bahan skripsi ir

Reading is a complex cognitive process of decoding symbols in order to construct or derive meaning

(reading comprehension). It is a means of language acquisition, of communication, and of

sharinginformation and ideas. Like all language, it is a complex interaction between the text and the

reader which is shaped by the reader’s prior knowledge, experiences, attitude, and language community

which is culturally and socially situated. The reading process requires continuous practice, development,

and refinement.

Readers use a variety of reading strategies to assist with decoding (to translate symbols into sounds or

visual representations of speech) and comprehension. Readers may use morpheme, semantics,syntax and

context clues to identify the meaning of unknown words. Readers integrate the words they have read into

their existing framework of knowledge or schema (schemata theory).

Other types of reading are not speech based writing systems, such as music notation or pictograms. The

common link is the interpretation of symbols to extract the meaning from the visual notations.

A. Metode dan Instrumen Pengumpulan Data

Dari arti kata kedua istilah tersebut segera dapat dikemukakan pengertiannya demikian:

"Metode pengumpulan data adalah cara-cara yang dapat digunakan olph peneliti untuk mengumpulkan data"

"Cara" menunjuk pada sesuatu yang abstrak, tidak dapat diwujudkan dalam benda yang kasat mata, tetapi hanya dapat dipertontonkan penggunaannya. Terdaftar sebagai metode-metode penelitian adalah: angket (questionnaire), wawancara atau interviu (interview), pengamatan (observation), ujian atau tes (test), dokumentasi (documentation), dan lain sebagainya.

2.  Instrurnen pengumpulan data adalah alat bantu yang dipilih dan digunakan oleh peneliti dalam kegiatannya mengumpulkan agar kegiatan tersebut menjadi sistematis dan dipermudah olehnya.

"Instrumen penelitian" yang diartikan sebagai "alat bantu" merupakan saran yang dapat diwujudkan dalam benda, misalnya angket (questionnaire), daftar cocok (checklist) atau pedoman wawancara (interview guide atau interview schedule), lembar pengamatan atau panduan pengamatan (observation sheet atau observation schedule) soal tes (yang kadang-kadang hanya disebut dengan "ter" saja, inventors(invertory), skala (scale), dan lain sebagainya.

Melihat daftar jenis-jenis metode dan daftar jenis-jenis instrumen tersebut diatas, terdapat istilah-istilah yang sama, yaitu angket dan tes. Dengan demikian ada metode angket dan instrumen angket. Demikian juga ada metode tes dan instrumen tes. Memang instrumen angket digunakan sebagai alat bantu dalam penggunaan metode angket; demikian juga halnya dengan tes. Namun ada kalanya peneliti memilih metode angket tetapi menggunakan daftar cocok sebagai instrumen.

Menurut pengertiannya, angket adalah kumpulan dari pertanyaan yang diajukan secara tertulis kepada seseorang (yang dalam hal ini disebut responden), dan cara menjawab juga dilakukan

dengan tertulis. Daftar cocok, menunjuk pada namanya, merupakan kumpulan dari pernyataan atau pertanyaan yang pengisiannya oleh responder dilakukan dengan memberikan tanda centang atau tanda cocok (ü) pada tempat-tempat yang sudah disediakan. Jadi "daftar cocok" sebenarnya merupakan semacam angket juga tetapi cara pengisiannya dengan memberikan tanda cocok itulah yang menyebabkan ia disebut demikian.

Instrumen merupakan alat bantu bagi peneliti di dalam menggunakan metode pengumpulan data. Dengan demikian terdapat kaitan antara metode dengan instrumen pengumpulan data. Pemilihan satu jenis metode pengumpulan data kadang-kadang dapat memerlukan lebih dari satu jenis instrumen. Sebaliknya satu jenis instrumen dapat digunakan untuk berbagai macam metode.

Jika daftar metode dan daftar instrumen tersebut dipasangkan, akan terlihat kaitan dalam tabel berikut ini.

Tabel 1. Pasangan Metode dan Instrumen Pengumpulan Data

No. Jenis Metode Jenis Instrumen

1 Angket (questionnaire) Angket (questionnaire)

Daftar cocok (checklist)

Skala (scala), inventori (inventory)

2 Wawancara (interview) Pedoman wawancara (interview guide)

Daftar cocok (checklist)

3 Pengamatan/Observasi (Observation)

Lembar Pengamatan, panduan pengamatan, panduan observasi (observation sheet, observation schedule), (checklist).

4 Ujian/Tes (test) Soal ujian, soal tes atau tes (test), inventori(inventory).

5 Dokumentasi Daftar cocok (checklist)

Tabel

Dari tabel tersebut dapat diketahui bahwa:

1. Inventors dapat digunakan sebagai angket (tidak digunakan untuk mengetahui sesuatu yang sifatnya "ketat" seperti tes, (misalnya angket minat) tetapil ada yang berkedudukan seperti tes.

2. Daftar cocok (checklist) dapat digunakan dalam berbagai metode, karena nama "daftar cocok" lebih menunjuk pada cara mengerjakan dan wujud tampiIan instrumen dibandingkan dengan jenis instrumen sendiri.

Mengenai jenis-jenis instrumen yang disebutkan di atas, penulis yakin bahwa para pembaca telah mengenalnya. Dalam buku-buku penelitian sudah banyak diuraikan. Meskipun demikian untuk memperoleh penjelasan menyeluruh tentang metode dan instrumen pengumpul data ini, dalam bagian berikut diberikan sekadar gambaran singkat tentang pengertian dan contoh-contoh instrumen terutama dalam mengenai persamaan dan perbedaannya.

1. Angket

Angket, seperti telah dikemukakan pengertiannya di atas, merupakan daftar pertanyaan yang diberikan kepada orang lain dengan maksud agar orang yang yang diberi tersebut bersedia memberikan respons sesuai dengan permintaan pengguna. Orang yang diharapkan memberikan respons ini disebut responden. Menurut cara memberikan respons, angket dibedakan menjadi dua jenis yaitu: angket terbuka dan angket tertutup.

a.  Angket terbuka

adalah angket yang disajikan dalam bentuk sedemikan rupa sehingga responden dapat memberikan isian sesuai dengan kehendak dan keadaannya.

Angket terbuka digunakan apabiia peneliti belum dapat memperkirakan atau menduga kemungkinan altematif jawaban yang ada pada responden.

Contoh pertanyaan angket terbuka:

Penataran apa saja yang pernah Anda ikuti yang menunjang tugas Anda mengajarkan bidang studi yang sekarang Anda ajarkan? Tuliskan apa, di mana, dan berapa lama!

Jawab:

No. Jenis Penataran        Tempat Penataran Berapa Hari

1. ...............................   ...............................   .......................   

2. ............................... ............................... ....................... 

3. ............................... ...............................  .......................

4. dan seterusnya kira-kira 5-7 nomor

Menggali informasi mengenai identitas responden biasanya dilakukan dengan membuat pertanyaan terbuka. Keuntungan pertanyaan terbuka terdapat pada dua belah pihak yakni pada responden dan pada peneliti:

(1).  Keuntungan pada responden: mereka dapat mengisi sesuai dengan keinginan atau keadaannya.

(2).  Keuntungan pada peneliti: mereka akan memperoleh data yang bervariasi, bukan hanya yang sudah disajikan karena sudah diasumsikan demikian.

 b.  Angket tertutup

adalah angket yang disajikan dalam bentuk sedemikian rupa sehingga responden tinggal memberikan tanda centang (x) pada kolom atau tempat yang sesuai.

Contoh pertanyaan angket tertutup:

1)  pernahkan Anda memperoleh penataran yang menunjang tugas Anda mengajarkan bidang studi yang sekarang Anda ajarkan?

Jawab: ..................................  ....a. Pernah ....b. Tidak

1. Jika pernah, penataran tentang apa saja? (dapat memberikan centang lebih dari satu)

....a.    materi bidang studi

....b.    metode mengajar/strategi belajar-mengajar

....c.    memilih dan penggunaan media/alat pelajaran

....d.    menyusun alat evaluasi

c. Angket campuran

yaitu gabungan antara angket terbuka dan tertutup.

Contoh pertanyaan angket campuran:

1)  Pernahkah Anda memperoleh penataran yang menunjang tugas Anda mengajarkan bidang studi yang sekarang Anda ajarkan? Jika pernah berapa kali?

....a.    Tidak pernah (langsung ke nomor 3)

....b.    Pernah, yaitu ...kali (teruskan nomor 2)

2)  Penataran tentang apa saja yang Anda ikuti dan berapa hari lamanya?

1. Materi pelajaran                                           .....hari2. Metode mengajar                                         .....hari3. Pemilihan dan penggunaan media                  .....hari4. Penyusunan alat evaluasi                               .....hari

2. Daftar Cocok (Checklist)

Di dalam penjelasan mengenai angket dikemukakan juga bahwa dalam mengisi angket tertutup responden diberi kemudahan dalam memberikan jawabannya. Di lain tempat, yakni di dalam penjelasan umum mengenai instrumen disebutkan bahwa daftar cocok adalah angket yang dalam pengisiannya responden tinggal memberikan tanda cek (ü). Dengan keterangan tersebut tampaknya angket tertutup dapat dikategorikan sebagai checklist. Namur demikian angket bukan khusus merupakan daftar. Daftar cocok mempunyai pengertian tersendiri. Daftar cocok bukanlah angket. Daftar cocok mempunyai bentuk yang lebih sederhana karena dengan daftar cocok peneliti bermaksud meringkas penyajian pertanyaan Berta mempermudali responden dalam memberikan respondennya. Daftar cocok memuat beberapa pertanyaan yang bentuk dan jawabannya seragam. Agar responden tidak diharapkan pada beberapa pertanyaan mengenai berbagai hal tetapi dalam bentuk membaca, maka disusunlah daftar cocok tersebut sebagai pengganti.

Contoh:

Berikan tanda silang tepat pada kolom yang menunjukkan kebiasaan Anda melakukan pekerjaan di rumah yang tertera di bawah ini.

No. Jenis kegiatan di rumah Dikerjakan oleh Anda

Dikerjakan bersama

Dikerjakan pembantu

1. Menyiapkan makan pagi      

2. Membersihkan rumah      

3. Mencuci pakaian sendiri      

4. Mencuci sprei, korden, dan seterusnya.

5. Mencuci alat-alat makan ...dan seterusnya

Dari contoh di atas dapat diketahui bahwa variasi jawaban yang harus diberikan oleh responden hanya empat macam yakni:. "Dikerjakan oleh Anda", “Dikerjakan bersama", dan "Dikerjakan pembantu". Dengan daftar cocok ini barang kali peneliti hendak mengungkap seberapa besar tanggung jawab responden terhadap pekerjaan di dalam rumah tangga. Jika pertanyaan dan alternatif jawaban tersebut disajikan dalam bentuk angket, alternatif jawaban hanya tiga macam itu akan disebutkan secara berulang-ulang dengan bentuk dan isi yang sama. Daripada memakan tempat padahal responden sudah tahu (dan hafal!) apa yang harus dipilih maka altematif tersebut disingkat dalam bentuk kolom-kolom yang apabila sudah diisi oleh responden terlihat adanya daftar tanda centang yang disebut daftar cocok. Istilah "daftar cocok" juga dapat datang dari apa yang diharapkan dari responden, yakni memberi tanda cocok atau tanda centang pada daftar pernyataan yang disediakan.

3. Skala (scale)

Skala menunjuk pada sebuah instrumen pengumpul data yang bentuknya seperti daftar cocok tetapi alternatif yang disediakan merupakan sesuatu yang berjenjang. Di dalam Encyclophedia of Educational Evaluation disebutkan: The term scale in the measurement sense, comes from the Latin word scale, meaning "ladder" or “flight of stairs". Hence, anything with gradation can be thought of as "scaled". 

Contoh:

Peneliti ingin mengungkapkan bagaimana seseorang mempunyai sesuatu kebiasaan. Alternatif yang diajukan berupa frekuensi orang tersebut dalam melakukan suatu kegiatan. Gradasi frekuensi dibagi atas: "Selalu", "Sering",. "Jarang", "Tidak pernah". Skala yang diberikan kepada responden adalah sebagai berikut:

No. Jenis kegiatan di rumah Selalu Sering Jarang Tidak Pernah

1. Bangun sebelum jam 5 pagi        

2. Menyiapkan makan pagi        

3. Membersihkan rumah        

4. Mencuci pakaian sendiri        

5. Mencuci perabot rumah tangga... dan seterusnya

Skala banyak digunakan untuk mengukur aspek-aspek kepribadian atau aspek kejiwaan yang lain. Selain skala, penelitian yang berhubungan dengdn aspek-aspek kejiwaan memerlukan jenis instrumen-instrumen pengumpul data lain, baik yang berupa tes, inventori untuk hal-hal umum (general inventories, misalnya Minnesota Multiphasic Personality Inventory - MMPI, dan inventori untuk aspek-aspek khusus (Specific Inventories seperti: Rokeach Dogmatism Scala, Fundamental Interpersonal Relations Orientation - Behavior - FIRO - B, Study of Values, dan lain-lain). Untuk penelitian pendidikan, walaupun dapat dikatakan tidak terlalu sering menggunakan instrumen-instrumen seperti disebutkan, tetapi bagi penelitinya perlu juga mengenal ragam alat pengumpul data aspek-aspek psikologi tersebut.

Problematika pendidikan seperti kerancuan dalam mengikuti pelajaran, lambatnya siswa menyelesaikan studi serta masalah-masalah yang berhubungan dengan proses belajar, menjadi topik yang tetap aktual di kalangan pendidikan sekolah formal. Selain penelitian yang tidak terlalu menyangkut aspek-aspek kejiwaan secara langsung, masih banyak problem pendidikan yang terkait dengan aspek kejiwaan tersebut, misalnya rendahnya prestasi disebabkan rendahnya harga diri siswa. Lemahnya semangat belajar dikarenakan adanya lesu kreativitas dan seterusnya. Itulah sebabnya dalam bagian ini akan disajikan pula beberapa contoh instrumen untuk mengungkap aspek-aspek kejiwaan agar para peneliti pendidikan dapat terperinci menggali penyebab timbulnya masalah pendidikan melalui aspek kejiwaan siswa dan guru yang terlibat di dalam kegiatan pendidikan tersebut. Namun demikian untuk dapat menggunakan alat-alat pengungkap gejala kejiwaan seperti tes, inventori khusus dan lain-lain, diperlukan suatu kemampuan khusus. Pada umumnya mahasiswa lulusan faktultas Psikologi dapat diminta untuk membantu melaksanakan pengumpulan data yang diungkap melalui instrumen-instrumen tersebut.

Skala seperti dicontohkan di atas merupakan skala bentuk gradasi dari satu jenis kualitas. Dalam contoh di atas, alternatifnya ada empat sehingga terdapat empat tingkatan kualitas kes eringan. Skala yang berasal dari ide yang dikemukakan oleh Likert dan dikenal dengan skala Likert ini biasanya menggunakan lima tingkatan. Tentu saja peneneliti dapat membuat variabel dengan menyingkat menjadi tiga tingkatan:

Selalu          -  Kadang-kadang       - Tidak Pernah

Baik             -  Cukup                   - Jelek

Besar           -  Sedang                  -  Kecil

Jauh            -   Cukup                  -  Dekat

dan dapat pula memperbesar rentangan menjadi lima tingkatan:

Selalu          - Sering Sekali     -  Sering      - Jarang   - Jarang Sekali

Selalu          -  sering sekali      -  Sering      -  Jarang   - Tidak Pernah

Baik Sekali    - Baik                -  Cukup      -  Jelek      -  Jelek Sekali

Besar Sekali  - Besar              -   Cukup       -  Kecil      - Kecil Sekali

Misalnya:

Sangat setuju Setuju Abstain Tidak Setuju Sangat Tidak Setuju

(SS) (S) (A) (TS) (STS)

Pemilihan alternatif diserahkan pada keinginan dan kepentingan peneliti yang menciptaka instrumen tersebut. Ada Jenis lain yang telah dikembangkan oleh Inkels, bukan menyajikan alternative jenjang kualitas untuk sesuatu predikat, tetapi jenjang dari kualitas mini suatu perbuatan. Bentuk skala model. indeks ini menyerupai tes objektif bentuk pilihan ganda, tetapi alternatifnya menunjuk pada gradasi.

Langkah-Langkah Dalam Menyusun Instrumen

Secara umum penyusunan instrumen pengumpul data dilakukan dengan penahapan sebagai berikut:

1.  Mengadakan identifikasi terhadap variabel-variabel yang ada di dalam rumusan judul penelitian atau yang tertera di dalam problematika penelitian.

2.  Menjabarkan variabel menjadi sub atau bagian variabel.

3.  Mencari indikator setiap sub atau bagian variabel.

4.  Menderetkan deskriptor dari setiap indikator.

5.  Merumuskan setiap deskriptor menjadi butir-butir instrumen.

6.  Melengkapi instrumen dengan (pedoman atau instruksi) dan kata pengantar.

Quantitative Methods in Education Research

Dr Ulrike Hohmann

Originally prepared by Professor John Berry© J Berry, Centre for Teaching Mathematics, University of

Plymouth, 2005

(links updated August 2006)

CONTENTS

A.       INTRODUCTION B.       QUANTITATIVE AND QUALITATIVE RESEARCH C.       INGREDIENTS OF QUANTITATIVE RESEARCH D.       STATISTICS E.       STATISTICAL CONCEPTS & QUANTITATIVE

PROCEDURES F.       TASKS G.       FURTHER READING

H.       WEBSITES

A.    INTRODUCTION

This component is unable to do more than help you to begin thinking about quantitative methods in educational research. Its aim is to give you an insight into the issues should you choose quantitative methods as part of your research methodology.

We will briefly address the following questions:-

What are quantitative methods? What are the ingredients of quantitative methods? How do you go about research design?

Education research has moved away largely from the numbers approach in recent years, and the emphasis has been on qualitative methods. However, the use of numbers can be a very useful tool, either as part of a larger project that employs many different methods or as a basis for a complete piece of work. With the use of sophisticated software packages such as SPSS it is relatively easy to deal with the computation side of things and it is possible to come up with numerous tables and charts almost instantly once your data is installed. However, it is very important that the underlying principles of statistical analysis are understood if sense is to be made of the results spewed out by such a package in terms of your research.

This component consists of two sections; we begin with an overview of quantitative methods and finish with a brief introduction to some of the basic statistical concepts to be looking for when you read research papers that use quantitative methods of research.

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B.     QUANTITATIVE AND QUALITATIVE RESEARCH

In simple terms we can think of two approaches to investigations in educational research: qualitative and quantitative. In the former we use words to describe the outcomes and in the latter we use numbers.

Quantitative research methods were originally developed in the natural sciences to study natural phenomena. However examples of quantitative methods now well accepted in the social sciences and education include:

surveys ; laboratory experiments; formal methods such as econometrics: numerical methods such as mathematical modelling.

Qualitative research methods were developed in the social sciences to enable researchers to study social and cultural phenomenon. Examples of qualitative methods include:

action research aims to contribute both to the practical concerns of people in an immediate problematic situation and to the goals of social science by joint collaboration within a mutually acceptable ethical framework;

case study  research - a case study is an empirical enquiry that investigates a contemporary phenomenon within its real-life context;

ethnography - the ethnographer immerses her/himself in the life of people s/he studies and seeks to place the phenomena studied in its social and cultural context.

Other components of this module cover various qualitative research methods.

Structure of Research Papers

When setting out on educational research you will be (have been) encouraged by your supervisor to read appropriate publications and this is a good way of identifying the methods of research that seem most used in your research area. A typical structure for a research paper is summarised in the table below:

literature survey other people’s work

methodology qualitative or quantitative

results your work

discussion/conclusions your discussion and reference to others

Table 1 Structure of typical research paper

Activity 1

Scan read the following three papers:

Mark Cosgrove: A study in science-in-the-making as students generate an analogy for electricity.   International Journal of Science Education . 17 No 3, pp 295-310, 1995

Susan Picker: Using Discrete Mathematics to give Remedial Students a Second Chance.   DIMACS , 36, pp 35-41, 1997

John Berry and Pasi Sahlberg: Investigating Pupils' Ideas of Learning.   Learning and Instruction.   6 No 1, pp 19-36, 1996

Identify the research method being used in each paper.   Answer the question 'Why use numbers in education research?' with reference to these examples.

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C.     INGREDIENTS OF QUANTITATIVE RESEARCH

As part of your research you will be looking at certain characteristics (variables) and endeavouring to show something interesting about how they are distributed within a certain population. The nature of your research will determine the variables in which you are interested. A variable needs to be measured for the purpose of quantitative analysis.

We may collect data concerning many variables, perhaps through a questionnaire, or choose to measure just two or several variables by observation or testing. The variables we are interested in may be dependent or independent. There will be other features present in the problem that may be constant or confounding.

Using the data that you have collected then you can:

Describe variables in terms of distribution: frequency, central tendency and measures and form of dispersion. Descriptive statistics include averages, frequencies, cumulative distributions, percentages, variance and standard deviations, associations and correlations. Variables can be displayed graphically by tables, bar or pie charts for instance. This may be all the statistics you need and you can make deductions from your descriptions. In fact univariate(one variable) analysis can only be descriptive. But descriptive statistics can be used to describe a significant relationship between two variables (bivariate data) or more variables (multivariate). 

Infer significant generalisable relationships between variables. The tests employed are designed to find out whether or not your data is due to chance or because something interesting is going on.

See the section on Variablesin The Research Methods Knowledge Base.

Often it is not possible to undertake a true experiment as part of your research and a common research approach in educational research is called  quasi-experimental design  represented in the following diagram:

Experimental Group O1 X O2

Control Group O3   O4

In this figure O1 and O3 represent initial testing of the two groups; X represents some intervention or experimentation strategy with one of the groups and O2 and O4 represent final testing of the two groups. We would use the test results to investigate whether the experimental teaching approach has led to an improvement in the feature being tested.

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D.    STATISTICS

Perhaps the best way to begin to appreciate the kind of statistics that you might employ in your own research is to have a look at what others have done.

Read the paper by  John Berry and Pasi Sahlberg: Investigating Pupils' Ideas of Learning.   Learning and Instruction.   6 No 1, pp 19-36, 1996  and attempt to identify the statistics that are used.

The section ‘Findings’ on pages 28 – 32 contains some of the important measures that we use in quantitative research methods. The mean and standard deviation tell us about the average and spread of the data. The bar graphs on page 31 allow a visual comparison of the means between the two groups of samples.

The symbol ‘p’ represents the probability of a significant difference between the two groups. This is probably the most difficult concept to grasp because in some senses it is counter intuitive. The probability of an event happening lies between 0 and 1. A large probability (i.e. p close to 1) implies a high likelihood of the event happening. For example if you are told that there is a 95% chance of winning a game (p = 0.95) then put your money on winning! On the other hand if there is a 5% chance of you winning (p = 0.05) it’s probably best not to bet on yourself!

So if p is small (close to 0) the event is less likely to happen than if p is large (close to 1). Let’s see what this means in the context of educational research. As an example look at page 31 Berry and Sahlberg. If we compare the means of the scores of the UK and Finland pupils for the statement "I learn better by doing work by myself than by watching the teacher", then there is a probability of less than 0.001 (i.e. 0.1%) that the difference in the means will occur by chance. In ordinary language the probability of it happening by chance is so small that we say it is a significant result. Because it is unlikely to occur, the reason that it does is significant.

In our analysis we look for small probabilities usually less than 5% or p = 0.05. Then we say that the result is significant at the 5% level of significance.

There is also evidence from Figure 3 that there is some difference between the UK and Finland pupils. For Q2 the UK rating is positive and the Finland rating is negative.

Very often a comparison of the means as in Figure 3 is a clue. For example look at Q7 in Figure 3. For the UK pupils the rating is positive whereas for the Finland pupils it is negative. The ‘p – value’ for this feature is less than 0.001 (0.1%) and so we conclude that there is a significant difference between the two groups of pupils for the statement "I like most teachers in my school".

One of the first steps in the design of a piece of quantitative research is setting up what is called a hypothesis. For example, we might propose the hypothesis that there is no difference between the UK pupils and Finnish pupils views of their teachers (item 7 in Berry and Sahlberg). This is

called the null hypothesis. We then need to use the pupil responses to try to disprove this hypothesis.

To be able to compare quantities we need to define a statistic whose distribution is known. In the paper by Berry and Sahlberg the t-statistic is used as a measure of the difference between the means of the two groups of pupils. Having calculated the value of the t-statistic for the feature under investigation we then look up in tables the probability of the feature occurring. (At this stage don’t worry about how it is calculated!) You can see on page 31 the t-statistic and its associated p-value.

At the heart of quantitative research methods is some very sound statistical theory. If you are planning to carry out a research investigation using quantitative research methods you do not need a thorough grounding in this theory but you will need an understanding of the statistical methods. We use statistical software packages to do the arithmetical calculations so the important skill is not doing the mathematics but is interpreting the results.

In what follows we have gathered together some of the essential statistical ideas needed for quantitative research. It is a summary with some examples to provide a flavour of the ideas.

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E.     STATISTICAL CONCEPTS & QUANTITATIVE PROCEDURES

NB   What follows is merely an introductory overview of some of the relevant concepts and procedures.  To find out more go first to a general textbook such as Denscombe (1998), Chapter 10, and then, for a much fuller account, tryPeers (1996).

1.    Variables

numerical measurements: person’s age or weight; size of a family

non-numerical measurements: position on a scale indicating a level of agreement e.g. Likert rating scale

continuous data: measurement that can, in principle, take any value within a certain range e.g. time, age and weight

categorical data: (or discrete data): measurements that can take only known discrete values e.g. the number of rooms in a house, the number of children in a family

o nominal data: numerical values are assigned to categories as codes e.g. in coding a questionnaire for computer analysis, the response ‘male’ might be coded as ‘1’; and ‘female’ as ‘2’. No mathematical analysis is usually possible and no ordering is implied.

o ordinal data: numerical values are assigned in accordance with a qualitative scale e.g. in coding a questionnaire for computer analysis, the responses ‘very good’, ‘good’, ‘poor’ and ‘very poor’ are coded ‘4’, ‘3’, ‘2’ and ‘1’ respectively.

See also the section on  Variables  in The Research Methods Knowledge Base.

2.    Basic Measures

mean: is a measure of the central location or average of a set of numbers, e.g. the mean of 2 7 2 1 8 2 6 9 10 5 1 4 is 4.75

standard deviation: is the square root of the variance!!

variance:   is a measure of dispersion (or spread) of a set of data calculated in the following way:   

median: is the centre or middle number of a data set, e.g. the median of 2 7 2 1 8 2 6 9 10 5 1 4 is 4.5

quartiles: divide a distribution of values into four equal parts. The three corresponding values of the variable are denoted by Q1, Q2 (equal to the median) and Q3

range: is a measure of dispersion equal to the difference between the largest and smallest value.

3.    Frequency Distribution

Example A

A new hybrid apple is developed with the aim of producing larger apples than a particular previous hybrid. In a sample of 1000 apples, the distribution of weights was as follows:

Weight (g) 0-50 50-100 100-150

150-200

200-250

250-300

300-350

350-400

frequency 20 42 106 227 205 241 106 53

1. Apples can only be sold to a particular supermarket with a weight greater than 150g. What proportion of the new hybrid apples would be rejected by this supermarket.

2. How many grams above this weight of 150 g is the mean weight of apples?3. What is this difference in weights in units of the standard deviation of apple weights?

Suppose that we graph the data using columns to show the amount in each group. We get a frequency distribution.

From the data there are 168 apples whose weight is less than 150 g and 832 apples whose weight is greater than 150 g. There are 1000 apples altogether.

We can deduce that the proportion  = 16.8 % cannot be sold to the supermarket.

The mean weight of the apples is 223.35 g and the standard deviation is 78.9 g.

The difference in weight between 150 g and the mean is 73.35 g and this is   of a standard deviation.

4.    Measures of Location and Dispersion

A distribution is symmetrical if the difference between the mean and the median is zero. An appropriate pictorial representation of the data, (histogram, stem and leaf diagram etc.) would produce a mirror image about the centre:

A distribution is positively skewed (or skewed to the right) if the mean - median is greater than zero. Such data when represented by a histogram would have a right tail that is longer than the left tail:

A distribution is negatively skewed (or skewed to the left) if the mean - median is less than zero. Such data when represented by a histogram would have a left tail that is longer than the right tail:

If data are skewed then the best measure of location is the median and the best measure of dispersion is the interquartile range.

If data are symmetrical then the best measure of location is the mean and the best measure of dispersion is the standard deviation or variance.

5.    Probability

This is an important concept in statistics and is an important part of our story.

It is defined in the following way: if an experiment has n equally likely outcomes and q of them are the event E, then the probability of the event E, P(E), occurring is

P(E) = 

Some simple examples:

the probability of getting a head from the toss of a fair coin is 

the probability of getting a six from the throw of a fair die is 

the probability of getting the ace of spades from cutting of a pack of cards is 

The smaller the probability the more unlikely the event is to happen. This is an important concept in quantitative methods in education research as we shall see.

There is an important link between probability and the frequency distribution. Consider again the hybrid apple example above. We saw that the proportion of apples weighing less than 150 grams was 0.168. If we pick up one of the apples ‘at random’ then it could weigh less than 150 g and be rejected or it could weigh more than 150 g and be accepted by the supermarket.

The probability of picking such an apple is 0.168.

Exercise

To illustrate this idea further complete the following:

the probability of picking an apple in the weight range 200-250 g is the probability of picking an apple in the weight range 350-400 g is the probability of picking an apple with a weight greater than 300 g is

The important idea here is that the probability is associated the amount of data under the distribution graph.

6.    Testing an hypothesis

There are two basic concepts to grasp before starting out on testing an hypothesis.

Firstly, the tests are designed to disprove hypotheses. We never set out to prove anything; our aim is to show that an idea is untenable as it leads to an unsatisfactorily small probability.

Secondly, the hypothesis that we are trying to disprove is always chosen to be the one in which there is no change. For example there is no difference between the two population means.

This is referred to as the null hypothesis and is labelled H0. The conclusions of a hypothesis test lead either to acceptance of the null hypothesis or its rejection in favour of the alternative hypothesis H1.

Hypothesis testing: a hypothesis test or significance test is a rule that decides on the acceptance or rejection of the null hypothesis based on the results of a random sample of the population under consideration.

step 1: Formulate the practical problem in terms of hypotheses. The null hypothesis needs to be very simple and represents the status quo, i.e. there is no difference between the processes being tested.

step 2. Calculate a statistic that is a function purely of the data. All good statistics should have two properties: (i) they should tend to behave differently when H0 is true from when H1 is true; and (ii) its probability distribution should be calculable under the assumption that H0 is true.

step 3: Choose a critical region. We must be able to decide on the kind of values of the test statistic, which will most strongly point to H1 being true rather than H0. The value of the test statistic in this critical region will lead us to reject H0 in favour of H1; otherwise we are not able to reject H0 in favour of H1. We should never conclude by accepting H0.

step 4: Decide the size of the critical region. i.e. a 1% probability of H0 being rejected etc.

For more on this see the section on Hypotheses in The Research Methods Knowledge Base.

Example B

In an educational research programme two groups of students are taught a topic in different ways. An experimental group uses a spreadsheet to explore the topic and a control group uses a more traditional pen and paper activity. Each group contains 20 students. At the end of the topic the teacher tests the two groups on their understanding of the topic and obtains the following data:

Experimental 5 11 25 33 35 40 45 46 52 55

  56 56 57 59 69 74 75 89 92 97

Control 33 39 44 45 45 46 47 48 49 49

  53 54 54 55 58 60 61 63 65 69

Extra data:

  mean standard deviation

Experimental 53.6 25.0

Control 51.8 9.1

Experimental Group                                 Control Group

How would you interpret these findings?

Some analysis

The researcher might be tempted to conclude that the experiment has had little or no effect on the performance of the experimental group as judged by the means. However the large difference in standard deviations might suggest that the experimental group is much more variable in performance than the control group.

The researcher might also be tempted to deduce that the experiment has turned some of the pupils off the task. Look at the three low scores!

Suppose that we investigate the difference in the means: Let

H0: there is no difference between the means of the two groups: m 1 = m 2

H1: the score of the experimental group is greater than the score of the control group:

m 1 > m 2

We use a two-sample t-test to get

The p value is 0.39 (39%) so we deduce that there is not enough evidence to reject the null hypothesis. The researcher could not deduce that there was an improvement in student performance.

7.    Statistical tests

t tests

In hypothesis testing, the t test is used to test for differences between means when small samples are involved. (n £ 30 say). For larger samples use the z test.

The t test can test

i) if a sample has been drawn from a Normal population with known mean and variance. (Single sample)

ii) if two unknown population means are identical given two independent random samples. (Two unpaired samples)

iii) if two paired random samples come from the same Normal population. (Two paired samples (paired differences))

Any hypothesis test can be one tailed or two tailed depending on the alternative hypothesis, H1.

Consider the null hypothesis, H0: m =3

A one tailed test is one where H1 would be of the form m > 3.

A two-tailed test is one where H1 would be of the form m ¹ 3.

Click here for more information on t-tests.

Single sample test

Let X1, X2, ¼ , Xn be a random sample with mean   and variance s2. To test if this sample comes from a Normal population with known mean m and unknown variance s2, the test statistic

is used to test the null hypothesis H0: the population mean equals m. If the test statistic lies in the critical region whose critical values are found from the distribution of Tn, a, H0 is rejected in favour of the alternative hypothesis H1.n are the degrees of freedom and for a single sample test n = n-1, and a is the significance level of the test.

Two unpaired samples

Let X1, X2, ¼ , Xm be a random sample with mean   and variance sx2 drawn from a Normal population

with unknown mean mx and unknown variance sx2.

Let Y1, Y2, ¼ , Yn be a random sample with mean   and variance sy2 drawn from a Normal population with

unknown mean my and unknown variance sy2.

To test the null hypothesis that the two unknown population means are the same we use the test statistic

where   , the estimate of the common population standard deviation.

The test statistic T is distributed Tn, where n =(m-1)+(n-1) for two unpaired samples. If the test statistic lies in the critical region whose critical values are found from the distribution of Tn, a, H0 is rejected in favour of the alternative hypothesis H1.

Example C

It is claimed that the concentration period of students doing a particular task is normally distributed with mean 44mins. A sample of 21 students were taken, and their concentration period measured. The mean time of the sample was found to be 42mins and the sample variance was calculated to be 36min. Is there any evidence at the 5% level of significance against the claim that the population mean is 44min?

Solution

Here m  = 44,   = 42, s = Ö 36 = 6 and n = 21.

This is a two-tailed test since we are looking for any difference.

H0: m  = 44

H1: m  ¹ 44

Since p = 0.1423 = 14.23% there is insufficient evidence to reject the null hypothesis. We therefore conclude that the population mean concentration period is 44 minutes.

Example D

A researcher investigating the effects of pollution on two rivers takes an independent random sample of fish of a certain species from each river, measures their mass in ounces and obtains the following results.

Test at the 5% level of significance if there is any evidence of a difference in the mean weight between the two rivers.

Solution

Assume that each sample is taken from a normal population.

Here m = 6,   = 13.667, sx2 = 23.556

River 1 20 10 17 7 10 18      

River 2 14 6 10 8 9 7 7 6 8

n = 9,   = 8.333, sy2 = 5.556.

Let m 1 be the population mean of river 1, and m 2 be the population mean of river 2.

H0: m 1 = m 2

H1: m 1 ¹ m 2

Since p = 0.043 = 4.3% < 5% we deduce that there is sufficient evidence to reject the null hypothesis at the 5% level of significance. We therefore conclude that the mean weight of fish in River 1 is not equal to the mean weight of fish in River 2.

Back to CONTENTS list

F.    TASKS

(NB: only for those University of Plymouth students undertaking the ‘Research in Education’ module as part of the preparation for the submission of a MA dissertation proposal)

Tasks, once completed, should be sent to [email protected], making clear:

which component it is from; which task it is (B or C); the name of your dissertation supervisor.

It will then be passed on to the component leader (and copied to your supervisor). The component leader will get back to you with comments and advice which we hope will be educative and which will help you in preparing your dissertation proposal once you are ready. (Remember that these tasks are formative and that it is the proposal which forms the summative assessment for the MERS501 (resined) module.) This email address is checked daily so please use it for all correspondence about RESINED other than that directed to particular individuals for specific reasons.

Before tackling either of the two tasks (B or C) you might want to consider ...

whether research has to be quantitative to be scientific?  See Cohen et al (2000), Chapter 1, and the section on Positivism and Post-Positivism in  Trochim (2000), examine the terms used in this question and discuss with reference to examples.

TASK B (DATA COLLECTION)

A research group suspects that girls and boys adopt different styles of working when using ICT in mathematics lessons. Outline a quantitative research method to investigate the effect of gender on the use of calculators on children’s understanding of mathematics.  What are your variables?  Formulate an experimental hypothesis for the research.What features might cause problems in your research?

TASK C (DATA ANALYSIS)

Twelve students fail an examination. After a period of revision and tutorial support they resit by taking a new examination. The marks for the original examination and the resit are shown in the table below.

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

Examination score 30 31 20 17 25 32 35 29 30 27 32 30

Resit score 42 38 30 21 40 45 31 32 38 50 34 40

What would you do with the data now?

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G.    FURTHER READING

Blaxter, I., Hughes C. and Tight M. (1996) How to Research.  Buckingham, Open University Press.

Bryman, A. and Cramer D. (1999) Quantitative Data Analysis with SPSS 8 Release for Windows: a guide for social scientists.  London, Routledge.

Cohen, L ; Manion, L & Morrison, K (2000) Research Methods in Education (5th edition). London, RoutledgeFalmer.

Denscombe, M. (1998) The Good Research Guide.   Buckingham, Open University Press.

Greenfield, Tony (ed) (1996) Research Methods – Guidance for Postgraduates. London, Arnold.

Kanji, Gopal (1993) 100 Statistical Tests. London, SAGE Publications.

Peers, Ian (1996) Statistical Analysis for Education & Psychology Researchers. London, Falmer.

Plewis, Ian (1997) Statistics in Education.  London, Arnold.

Robson,  C. (1990) Experiment, Design and Statistics in Psychology. Middlesex, Penguin Books.

Rose, D. and Sullivan, O. (1993) Introducing Data Analysis for Social Scientists.   Buckingham, Open University Press.

MORE ADVANCED READING

http://www.cf.ac.uk/socsi/capacity/References.html