ELEMENTS OF MATHEMATICAL

STATISTICS

ELEMENTS OF MATHEMATICAL

STATISTICSGENERAL NOTIONS

Mathematical statistics is a set of methods that are used to collect, analyzegroup and interprete data concerning a certain phenomenon and predict its future evolution.THE STATISTICAL POPULATION is a definite set of similar objects.Elements of a statistical population are called statistical units or individuals.The part of the entire population for which the statistical analysis is made is called sample.CHARACTERISTIC ( or variable ) is the common feature of statistical population which we study in the statistical analysis. Characteristics are : - quantitative – if we can measure them (average , height) - qualitative – if we can`t measure them ( eyes color) .

ELEMENTS OF MATHEMATICAL

STATISTICS THE QUANTITATIVE CHARACTERISTICS can be :

- discontinuous – if the variable takes a finite number of values- continuous – if the variable takes any value from a finite or infinite interval

For example : the average mean in mathematics takes values: 1, 2, …, 10and is a discontinuous characteristic and height can take any value in a finite interval and is a continuous characteristic.STAGES OF A STATISTICAL STUDY (survey)

- collecting data- presenting data in a table from which expresses answers, allows

comparing and classifying- representing data- calculating specific parameters- interpreting data

ELEMENTS OF MATHEMATICAL

STATISTICS TOTAL EFFECTIVE is the total number of the units of a statistical population.

ABSOLUTE FREQUENCY of a value of the characteristic is the number of unitscorresponding to this value

RELATIVE FREQUENCY of a value is the fraction between the absolute frequency and the total effective.

CUMULATIVE FREQUENCY ( increasing ) of a value of the characteristic is the sum of all relative frequencies of values up to and including this value

ELEMENTS OF MATHEMATICAL

STATISTICS

EXEMPLES

GRAPHIC REPRESENTATION OF DATA

Representation of statistical series with discontinuous quantitative variable:

- by columns (on the horizontal axis – values of the variable

on the vertical axis - values of absolute frequency ).

Representation of statistical series with continuous quantitative variable: - by histogram ( on the horizontal axis- succession of segments proportional to the length of classes; there are built rectangles having these segments as bases and height proportional to the absolute frequencies ) .

1 2 3

ELEMENTS OF MATHEMATICAL

STATISTICSRepresentation of statistical series with qualitative variable : - by structural rectangle ( it is divided into rectangles of areas proportional to relative frequencies ) ,

- by structural circle ( it is divided into circle sectors with areas proportional to relative frequencies ) ,

- diagrams with columns or bands ( rectangles with equal bases and heights proportional to the absolute frequencies, placed at equal distances from each other ).

4 65 7EXEMPLES

ELEMENTE DE STATISTICĂ

MATEMATICĂ

EXEMPLU

CHARACTERISTIC ELEMENTS OF A STATISTICAL SERIESThe tables with the values of the characteristic and the correspondingabsolute frequencies define the distribution of the statistical repartition of the statistical variable and represent a statistical series with only one variable.

THE AVERAGEConsidering the statistical variable x taking the values x1 , x2 , x3 , … xk with the corresponding effectives n1 , n2 , n3 , … nk

Definition We call the average of the characteristics the number

_ n1 x1 +n2 x2+…+nkxk

x= ——————————— n1 +n2 +…+nk

ELEMENTE DE STATISTICĂ

MATEMATICĂ

EXEMPLU

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Definition THE MEDIAN x , is a value for which half of the values xi are smaller or equal and the other half of the values xi are bigger or equal to that.

Definition THE MODULUS ( or THE DOMINANT ) of a statistical series is the value of the characteristic corresponding to the highest frequency, in case of discontinuous values and the main value of the class corresponding to the highest frequency in case of continuous values.

Marks in MathematicsMarks in Mathematics

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4

5

6 7

8 9

10

0

1

2

3

4

5

6

7

8

9

10

1 8 10 10 9 9 3

4 5 6 7 8 9 10

Marks in MathematicsMarks in Mathematics

MarksMarksStudentsStudents

Absolute Absolute frequency(ffrequency(fa a ))

Relative Relative frequency(ffrequency(fr r

) )

Cumulative Cumulative frquency(ffrquency(fc c ))

44 11 0.020.02 0.020.02

55 88 0.160.16 0.180.18

66 1100 0.200.20 0.380.38

77 1100 0.200.20 0.580.58

88 99 0.180.18 0.760.76

99 99 0.180.18 0.940.94

1010 33 0.060.06 1.001.00

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HeightHeight

TABLEBACK

0123456789

101112131415

150-155

155-160

160-165

165-170

170-175

175-180

HeightHeight

CHARTBACK

HeightHeight

StudentsStudents

Absolute Absolute frequency(ffrequency(fa a

))

Relative Relative frequency(ffrequency(fr r

) )

Cumulative Cumulative frquency(ffrquency(fc c ))

44 11 0.020.02 0.020.02

55 88 0.160.16 0.180.18

66 1100 0.200.20 0.380.38

77 1100 0.200.20 0.580.58

88 99 0.180.18 0.760.76

99 99 0.180.18 0.940.94

1010 33 0.060.06 1.001.00

Distance from home to Distance from home to schoolschool

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6

910

9

4

7

5

0

1

2

3

4

5

6

7

8

9

10

0-300 300-600 600-900 900-1200 1200-1500

1500-1800

1800-2100

Distance from home to Distance from home to schoolschool

CHARTBACK

DistanceDistance

StudentsStudents

Absolute Absolute frequency(ffrequency(fa a

))

Relative Relative frequency(ffrequency(fr r

) )

Cumulative Cumulative frquency(ffrquency(fc c ))

0-3000-300 66 0.120.12 0.120.12

300-600300-600 99 0.180.18 0.300.30

600-900600-900 1100 0.200.20 0.500.50

900-1200900-1200 99 0.180.18 0.680.68

1200-15001200-1500 44 0.080.08 0.760.76

1500-18001500-1800 77 0.140.14 0.900.90

1800-21800-2110000 55 0.100.10 1.001.00

Eye colourEye colour

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Blue Green Brown Black

10% 28% 52% 10%

Eye colourEye colour

Eye colourEye colour StudentsStudents

BlueBlue 55

GreenGreen 1414

BrownBrown 2626

BlackBlack 55

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Favourite sportFavourite sport

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0 1 2 3 4 5 6 7 8 9 10111213141516

Voleyball

Handball

Basketball

Football

Tennis

Other

Favourite sportFavourite sport

SportSport StudentsStudents

VoleyballVoleyball 1515

HandballHandball 33

BasketballBasketball 44

FootballFootball 1515

TennisTennis 88

OtherOther 66

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Favourite musicFavourite music

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House22%

Dance24%

Rock12%

Symphonic8%

HeavyMetal6%

Rap14%

Other14%

Favorite musicFavorite music

TypeType StudentsStudents

HouseHouse 1111

DanceDance 1212

RockRock 66

SymphonicSymphonic 44

Heavy metalHeavy metal 33

RapRap 77

OtherOther 77

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Birth monthBirth month

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012345678

Janu

ary

Febr

uary

Mar

chAp

rilMay

June Ju

ly

Augu

st

Sept

embe

r

Octob

er

Novem

ber

Decem

ber

Birth monthBirth month

MonthMonth StudentsStudents

JanuaryJanuary 33

FebruaryFebruary 22

MarchMarch 77

AprilApril 44

MayMay 55

JuneJune 44

JulyJuly 44

AugustAugust 22

SeptemberSeptember 44

OctoberOctober 66

NovemberNovember 44

DecemberDecember 55

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Survey carried out bySurvey carried out bystudents and staff of students and staff of

National College National College “August Treboniu Laurian” “August Treboniu Laurian”

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