Analysing a single statistical variable

Analysing a single statistical variable

HDL
HOCHSCHULVERBUND DISTANCE LEARNING
BUSINESS STATISTICS
Götze
1. Auflage 2000
Studienbrief
2 – 030 – 0901 E
Analysing a single
statistical variable
Verfasser:
Prof. Dr. Wolfgang Götze
Professor für Mathematik, Statistik, Informatik
im Fachbereich Wirtschaft
an der Fachhochschule Stralsund
Übersetzer:
Prof. Dr. Wolfgang Götze,
Urta Schünemann
Fachhochschule Stralsund
Der Studienbrief wurde auf der Grundlage des Curriculums für das Studienfach Wirtschaftsstatistik verfasst. Die Bestätigung des Curriculums erfolgte durch den
Fachausschuss für das modulare Fernstudienangebot Betriebswirtschaftslehre,
dem folgende Professoren angehören:
Dr. Albers (FH Magdeburg), Dr. Bathe (HTWS Zittau/Görlitz), Dr. Greife (FH Brandenburg),
Dr. Götze (FH Stralsund), Dr. Hannemann (HS Anhalt), Dr. Heger (FHTW Berlin), Dr. Heße
(FH Jena), Dr. Hofmeister (FH Erfurt), Dr. John (HTWK Leipzig), Dipl.-Ök. Schindler (FH
Merseburg), Dr. Strunz (Westsächsische HS Zwickau), Dr. Teichmann (TFH Wildau), Dr. Witt
(HS Wismar).
1. Auflage 2000
Redaktionsschluss: August 2000
 2000 by Service-Agentur des Hochschulverbundes Distance Learning mit Sitz an der FH Brandenburg.
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(Hochschulverbund Distance Learning)
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Business Statistics
Analysing a single statistical variable
Table of contents
Statistical symbols...................................................................................................................... 4
Table with icons ......................................................................................................................... 5
Literature suggested .................................................................................................................. 7
1
Aims and methods of descriptive statistics ................................................................... 7
1.1
1.2
1.3
1.4
1.4.1
1.4.2
1.5
1.6
Fields of applications ....................................................................................................... 7
History............................................................................................................................. 8
The approach of a statistical survey ................................................................................. 8
Units, variables, values .................................................................................................... 9
Scaling and typing ........................................................................................................... 9
Cross section and longitudinal section ........................................................................... 10
Statistical reliability....................................................................................................... 10
Study objectives and examples for basic terms............................................................... 11
2
Empirical distributions ................................................................................................ 13
2.1
2.2
2.3
2.4
Frequency distribution table........................................................................................... 13
Stick diagrams and polygons.......................................................................................... 13
Grouping data ................................................................................................................ 15
Histogram and empirical distribution function ............................................................... 17
3
Measures of central tendency ...................................................................................... 21
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Mode ............................................................................................................................. 22
Median........................................................................................................................... 22
Arithmetic mean ............................................................................................................ 24
Relations between the several means ............................................................................. 25
Weighted arithmetic mean ............................................................................................. 27
Geometric mean ............................................................................................................. 27
Harmonic mean.............................................................................................................. 28
4
Measures of variability ................................................................................................ 30
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
Range............................................................................................................................. 31
Distance of quartiles ...................................................................................................... 31
Variance ........................................................................................................................ 32
Relations between variance and arithmetic mean ........................................................... 33
Standard deviation ......................................................................................................... 33
Further characteristics of variance and standard deviation ............................................. 34
Relative dispersion......................................................................................................... 35
Further statistical measures ............................................................................................ 35
Summery........................................................................................................................ 37
5
Concentration............................................................................................................... 39
5.1
5.2
Measuring the relative concentration using Lorenz curves ............................................. 40
Gini’s coefficient ........................................................................................................... 43
Analysing a single statistical variable
Business Statistics
5.3
5.4
5.5
Measuring the relative concentration for grouped data ...................................................44
Measuring the absolute concentration.............................................................................45
Summery ........................................................................................................................48
6
Indices ...........................................................................................................................51
6.1
6.2
6.2.1
6.2.2
6.2.3
6.2.4
Simple Indices................................................................................................................51
Compounded indices ......................................................................................................52
Price indices ...................................................................................................................53
Set indices ......................................................................................................................54
Sales indices...................................................................................................................55
Stock exchange indices................................................................................................... 55
Solutions of exercises and control questions ...........................................................................59
List of further literature ..........................................................................................................62
Statistical symbols
xi
observations or values of a variable X for i = 1,...,n
aj
different values of a variable X for j = 1,...,k
Hn (aj) absolute frequency
hn (aj) relative frequency
Sn (aj) cumulated relative frequencies
Fn (x) empirical distribution function of a discrete variable
F(x)
empirical distribution of a continuous variable
gj
class limits for classes j = 1,..,q
bj
width of a class
mj
midpoint of a class
Hj
absolute frequency of a class
hj
relative frequency of a class
Sq (j)
cumulated relative frequencies of grouped data
Fq (x) empirical distribution in the case of grouped data
x mod
mode
x (i)
ascending sorted observations
x med
median
xα
α-quantile
x
arithmetic mean
x∗
arithmetic mean for grouped data
4
Business Statistics
Analysing a single statistical variable
_
xw
weighted arithmetic mean
x harm
weighted harmonic mean
x geom
geometric mean
R
range
∆
distance of quartiles
s
2
mean square deviation from an arithmetic mean
v
variation coefficient
G
Gini’s coefficient
H
Herfindahl’s index
P L0,1
Laspeyres’ price index
PP0,1
Paasche’s price index
Q
L
Q
P
0,1
Laspeyres’ set index
0,1
Paasche’s set index
0,1
Fisher’s Ideal-Index for prices
F
P
F
Q
0,1
U0,1
Fisher’s Ideal-Index for sets
sales index
Table with icons
N
D
E
Q
O
S
Note
Definition
Example
Questions
Objectives
Summary
5