can you calculate the mean?
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Central Tendency - Mean and Mid-point (contd.)

• Lets calculate
• Town A (228450)/2 339
• Town B (320430)/2 375
• Are these figures means?

M ½(Min Max)
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Central Tendency - Mean and Mid-point (contd.)

• Lets say we have price data as follows
• Town A 228, 295, 310, 420, 450
• Town B 320, 295, 310, 400, 430
• Calculate the means?
• Town A
• Town B
• Are the results same as previously?
• ? Be careful about mean and mid-point!

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Central Tendency Mean of Grouped Data

• House rental or prices in the PMR are frequently
  tabulated as a range of values. E.g.
• What is the mean rental across the areas?
• 23 3317.5
• Thus, 3317.5/23 144.24

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Central Tendency Median

• Let say house rentals in a particular town are
  tabulated
• Calculation of median rental needs a graphical
  aids?

• Median (n1)/2 (251)/2 13th. Taman
• 2. (i.e. between 10 15 points on the vertical
  axis of ogive).
• 3. Corresponds to RM 140-145/month on the
  horizontal axis
• 4. There are (17-8) 9 Taman in the range of RM
  140-145/month

5. Taman 13th. is 5th. out of the 9
Taman 6. The rental interval width is 5 7.
Therefore, the median rental can be
calculated as 140 (5/9 x 5) RM 142.8
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Central Tendency Median (contd.)
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Central Tendency Quartiles (contd.)
Following the same process as in calculating
median
Upper quartile ¾(n1) 19.5th. Taman UQ 145
(3/7 x 5) RM 147.1/month Lower quartile
(n1)/4 26/4 6.5 th. Taman LQ 135 (3.5/5
x 5) RM138.5/month Inter-quartile UQ LQ
147.1 138.5 8.6th. Taman IQ 138.5 (4/5 x
5) RM 142.5/month
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Variability

• Indicates dispersion, spread, variation,
  deviation
• For single population or sample data
• where s2 and s2 population and sample
  variance respectively, xi individual
  observations, µ population mean, sample
  mean, and n total number of individual
  observations.
• The square roots are
• standard deviation standard deviation

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Variability (contd.)

• Why measure of dispersion important?
• Consider yields of two plant species
• Plant A (ton) 1.8, 1.9, 2.0, 2.1, 3.6
• Plant B (ton) 1.0, 1.5, 2.0, 3.0, 3.9
• Mean A mean B 2.28
• But, different variability!
• Var(A) 0.557, Var(B) 1.367
• Would you choose to grow plant A or B?

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Variability (contd.)

• Coefficient of variation CV std. deviation as
  of the mean
• A better measure compared to std. dev. in case
• where samples have different means. E.g.
• Plant X (ton/ha) 1.2, 1.4, 2.6, 2.7, 3.9
• Plant Y (ton/ha) 1.4, 1.5, 2.1, 3.2, 3.9

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Variability (cont.)
Calculate CV for both species.
CVx (1.2/2.36) x 100
50.97 CVy (1.2/2.42) x 100 49.46
? Species X is a little more variable than
species Y
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Variability (cont.)

• Std. dev. of a frequency distribution
• E.g. age distribution of second-home buyers
  (SHB)

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Probability distribution

• If there 20
  lecturers, the probability that A becomes a
  professor is p 1/20 0.05
• Out of 100
  births, half of them were girls (p0.5), as the
  number increased to 1,000, two-third were girls
  (p0.67) but from a record of 10,000 new-born
  babies, three-quarter were girls (p0.75)
• The
  probability of a drug addict recovering from
  addiction is 5050
• General rule
• No. of times event X
  occurs
• Pr (event X) --------------------------------
  -----
• Total number of
  occurrences
• Probability of certain event X to occur has a
  specific form of distribution

Logical probability
Experiential probability
Subjective probability
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Probability Distribution
Classical example of
tossing
What is the distribution of the sum of tosses?
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Probability Distribution (contd.)
Discrete variable
Values of x are discrete (discontinuous) Sum of
lengths of vertical bars ?p(Xx) 1
all x
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Probability Distribution (cont.)
Continuous variable
Mean 39.5 Std. dev 2.45
Pr (Area under curve) 1
Pr (Area under curve) 1
Age distribution of second-home buyers in
probability histogram
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Probability Distribution (cont.)

• Pr (Age 36) 0.02
• Pr (Age 37) Pr (Age 36) Pr (Age 37)
  0.02 0.07 0.09
• Pr (Age 38) Pr (Age 37) Pr (Age 38)
  0.09 0.04 0.13
• Pr (Age 39) Pr (Age 38) Pr (Age 39)
  0.13 0.18 0.31
• Pr (Age 40) Pr (Age 39) Pr (Age 40)
  0.31 0.36 0.67
• Pr (Age 41) Pr (Age 40) Pr (Age 41)
  0.67 0.14 0.81
• Pr (Age 42) Pr (Age 41) Pr (Age 42)
  0.81 0.10 0.91
• Pr (Age 43) Pr (Age 42) Pr (Age 43)
  0.91 0.09 1.00

?Cumulative probability corresponds to the
left tail of a distribution
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Probability Distribution (cont.)
Larger sample

• As larger and larger samples are drawn, the
  probability distribution is getting smoother
• Tens of different types of probability
  distribution Z, t, F, gamma, etc
• Most important normal distribution

Very large sample
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Normal Distribution - ND

• Salient features of ND
• Bell-shaped, symmetrical
• Total area under curve 1
• Area under curve between
• any two points prob. of
• values in that range (shaded area)
• Prob. of any exact value 0
• Has a function of

µ mean of variable x s std. dev. of x p
ratio of circumference of a circle to its
diameter 3.14 e base of natural log
2.71828.
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Normal Distribution - ND
Population 2
Population 1
?2
?1
?1 ?2
A larger population has narrower base
(smaller variance)
? determines location while ? determines
shape of ND
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Normal Distribution (cont.)
Has a mean ? and a variance ?2, i.e. X ? N(?,
?2 ) Has the following distribution of
observation
Home-buyers example
Mean age 39.3 Std. dev 2.42
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Standard Normal Distribution (SND)

• Since different populations have different ? and
  ? (thus, locations and shapes of distribution),
  they have to be standardised.
• Most common standardisation standard normal
  distribution (SND) or called Z-distribution
• ?(Xx) is given by area under curve
• Has no standard algebraic method of integration
• ? Z N(0,1)
• To transform f(x) into f(z)
• x - µ
• Z ------- N(0, 1)
• s

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Z-Distribution

• Probability is such a way that
• Approx. 68 -1lt z lt1
• Approx. 95 -1.96 lt z lt 1.96
• Approx. 99 -2.58 lt z lt 2.58

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Z-distribution (cont.)

• When X µ, Z 0, i.e.
• When X µ s, Z 1
• When X µ 2s, Z 2
• When X µ 3s, Z 3 and so on.
• It can be proven that P(X1 ltXlt Xk) P(Z1 ltZlt Zk)
• SND shows the probability to the right of any
  particular value of Z.

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Normal distributionQuestions

• A study found that the mean age, A of second-home
  buyers in Johor Bahru
• is 39.3 years old with a variance of RM
  2.45.Assuming normality, how sure
• are you that the mean age is (a) 40 years old
  (b) 39 to 42 years old?
• Answer (a) P(A 40)
• PZ (40 39.3)/2.4
• P(Z 0.2917? 0.3000)
• 0.3821
• (b) P(39 A 42)
• P(A 39) P(A 42)
• 0.45224 PA
  (42-39.3)/2.4
• 0.45224 P(A 1.125)
  
• 0.45224 0.12924
• 0.3230

Use Z-table!
Always remember to convert to SND, subtract the
mean and divide by the std. dev.
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Students t-Distribution

• Similar to Z-distribution (bell-shaped,
  symmetrical)
• Has a function of
• where ? gamma distribution v n-1
  d.o.f ? 3.147
• Flatter with thicker tails
• Distributed with t?(0,s) and -8 lt t lt 8
• As n?8 t?(0,s) ? N(0,1)
• Probability calculation requires
• information on d.o.f.

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How Are t-dist. and Z-dist. Related?

• Using central limit theorem, ?N(?, ?2/n) will
  become
• z?N(0, 1) as n?8
• ?For a large sample, t-dist. of a variable or a
• parameter is given by
• The interval of critical values for variable, x
  is

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Skewness, m3 Kurtosis, m4

• Skewness, m3 measures degree of symmetry of
  distribution
• Kurtosis, m4 measures its degree of peakness
• Both are useful when comparing sample
  distributions with different shapes
• Useful in data analysis

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Skewness
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Kurtosis
Mesokurtic distributionkurtosis 3 Leptokurtic
distributionkurtosis lt 3 Platykurtoc
distributionkurtosis gt 3
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Occurrence of ganoderma
Occurrence of ganoderma
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Aluminium residues in the soil
E.g. Al2 H2O--
? Al2O H2
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Measures of spatial separation

• E.g. WCM ((545.10-542.86)2 (105.90-105.48)2)0.
  5
• (5.0176 0.1764)0.5
• 2.28 (i.e. 2,280 m)

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Spatial distribution
Occurrence of ganoderma
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Spatial distribution point data
Ethnic distribution of residence
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Ethnic distribution of residence