Mathematical Statistics

Anna Janicka

Lecture II, 24.02.2020

DESCRIPTIVE STATISTICS, PART II

Plan for today

1. Descriptive Statistics, part II:

mode

quantiles

measures of variability

measures of asymmetry

the boxplot

Measures of central tendency – reminder

Classic:

arithmetic mean

Position (order, rank):

median

mode

quartile

Example 1 – cont.

Mode – examples

Quartiles – example

Variance – example

Grade Number Frequency

2 74 29.84%

3 76 30.65%

3.5 48 19.35%

4 31 12.50%

4.5 9 3.63%

5 10 4.03%

Total 248 100%

Example 3 – cont.

Interval Class

mark Number Frequency

Cumulative

number cni

Cumulative

frequency cfi

(30,40] 35 11 0,11 11 0,11

(40,50] 45 23 0,23 34 0,34

(50,60] 55 33 0,33 67 0,67

(60,70] 65 12 0,12 79 0,79

(70,80] 75 6 0,06 85 0,85

(80,90] 85 8 0,08 93 0,93

(90,100] 95 3 0,03 96 0,96

(100,110] 105 2 0,02 98 0,98

(110,120] 115 2 0,02 100 1

Total 100 1 Mode – example

Quartiles – example

Variance – example

Mode

Mode

the value that appears most often

for grouped data:

Mo = most frequent value

for grouped class interval data:

where

nMo – number of elements in mode’s class,

cL, b – analogous to the median

bnnnn

nncMo

MoMoMoMo

MoMoL

)()( 11

1

Mode – examples

Example 1:

Mo = 3

Example 3:

the mode’s interval is (50,60], with 33 elements

nMo = 33, cL = 50, b = 10, nMo-1 = 23, nMo+1 = 12

23.5310)1233()2333(

233350

Mo

Example 1 –

cont.

Example 3 –

cont.

Which measure should we choose?

Arithmetic mean: for typical data series

(single max, monotonous frequencies)

Mode: for typical data series, grouped data

(the lengths of the mode’s class and

neighboring classes should be equal)

Median: no restrictions. The most robust (in

case of outlier observations, fluctuations

etc.)

Quantiles, quartiles

p-th quantile (quantile of rank p): number

such that the fraction of observations less

than or equal to it is at least p, and values

greater than or equal to it at least 1-p

Q1 : first quartile = quantile of rank ¼

Second quartile = median

= quantile of rank ½

Q3: Third quartile = quantile of rank ¾

Quantiles – cont.

Empirical quantile of rank p:

ZnpX

ZnpXX

Q

nnp

nnpnnp

p

:1][

:1:

2

Quartiles – cont.

Quantiles for p = ¼ and p = ¾.

For grouped class interval data –

analogous to the median

for k=1 or 3

where M1, M3 – number of the quartile’s class

b – length of quartile class interval

cL – lower end of the quartile class interval

1

14

k

k

M

i

i

M

Lk nnk

n

bcQ

Quartiles – examples

Example 1:

so

Example 3:

so

75100 25100 43

41

4M ,2 31 M

67,66)6775(12

106009,46)1125(

23

1040 31 QQ

Example1 –

cont.

Example 3 –

cont.

18624862248 41 4

3

5322321

2481872481862486324862 ., :::: XXXX

Q Q

Variability measures

Classical measures

variance, standard deviation

average (absolute) deviation

coefficient of variation

Measures based on order statistics

range

interquartile range

quartile deviation

coefficients of variation (based on order stats)

median absolute deviation

Measures based on order statistics

Range

the most simple measure, does not take into

account anything but the extreme values

Inter Quartile Range (midspread, middle fifty)

more robust than the range

nnn XXr :1:

13 QQIQR

may be further used to calculate quartile deviation Q= IQR/2, and

coefficients of variation VQ = Q/Med or VQ1Q3 = IQR/(Q3+Q1) (quartile

variation coefficient) or the typical range: [Med – Q, Med + Q]

length of the interval that covers the

middle 50% observations

Range, interquartile range – examples

Example 1:

Example 3:

(in reality

5.125.3

,325

IQR

r

58,2009,4667,66

)45,8645329118

9030120

IQR

,-,

r

Classical measures of dispersion

Variance

raw data

grouped data

grouped class interval data

+ Sheppard’s correction

in general

2

1

21

1

212 )()(ˆ

n

i

in

n

i

inXXXXS

2

1

21

1

212 )()(ˆ

k

i

iin

k

i

iinXXnXXnS

2

1

21

1

212 )()(ˆ

k

i

iin

k

i

iinXcnXcnS

12

22 2ˆ cSS

c=length of

class

interval (for

equal

intervals)

2

1

112122 )(ˆ

k

i

iiinccnSS

Variance – examples

Example 1:

Example 3:

in reality

1006359063543106344806353760633740632 222222

2481 ).()..().()..().().(

710

2

.

ˆ

S

98.32212

1031.331

31.331

ˆ

22

10012

S

S

)2)7.58115(2)7.58105(3)7.5895(8)7.5885(6)7.5875(

12)7.5865(33)7.5855(23)7.5845(11)7.5835((

22222

2222

85.333ˆ2 S

Example 1 – cont.

Example 3 – cont.

distrubution not normal or

sample too small for

Sheppard’s correction –

larger errors from small

sample size than from class

grouping.

Standard deviation

In the same units as the initial variable

Example 1:

Example 3:

22 ,ˆˆ SSSS

[grade] S 840.ˆ

][m 22.18ˆ S

Average (absolute) deviation, mean deviation

Nowadays seldom used. Simple

calculations.

for raw data

etc...

We have: d<S

n

i

inXXd

1

1 ||

Coefficient of variation (classical)

For comparisons of the same varaible

accross populations or different

variables for the same population

%)100( or

%),100(ˆ

X

dV

X

SV

d

S

Skewness (asymmetry)

left symmetry right

(negative) (zero) (positive)

(typical order)

MoMedX MoMedX MoMedX

Measures of asymmetry

Skewness

where M3 is the third

central moment

Skewness coefficient

Quartile skewness coefficient

3

3

S

MA

ˆ

or ˆ 11

S

MedXA

S

MoXA

13

132

2

QQ

QMedQA

measures skewness for the

middle 50% observations only

Interpretation

positive values= positive asymmetry (right

skewed distribution)

negative values = negative asymmetry

(left skewed distribution)

For the skewness coefficient (with the

median) and the quartile skewness

coefficient the strength of asymmetry

(absolute value):

0 – 0.33: weak

0.34 – 0.66: medium

0.67 – 1: strong

Asymmetry – examples

Example 1:

Example 3:

15.009.4667.66

09.4685.54267.66

24.02.18

85.547.583.0

2.18

23.537.58

,15.1

2

11

A

)( A or )( A

A

MedMo

3

1

253

23253

070840

3063

070840

3063

280

2

1

1

.

.

..

.

..

.

.

A

)Mo( A

)Med( A

A

0

0,05

0,1

0,15

0,2

0,25

0,3

0,35

25 35 45 55 65 75 85 95 105 115 125

Fre

qu

en

cy

Surface area

0%

5%

10%

15%

20%

25%

30%

35%

2 3 3.5 4 4.5 5

Fre

qu

en

cy

Grade

0%

5%

10%

15%

20%

25%

30%

35%

2 2.5 3 3.5 4 4.5 5

Fre

qu

en

cy

Grade

Boxplot

(Box and whisker plot)

Allows to compare two (or more)

populations

outliers:

xmax

outliers

X*

Q3

Med

Q1

X*

outliers

xmin

]},[:max{

]},[:min{

23

33

123

1

IQRQQXXX

QIQRQXXX

ii

ii

XxXx or

Boxpolot – example of comparison

05

10

15

1 2

Examples (1)

Source: CSO Poland, 2009

Examples (2)

Source: European

Commission

Dispersion (coeffcient of variability) of unemployment

rates

Examples (3)

Growth charts

Source: WHO

Examples (4)

Gross hourly earinings

Source: European Commission

Examples(5)

Salary by occupational group and gender

Source: New Zealand State Services