Introduction to Quantitative Methods PDF

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MKU: Quantitative Methods 2018

BMCU002: QUANTIATIVE METHODS Brief Course Outline  Introduction to Statistics  Probability  Correlation and Regression Analysis  Statistical Inference  Times Series Analysis and Index Numbers TOPIC ONE: INTRODUCTION TO STATISTICS 1.1 Introduction Statistics Statistics is the science of collecting, organizing, presenting, analyzing and interpreting of data. This definition clearly points out four stages in a statistical investigation, namely: i. Collection of data ii. Organization and Presentation of data iii. Analysis of data iv. Interpretation of data Uses of Statistics a) To present the data in a concise and definite form: Statistics helps in classifying and tabulating raw data for processing and further tabulation for end users. b) To make it easy to understand complex and large data: This is done by presenting the data in the form of tables, graphs, diagrams etc., or by condensing the data with the help of means, dispersion etc. c) For comparison: Tables, measures of means and dispersion can help in comparing different sets of data. d) In forming policies: It helps in forming policies related to the education environment. e) Enlarging individual experiences: Complex problems can be well understood by statistics, as the conclusions drawn by an individual are more definite and precise than mere statements on facts. f) In measuring the magnitude of a phenomenon (occurrence):- Statistics has made it possible to count the population of a country, the industrial growth, the agricultural growth, the educational level (of course in numbers)

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Definitions: Statistics is the art and science of collecting, analyzing, presenting and interpreting data. A branch of mathematics taking and transforming numbers into useful information for decision makers. It refers to methods used for helping reduce the uncertainty inherent in decision making Data are the facts and figures that are collected, summarized, analyzed, and interpreted. Raw data refers to unprocessed or unorganized data. Data can be broadly classified as being qualitative or quantitative. Quantitative data indicate either how many or how much. Are countable or numerical. – Quantitative data that measure how many are discrete i.e. take specific values e.g. whole numbers

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MKU: Quantitative Methods 2018 Quantitative data that measure how much are continuous because there is no separation between the possible values for the data i.e. can take any value including fractions. Qualitative data are labels or names used to identify an attribute of each element. Are non numerical and therefore not countable. Qualitative data use either the nominal or ordinal scale of measurement. The statistical analyses for qualitative data are rather limited The statistical analysis that is appropriate depends on whether the data for the variable are qualitative or quantitative. Attribute: A characteristic of an elementary unit that can only be observed as to its presence or absence. Variable: An observable quantitative characteristic of an elementary unit that vary from unit to unit. Discrete Variable: A variable whose values are restricted to integer values only i.e. takes whole numbers e.g. no. of students Continuous Variable: A variable that can assume any value within some interval i.e. can take even fractions e.g. height or size of a building, measurements, weights, age Population: the entire possible observations that may be made in the universe Sample: Any portion drawn from a population. Generally a sample consist of a fewer elementary units or observations than contained in a population. Thus a sample is a sub set of a population. Elementary units: Physical entity on which an observation is made. Survey: A planned and Systematic process of collecting statistical data Census: A survey in which observations are made on every elementary unit of the whole population Sample Survey: A survey in which observations are made on a sample of elementary unit drawn from the population. –

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Types of statistical data Data

Qualitative (Categorical)

Quantitative (Numerical)

Discrete (takes only whole Nos.)

Continuous (takes whole Nos. & fractions )

Classification of Statistics Broadly classified into two categories: i. Descriptive statistics: Refers to the collection, analysis and synthesis of data in order to come up with a better description of the situation. It is a branch of statistical which is concerned with collecting, describing and summarizing a set of data so as to derive meaningful information. It

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ii.

involves classification of data, presentation of data in tabular forms, graphs, charts and calculation of averages. Inferential statistics: Divided into two: a) Inductive statistics: Is concerned with the development of scientific criteria so that values of a group may be meaningfully estimated by examining only a small portion of that group. The whole group is known as population or universe whle the portion is known as sample. Values in the samples are known as statistics and values in the population are known as parameters. Thus, inductive statistics is concerned with estimating universe parameters from the sample statistics. A sample is chosen instead of considering the whole population because of:  Time limit: using a census survey based on the entire universe requires a lot of time which might not be available.  Costs: A sample survey is much cheaper compared to a census survey  Volatility: since the census survey is time consuming, the relevance of the research may not apply by the time of finishing the research. b) Deductive statistics: it is concerned with the establishing of laws and procedures for choosing one course from alternatives courses of actions under situations of uncertainty. Since deductive statistics uses probability theory, it provides a rational base for dealing with situations influenced by chances related factors.

Types of Statistics

Descriptive statistics

Inferential Statistics

Inductive Inferential Statistics

Deductive Inferential Statistics

Scales of measurement Nominal Scale Nominal measurement consists of assigning items to groups or categories. No quantitative information is conveyed and no ordering of the items is implied. Nominal scales are therefore qualitative rather than quantitative e.g. Religious preference, race, and gender. Variables measured on a nominal scale are often referred to as categorical or qualitative variables. Ordinal Scale Measurements with ordinal scales are ordered in the sense that higher numbers represent higher values. However, the intervals between the numbers are not necessarily equal. For example, on a fivepoint rating scale measuring attitudes towards whether the quality of education offered in M.K.U. is of standard. The rating on scale could be I strongly agree, I agree, am neutral, I disagree or I strongly disagree. The difference between a rating of 2 and a rating of 3 may not represent the same difference

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MKU: Quantitative Methods 2018 as the difference between a rating of 4 and a rating of 5. There is no "true" zero point for ordinal scales since the zero point is chosen arbitrarily. Interval Scale On interval measurement scales, one unit on the scale represents the same magnitude on the trait or characteristic being measured across the whole range of the scale. Interval scales do not have a "true" zero point, however, and therefore it is not possible to make statements about how many times higher one score is than another Ratio Scale Ratio scales are like interval scales except they have true zero points.

1.2 Organizing and Presenting Data It’s hard to interpret raw data in its original form. Hence it is always important to organize the data in a systematic way. • Organizing data refers to arranging data:  according to similarity or resemblance  according to the order of importance  in the descending or ascending order • The purpose for organizing data: To make the data easily understandable  In order to make comparison and draw meaningful conclusion easily  To eliminate unnecessary data Statistical Series: Refers to different ways of arranging data. •

a. Time Series: This is arranging data according to when they occur. This can be in terms of hrs, days, months or years b. Spatial Series: This is arranging data according to their geographical characteristics. c. Conditional Series: This is arranging data according to their specific characteristics. E.g male or female. NOTE: Refer to class exercises for different methods of presenting data.

1.3 Measures of Central Tendency These are statistical values which tend to occur at the centre of any well ordered set of data. These measures are as follows: i. The arithmetic mean ii. The weighted average iii. The mode iv. The median v. The geometric mean vi. The harmonic mean 1.

The aarith rith rithme me metic tic mea mean n

This is commonly known as avera average ge or mea mean. n. It is obtained by summing up the values given and by dividing the total value by the total no. of items. [email protected]

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X n Where x = values of items ∑ = summation n = no. of observations or items Example The mean of 60, 80, 90, 120

Mean ( x )

=

60 + 80 + 90 +120 4

=

350 4

= 87.5

The arithmetic mean is very useful because it represents the values of most observations in the population. The mean therefore describes the population quite well in terms of the values attained by most of the members of the population. Note: Refer to class exercises for further understanding. Calculating Arithmetic Mean for grouped data

x Where f=frequency The following statistical terms are commonly used in statistical calculations. They must therefore be clearly understood. i) Class limits These are numerical values which give a lower and upper limit for any given class i.e. all the observations in a given class are expected to fall within the interval which is bounded by the class limits. ii) Class boundaries These are statistical boundaries, which separate one class from the other. They are usually determined by adding the upper class limit to the next lower class limit and dividing by 2 e.g. in the above table the 19+ 20 . class boundary between 19 and 20 is 19.5 which is = 2 iii) Class mid points These are very important values which mark the center of a given class. They are obtained by adding together the two limits of a given class and dividing the result by 2. [email protected]

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MKU: Quantitative Methods 2018

iv) Class interval/width This is the difference between an upper class boundary and lower class boundary. The value usually measures the length of a given class. Note: Refer to class exercises for further understanding. 2. The mo mode de -

The mode is defined as the value of item which is repeated more than any other in a series. Sometimes a single value may not exist as such in which case we may refer to the class with the highest frequency. Such a class is known as a modal cl class. ass.

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The mode can be important statistical value in education activities e.g. most students finish O-level at the age of 16yrs. The mode can easily be determined for ungrouped data by determining the one with the highest frequency.

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When determining the values of the mode from the grouped data we may use the following methods;i. The graphical method which involves use of the histogram ii. The computation method which involves use of formula Exa Exampl mpl mple e In a social survey in which the main purpose was to establish the intelligence quotient (IQ) of resident in a given area, the following results were obtained as tabulated below: IQ 1 – 20 20 – 40 40 – 60 60 – 80 80 – 100 100 – 120 120 – 140

No. of residents 6 18 32 48 27 13 2

Re Require quire quired d Calculate the modal value of the IQ’s tabulated above using the formula method and by graphical method. For Formula mula mula.. Firs irstt ide iden n tif tify y the mo modal dal class i.e i.e.. the clas classs with the h highe ighe ighest st frequ freque ency ncy.. The use the foll follo owin wing g form formula. ula.

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Whe Where re ll=lo =lo =low wer llimit imit of the mo mod dal group f1=fre =freque que quency ncy o off th the e cla class ss pre rece ce ced din ing g the m mo odal cl cla ass f2=fre =freque que quency ncy o off th the e cla class ss ffollow ollow ollowin in ing g the m mo odal cl clas as asss i=cla i=class ss iinterva nterva ntervall

Gra Grap phical m me etho thod d

50 40 30 20 10

20

40

60

80

100

120

140

Value of the mode Note: Refer to class exercises for further understanding. 3. The m me edian -

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Refers to the value of the middle item in a series when the data is arranged in ascending or descending order. eg 14, 17, 9, 8, 20, 32, 18, 14.5, 13. When the data is ordered it will be 8, 9, 13, 14, 14.5, 17, 18, 20, 32 The middle number/median is 14.5 The importance of the median lies in the fact that it divides the data into 2 equal halves. The no. of observations below and above the median are equal. When data is grouped the median may be determined by using the following methods

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i. ii.

Graphical method using the cumulative frequency curve (o give) The formula

Example Referring to the table below, determine the median using the methods above The gra graphica phica phicall me method thod IQ 0 – 20 20 – 40 40 – 60 60 – 80 80 – 100 100 – 120 120 – 140 UCL- Upper Class limit

No of residents 6 18 32 48 27 13 2

Cumulative Frequency 6 24 56 104 131 144 146

x 160 140 120 100 80 60 40 20

20 40

60

80

100

120

140

160

Value of the median

The position of the median =

n+1 146+1 = 2 2

Median position = 73.5 Median value = 67

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Where l=lower limit of the middle group M=middle item C= cummulatve frequency of the class before the middle group F=frequency of the middle group i=class interval of the middle group

Note: Refer to class exercises for further understanding. 4. Weighte eighte ghted d mea mean n It is an average used to show the degree of importance for the varying proportions or weights of items. Exa Exampl mpl mple es -

The following table shows that marks scored by a student doing in an exam. Subject Communication Maths Statistics Psychology Educational Management School Management

Scores (x) 65 63 62 80 69

Weight (w) 50 40 45 35 55

wx 3250 2520 2340 2800 3795

55

60

3300

w = 285

wx = 18005

Weighted mean

Meri Merits ts aand nd dem demer er erits its o off the diffe differe re rent nt mea measure sure suress o off cen central tral te tendenc ndenc ndency y The ar arithme ithme ithmetic tic me mean an (a.m (a.m)) Meri Merits ts [email protected]

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i. ii. iii. iv.

It utilizes all the observations given It is a very useful statistic in terms of applications. It has several applications in business management e.g. hypothesis testing, quality control e.t.c. It is the best representative of a given set of data if such data was obtained from a normal population The a.m. can be determined accurately using mathematical formulas

Dem Demer er erits its of the a. a.m. m. i. ii.

If the data is not drawn from a ‘normal’ population, then the a.m. may give a wrong impression about the population In some situations, the a.m. may give unrealistic values especially when dealing with discrete variables e.g. when working out the average no. of children in a no. of families, it may be found that the average is 4.4 which is unrealistic in human beings

The mo mode de Meri Merits ts i. ii. iii. iv.

It can be determined from incomplete data provided the observations with the highest frequency are already known The mode has several applications in business eg stocking the most sold good. The mode can be easily defined It can be determined easily from a graph

Dem Demer er erits its If the data is quite large and ungrouped, determination of the mode can be quite cumbersome ii. Use of the formula to calculate the mode is unfamiliar to most business people iii. The mode may sometimes be non-existent or there may be two modes for a given set of data. In such a case therefore a single mode may not exist The me median dian i.

Meri Merits ts i. ii. iii. iv. v.

It shows the centre of a given set of data Knowledge of the determination of the median may be extended to determine the quartiles The median can easily be defined It can be obtained easily from the cumulative frequency curve It can be used in determining the degrees of skew-ness

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Dem Demer er erits its i.

In some situations where the no. of observations is even, the value of the median obtained is usually imaginary The computation of the median using the formulas is not well understood by most people. In the education environment the median has got very few applications

ii. iii.

Other Measures of Location Quartiles: Refers to values of items that divide a series into four equal parts when the series is arranged into ascending order. -

i. ii.

The quartiles normally used are three namely; a) The lower quartile (first quartile Q1) b) The median (second quartile Q 2) c) The upper quartile (third quartile Q3)

Deciles: Refers to values of items that divide a series into ten equal parts when the series is arranged in ascending order. Percentiles: Refers to values that divide a series into 100 equal parts when the series is arranged in ascending order.

Note: Refer to class exercises for further understanding. Measures of Dispersion - Also known as Measures of variations or variability. - They measure how much data spread out around a central measure. - The measures of dispersion are very useful in statistical work because they indicate whether the rest of the data are scattered around the mean or away from the mean. - If the data is approximately dispersed around the mean then the measure of dispersion obtained will be small therefore indicating that the mean is a good representative of the sample data. But on the other hand, if the figures are not closely located to the mean then the measures of dispersion obtained will be relatively big indicating that the mean does not represent the data sufficiently. - The measures of dispersion are expressed in two ways: i. Absolute measure: This is when the measures are expressed using the same units of measure as the original data. ii. Relative measure: This is when the measures are expressed as a fraction or percentage. Are also known as coefficients of dispersion. Methods of Measuring Dispersion The commonly used measures of dispers...