Math for XI Science Temp1(2)

8/8/2019 Math for XI Science Temp1(2)

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MATHEMATICS FOR SCIENCE XI GRADE

By: Syaiful Yazan

SMA NEGERI 1 MARTAPURAJl. A. Yani 59 A 0511-4721272 Martapura

Web Sekolah: www.sman1mtp.sch.idPrivate Web: www.msyna.com

e-mail: [email protected] e-mail: [email protected]


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INTRODUCTION

All gratefull for Allah, on His willing this mathematics learning material can bepresented. Shalawat and peacefull always address to my belove Prohet

Muhammad SAW.

This learning material consists of:1. Competency standard and basic competency as in Curriculum of Education

Unit Level, Kurikulum Tingkat Satuan Pendidikan (KTSP).2. Benefits and competencies that hoped can be reached by student after learn

each chapter.3. Material explanation, involves: 1) Theory that generally begin with the reality

or presenting the relevan illustration, 2) examples and 3) drill and testcompetency try out.

4. In some parts serves the computer program application to facilities theaccounting or contributes the related mathematical jobs. Its aimed to give the

wider perspectives for student in managing mathematical jobs, especially formathematics application.

5. Some enrichments are also given to strenghten and axpand the studentsknowledge.

This learning material especially is served for student in XI Grade Scienceprogram. It also as a reference for teachers.

The author thank to Mr. Drs. H. Busra as the head master of SMA Negeri 1Martapura, Mathematics Teachers of SMA Negeri 1 Martapura, my colleaguesinMathematics Teacher Conference (MGMP Matematika), and also to all studentsof SMA Negeri 1 Matapura for their loyalty in following my mathematicsinstructional. Those all give me stimulus to write this learning material. My thankalso to all people that have contributes me.

I hope this simple learning material can be benefits for all of us. I also hope thesuggestions for improving it.

Martapura, October 4th 2009Author,

Syaiful Yazan


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Contents

Introduction........................................................................................ 1

Contents ............................................................................................ 2

Chapter 1 Statistics ........................................................................... 3


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STATISTICSCompetency Standard

1. Use the rules of statistics and counting, and the natures ofprobability in solving problems.

Basic Competency

1.1 Reading data in form of table, bar, line, pie and ogive.

1.2 Presenting data in form of table, bar, line, pie and ogive andmaking the interpretation.

1.3 Calculating the central tendencies, position and distributiontendencies and making the interpretation.

Lesson BenefitCreate the objective behavior of student. Do action on base of

facts, collecting data, analyzing data and making interpretation fromthe analyzing data.

Will be Reached Competencies

Students are capable in:

1. Reading data in form of table, bar, line, pie and ogive.2. Presenting data in form of table, bar, line, pie and ogive and

making the interpretation.

3. Calculating the central tendencies, position and distributiontendencies and making the interpretation.

Material

A. What is Statistics?Statistics is applied mathematics that discusses about:1. Collecting and arranging data, analyzing data and presenting it in

form of curve or diagram.2. Testing a hypothesis based on of the analyzing of the data,

making the interpretation and conclusions. These results are

meaningful to take an action.Statistics is highly related to research activities, in which it try to findthe truth from the research object.

Chapter 1


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What is the statistical result?

Answer: The statistical results is the product of the analyzing datalike average, median, mode, skewness, maximum andminimum, variance, standard deviation etc.

B. Descriptive and Inferential Statistics

Descriptive Statistics gives the description of the real condition ofthe observed object, highly related to the work of collecting,summarizing and presenting the results. It usually manages thecentral tendencies: average, median and mode.

Inferential Statistics makes estimation to the population,estimating the parameter and testing the hypothesis.

Notice the following diagram:

If we observe a group of children and find the statistical results, forexample the average of their ages and intelligence quotient, withoutestimate the average of the ages and intelligence quotient ofchildren out of the group or in greater number of the children, thenits called as descriptive statistics. However, if we choose somechildren as the representatives of the population and the statisticalresults that found from the observation to the sample are used toestimate the condition of the population, then its called asinferential statistics.

Because the aim is to estimate the condition of the populationbased on limited observation, to the sample, then inferentialstatistics relatively insist of high rules. Some statistical tests mustbe conducted like instrument test, qualifications test to applying amodel of statistical test and doing the model of statistical test itself.


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C. Data Types

1. Qualitative Data

Non-numeral Data, like very good, good, enough good, not good.

2. Quantitative DataNumeral data, like sum, weight, height, temperature, velocity,time, power, frequency and intensity.

D. Data PresentingThere are many ways of presenting data. How is the data bepresented, the most important is the data or the information mustbe readable, meaningful, communicative and informative. Then, inorder, the presented data is attractive; it must be in good style,using adequate accessories and well layout setting.

1. Models of Graphs that served by ExcelAccording to the Information and Communication Technologydevelopment, for example the using of microsoft excel programand SPSS (Scientific Program for Social Science), many modelsof graphs being offered to serve data. Excel offers: column, line,pie, bar, area, XY (Scatter), Stock, Surface, Doughnut, Bubbleand radar(Picture 1). For each model insists sub models and besupported with very variety property settings.

Picture 1. Graph Models in Excel


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Example:

In one city consistssome models of pr

Answer:The given informatidata in narration. A

1) Table

2) Bar

3) Pie

10

0

5

10

15

20

25

SD

of 10 SD, 7 SMP and 5 SMA. Servesentation.

on above actually is a simple way ofnother ways are:

u School Level Sum

SD 10

SMP 6

SMA 5

Total 21

6 5

21

SMP SMA Jumlah

Number of School

Juml

SD

10

48%SMP

6

28%

SMA5

24%

Number of School

the data in

presenting

ah


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4) Line

2) Stem and leaf Diagram

The model of data presentation must be suitable with thedata type. For data likes: 6, 5, 5, 7, 8, 10, 11, 12, 12, 13, 24, 25,32, 34, 34, 35, 41, 5. It can be presented by Stem and LeadDiagram as the following:

Stem Leafl0 5 5 6 7 81 0 1 2 2 32 4 53 2 4 4 5

4 15 1Note:Stem : presenting tensLeaf : presenting ones

Example

Given two groups numbers

a. 121, 122, 234, 235, 317, 456, 457, 650b. 78, 100, 127, 138, 245, 345, 678, 679, 767

It can be presented in the way of Stem and Leaf Diagram with alittle modification, likes:

10

6

5

0

2

4

6

8

10

12

SD SMP SMA

Number of School

Series1


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Stem, Twig and Leaf

Part a.Stem Twig Leaf

1 2 1 22 3 4 53 1 74 5 6 76 5 0

Part b.

Stem Twig Leaf0 7 8

10 02 7

3 82 4 53 4 56 7 8 97 6 7

Note:Stem : presenting hundredsTwig : presenting tens

Leaf : presenting ones

3) Line Box Diagram

If the data has been in order from the smallest to thegreatest, then the data can be classified into four parts with thethree points of devisor called as first quartile (Q1, sub quartile),second quartile (Q2, middle quartile, median) and third quartile(Q3, up quartile). In presenting data that emphasize theindications of Q1, Q2 dan Q3 can use Line Box Diagram.

Example:

Given data: 2, 8, 4, 5, 7, 9, 10, 11, 13

The procedure to make the Line Box Diagram is:

a) Arrage data from the smallest to the greatest, that is 2, 4, 5,7, 8, 9, 10, 11, 13


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b) Devide the data into four parts to decide the points Q1, Q2,Q3, that is:2, 4, 5, 7, 8, 9, 10, 11, 13

Notice Q1 = 5, Q2 = 9 dan Q3 = 11

c) The data is distributed in form:

0 2 4 6 8 10 12 14

The quartiles are going to be discusses specifically in thenext explanation.

4) Frequency Distribution

From an observation of the height of 40 persons, has gottenthe raw data:

138 164 150 132 144 125 149 157146 158 140 147 136 148 152 144168 126 138 178 163 118 154 165

146 173 142 147 135 153 140 135162 145 135 142 150 156 145 128

The raw data need to be processed, the table of frequencydistribution will be made.

1) Arrange the data (from the smallest to the highest)118 135 138 144 146 150 156 164125 135 140 144 147 150 157 165126 135 140 145 147 152 158 168

128 136 142 145 148 153 162 173132 138 142 146 149 154 163 178

The range = 178 -118 = 60

2) The number of class (=k) use formula:

Sturgess, k = 1 +3,3 log n

Q2Q1 Q3

Q1 Q2 Q3


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For n = 40

k = 1 + 3,3 log 40 = 6,286

The number of class is round up to7.

3) The length of class is the range/number of class= 60/7= 8,571429

The length of class is round up to 9.

4) Stating each class:

Class I 118 126 mid point 122

Class II 127 135 mid point 131

Class III 136 144 mid point 140

Class IV 145 153 mid point 149

Class V 154 162 mid point 158

Class VI 163 171 mid point 167

Class VII 172 180 mid point 176

5) Making the table

Data Mid point Tally Frequency

118 126 122 III 3127 135 131 IIII I 6

136 144 1410 IIII IIII 10

145 153 149 IIII IIII I 11

154 162 158 IIII 5

163 171 167 III 3

172 180 176 II 2

Total 40


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For the firs class:

Class border: bottom top

118 126

Class edge: 118 0,5 126 + 0,5

117,5 126,5

The length of class Top edge bottom edge

126,5 117,5 = 9

D. Central TendencyFirst, should be considered that the data collection can be devidedinto two categories, individual data and group data. Individual datais relatively simple, the number of data is small, meanwhile groupdata is the data that distributed into several classes, and usually thenumber of data is big. Here, the discussion begins for the individualdata then for the group data.1. Central tendency for individual data

a. Average

Given data x1, x2, x3, ... , xn, then the average is

1 2 3...

n x x x x

xn

+ + + += or

1

1 n

i

i

x xn =

=

Example:

In a clinic, there are 10 babies with the average weight 3.2 kg.Then born a baby and the average weight for all the babiesbecome 3.19 kg. Find the weight of the newborn baby!

SolutionIf is the origin average is the origin data number is the new average

is the new data number

then the value of new data, is

. . In this case the asked weight = (11 x 3.19 10 x 3.2) kg =(35.09 32.0) kg = 3.09 kg


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Excell program to find the average of data 2, 3, 3, 5:

The result = 3,25.b. Mode

Mode is the value that most freguently appears in a set ofdataExample:1) 2, 3, 3, 4, 5, 6 the mode is 32) 3, 4, 3, 4, 5, 4, 6, 3 the modes are 3 dan 43) 3, 4, 5, 7, 6, 8, 9 no mode4) 3, 4, 3, 5, 4, 5, 6, 6 no mode

As long as there is an element of data that depeated byanother elements of data, then the mode is exist, maybe thenumber of mode is 1, 2, 3 or more. However, if each elementof data appears in same frequency, then there is no mode.

Excel program to find the mode of 2, 3, 3, 5, 6:

The result = 4.

c. Quartile (Q1, Q2 and Q3)It begins with the explanation of median from a set of datathat has been arraged from the smallest to the highest.Median is the second quartile (Q2). From this concept, hopelywill facilitate of understanding (Q1) dan ketiga (Q3).

Given data x1, x2 , x3 , ..., xn ordered from smallest to thehighest, then

1) If the data number (=n) is odd, the median located in datanumber

1

2

n +, that =

1

2

nx

+


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Example:

3, 5, 6, 7, 8, 10, 12, 13, 16,

n = 9, median in the 5th data (9 1

2

+= 10/2 = 5), x5 = 8.

2) If the data number (=n) is even, the median is the averageof the value of the n/2 th data and the (n/2)+1 th data.

The median = 12 2

1

2n n

x x+

+

Contoh:

3, 5, 6, 6, 7, 9, 10, 11, 13, 16, 17, 18n = 12, the median =

( ) ( ) ( )12 12 6 71

2 2

1 1 1 19 10 19 9,5

2 2 2 2 x x x x

+

+ = + = + = =

.

The ordered data from the smallest to the largest are devidedinto four parts. There are three devisor points, called as firstquartile (Q1), second quartile (Q2, median) and third quartile(Q3).

Ilustration

a. Given data: 3, 5, 6, 7, 9, 10, 12, 13, 16, find Q1, Q2 dan Q3.Answer:

Quartile Q1 =5 62 5,5

Q2

Q3 = 12,5

Data 3 5 6 7 9 10 12 13 16Data order 1 2 3 4 5 6 7 8 9

Part I = {3, 5}; Part II = {6, 7}; Part III = {10, 12} and Part IV = {13, 16}To state Q2 is clear. To state:1) Q1, take the ordered data before Q2 (data Q2 is not included),

from these data find the median as the previous way.


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2) Then for Q3, take the ordered data after Q2 (data Q2 is notincluded), from these data find the median as the previous way.

b. Given 3, 5, 6, 6, 7, 9, 10, 11, 13, 16, 17, 18, find Q1, Q2 danQ3.

Answer:

Quartile

Q2 =6 62 6Q2 =6 723,5

Q2 =1 3 1 6214,5Data 3 5 6 6 7 9 10 11 13 16 17 18Dataorder

1 2 3 4 5 6 7 8 9 10 11 12

d. Range (Extent)Range (J) is the highest value data (Xmax) minus the lowestvalue data (Xmin).

J = Xmax- Xmin

e. Range among quartiles antar kuartil (Field)Range among quartiles (H) is the third quartile minus the firstquartile.

H = Q3 Q1f. Quartile deviation

Quartile deviation (QD) is a half of range among quartiles.QD= ( Q3 Q1)

g. Average DeviationAverage deviation (Sr) from data x1, x2, x3, x4, x5, , xn isthe average of the difference of each data to the average

( ). In symbol:Sr =

|| Xi = the i

th datan = number of data

h. VariansVarians (R) from data x1, x2, x3, x4, x5, , xn is defined as

R =

Xi = the ith data

n = number of data


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i. Standard Deviation (s)

Standard deviation (S) is the root of varians, so:

S = Xi = the i th datan = number of data2. Central Tendency for Group Data

Data berkelompok, biasanya data yang disajikan dalam tabeldistribusi frekuensi, dan ini ada dua tipe:a. Tipe data berdistribusi individu

Contoh:Data Nilai Frekuensi

6 37 48 79 510 2

b. Tipe data berdistribusi intervalContoh:

Interval Nilai Tengah Frekuensi

21 - 25 23 426 - 30 28 631 - 35 33 936 - 40 38 841 - 45 43 3

Nilai tengah untuk tiap kelas sama dengan setengah dari hasilbatas atas ditambahkan dengan batas bawah, misal untukkelas pertama, nilai tengahnya = (21 + 25) = .46 = 23.

Sehingga dengan melihat hal di atas, perlu keseksamaandalam penggunaan dan pengembangan rumus ukuran-ukuranpemusatan lebih lanjut.a. Rerata


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1

1

.n

i i

i

n

i

i

f x

x

f

=

=

=

Jika data berdistribusi interval, xi adalah nilai tengah.

b. ModusUntuk data berdistribusi individu, modus dicari samadengan data tunggal, sedangkan untuk data berdistribusiinterval digunakan rumus:

1

1 2

dModus = L + c.

d + d

Keterangan:L : tepi bawah kelas modusd1 : selish kelas modus dengan frekuensi kelassebelumnyad2 : selisih kelas modus dengan frekuensi kelassesudahnya

Untuk pembuktian rumus di atas dapat dilihat pada bagianInventing Formula pada Gambar 2.. Dengan menerapkanprinsip perbandingan pada kesebangunan dapat ditemukan

rumus pencarian modus di atas.


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Gambar 2. Penemuan Rumus Modus

c. Kuartil (Q1, Q2 dan Q3)Untuk data berdistribusi individu, Kuartil dicari samadengan data tunggal, sedangkan untuk data berdistribusiinterval digunakan rumus:

qi

i=1,2,3 qi

qi

1in - F4Q = L + c.

f

Keterangan:Lqi : Tepi bawah kelas Kuartiln : jumlah frekuensi, ukuran datafqi : frekuensi pada interval kelas Kuartil

Fkqi : frekuensi komulatif sebelum kelas Kuartil

Untuk pembuktian rumus di atas dapat dilihat pada bagianInventing Formula pada Gambar 3.. Dengan menerapkanprinsip perbandingan senilai dapat ditemukan rumuspencarian Q2 (median), dengan cara yanga sama makaditemukan juga untuk Q1 dan Q2 di atas.


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d. Simpangan Rata-rataPerlu sedikit perubahan dari rumus untuk data tunggal, karena

menyesuaikan dari karakter data berkelompok, yakni:

Sr = || Bila data berdistribusi interval maka xi adalah nilai tengah.e. Ragam (Varians)Analog dengan penjelsasan sebelumnya, maka ragamnya adalah

R = . 1

f. Simpangan Baku (s)Analog dengan penjelsasan sebelumnya, maka simpangan baku

adalah

S = . 21 1


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Contoh:Pada soal 1) sampai dengan 3) berikut tentukan berturut-turut:a) Rerata, modusb) Q1, Q2, Q3,c) Jangkauan Data

d) Jangkauan Antar Kuartil dan Simpangan Kuartile) Simpangan rata-rataf) Ragam dan Simpangan Baku

Soal:

1) 3, 4, 5, 6, 7, 6, 8, 9, 102)

Data Frekuensi

5 3

6 47 7

9 6

3)

Data Frekuensi

24 - 33 8

34 - 43 12

44 - 53 1354 - 63 10

64 - 73 7

Pembahasan:

Soal 1):

a) Rerata = 6,4

modus = 6

b)

Q1=4 52 4,5 Q2Q3=8 92 8,5

3 4 5 6 6 7 8 9 10

1 2 3 4 5 6 7 8 9


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c) Jangkauan Data = 10 3 = 7d) Jangkauan Antar Kuartil = Q3 Q1 = 8,5 4,5 = 4

dan Simpangan Kuartil = (Q3 Q1) = . 4 = 2e) Simpangan rata-rata

=|,||,||,||,||,||,||,||,||,| = , , , , , , , , , , 1,83f) Ragam

=,,,,,,,,, =

,,,,,,,,, =

, , , , , , , , ,

, 4,7

dan Simpangan Baku

= 4,72,2Soal 2): Kita kembangkan tabel untuk membantu analisis, sebagaiberikut:

Data(= x)

Frekuensi(= f)

x.f fk | | . | | )2 . )25 3 15 3 2,1 6,3 4,41 13,23

6 4 24 7 1,1 7,7 1,21 4,84

7 7 49 14 0,1 1,4 0,01 0,07

9 6 54 20 1,9 38 3,61 21,66

20 142 5,2 53,4 9,24 39,8

a) Rerata = =7,1

modus = 7

b) Kelas Q1 pada data ke = . 2 0 5, lihat frekuensi kumulatif, fk

= 7, berarti pada kelas dengan data 6, jadi Q1 = 6.

Kelas Q2 pada data ke = . 2 0 1 0, lihat frekuensi kumulatif,

fk = 14, berarti pada kelas dengan data 7, jadi Q2 = 7.

Kelas Q3 pada data ke = . 2 0 1 5, lihat frekuensi kumulatif,fk = 20 berarti pada kelas dengan data 9, jadi Q3 = 9.c) Jangkauan Data = 9 5 = 4d) Jangkauan Antar Kuartil = Q3 Q1 = 9 6 = 3

dan Simpangan Kuartil = (Q3 Q1) = .3 = 1,5.

e) Simpangan rata-rata, perhatikan tabel


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=, 1,14

f) Ragam, perhatikan tabel

=

, 1,99

dan Simpangan Baku = 1,99 1,41Soal 3): Analog dengan soal 2), maka

x xi f f.xi fk | | . | | )2 . )224 - 33 28,5 8 228 8 19,2 153,6 368,64 2949,12

34 - 43 38,5 12 462 20 9,2 110,4 84,64 1015,68

44 - 53 48,5 13 630,5 33 0,8 10,4 0,64 8,32

54 - 63 58,5 10 585 43 10,8 108 116,64 1166,4

64 - 73 68,5 7 479,5 50 20,8 145,6 432,64 3028,48

50 2385 60,8 528 1003,2 8168

a) Rerata = 47,7modus =

1

1 2

dL + c.

d + d

=1 1

43,5 + 10. 43,5 + 10. 43,5 + 2,5=461+ 3 4

= =

b) Kelas Q1 pada data ke = .5012,5, lihat frekuensi kumulatif,fk = 20, sehinggaq1

1 q1

q1

1n - fk

4Q = L + c.f

150 - 8

12,5 - 8433,5+ 10. 33,5+ 10.

12 12

= =

4533,5+ 33,5+ 3,75=37.25

12= =


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Kelas Q2 pada data ke = . 5 0 2 5, lihat frekuensi kumulatif,

fk = 33, sehingga

q2

2 q 2q2

2n - fk

4Q = L + c. f

250 - 20

25 - 20443,5+ 10. 43,5+ 10.13 13

= =

50

43,5 + 43,5 + 3,85 = 47,3513= =

Kelas Q3 pada data ke = .5037,5, lihat frekuensi

kumulatif, fk = 43, sehingga

q3

3 q3

q3

3n - fk

4Q = L + c.f

3 50 - 3337,5 - 33453,5+ 10. 53,5+ 10.

10 10

= =

4553,5 + 53,5 + 4,5 = 58,0

10= =

c) Jangkauan Data = 73 24 = 49

d) Jangkauan Antar Kuartil = Q3 Q1 = 58,0 37,25 = 10,75dan Simpangan Kuartil = (Q3 Q1) = .10,75 = 5,375.

e) Simpangan rata-rata, perhatikan tabel

= 10,56

f) Ragam, perhatikan tabel


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= = 163,36

dan Simpangan Baku = 163,36 12,7812362Dari pekerjaan di atas, nampaknya analisis akan lebih mudah

jika pengolahan data melalui tabel (tabelisasi), dan akan lebihterbantu dengan bantuan program excel.

Latihan Uji Kompetensi:1. Median dari nilai ulangan matematika suatu kelas yang disajikan pada

diagram di bawah adalah 1

2 1 1 2 3 5

2 2 2 3 4 5 6

4 0 1 2 2 4 5 6 8

5 3 3 4 4 5 7 9

6 5 5 7 8

7

2. Buatkan diagram kotak garis dari data5, 3, 4, 3, 6, 5, 7, 4, 6!

3. Rataan nilai ulangan 36 siswa adalah 6,5. Dua orang siswa ikut ulangansusulan. Jika rataan nilai ulangan sekarang menjadi 6,56, hitunglah

rataan nilai ulangan kedua siswa tersebut!4. Hitunglah nilai rataan hasi ulangan matematika pada tabel berikut!

Nilai f

50 57 258 65 6

66 73 11

74 81 7

82 89 3

90 95 1

5. Carilah modus pada histogram berikut!

109

8

6 65

44

3

22

1

040 45 55 60 65 70 75 80


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6. Kuartil atas dari data berikut adalah .Nilai f

30 37 5

38 45 7

46 53 10

54 61 14

62 69 12

70 77 9

78 88 3

fi = 60

7. Hitunglah desil ke-2 dari data di bawah!Nilai f

60 64 12

65 69 14

70 74 21

75 79 28

80 84 1985 89 15

90 94 11

fi =

8. Carilah simpangan kuartil dari data berikut!x 2 3 4 5 6 7 8 9 10

f 5 6 7 11 12 18 10 8 3

9. Hitunglah simpangan baku dari2, 3, 4, 5, 6, 7, 8, 9, 10.

10. Data berikut merupakan hasil ulangan umum matematika dari duakelas. Jumlah siswa yang nilainya 71,5 adalah .Nilai f

45 49 3

50 54 6

55 59 8

60 64 11

65 69 14

70 74 15

75 79 12

80 84 6

85 89 5fi =

Math for XI Science Temp1(2)

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