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- 1. www.crescent-university.edu.ng L ECTURE N OTE ON DESCRIPTIVE STATISTICS (STS 102) BY ADEOSUN SAKIRU ABIODUN E-mail: adeosunsakiru@gmail.com
- 2. www.crescent-university.edu.ng COURSE CONTENTS Statistical data: types, sources and methods of collection. Presentation of data: tables, charts and graphs. Error and Approximations. Frequency and cumulative distributions. Measures of location, partition, dispersion, Skewness and kurtosis. Rates, Proportion and index numbers. READING LISTS 1. Adamu S.O and Johnson Tinuke L (1998): Statistics for Beginners; Book 1. SAAL Publication. Ibadan. ISBN: 978-34411-3-2 2. Clark G.M and Cooke D (1993): A Basic course in statistics. Third edition. London: Published by Arnold and Stoughton. 3. Olubosoye O.E, Olaomi J.O and Shittu O.I (2002): Statistics for Engineering, Physical and Biological sciences. Ibadan: A Divine Touch Publications. 4. Tmt. V. Varalakshmi et al (2005): Statistics Higher Secondary - First year. Tamilnadu Textbook Corporation, College Road, Chennai- 600 006 2
- 3. www.crescent-university.edu.ng INTRODUCTION In the modern world of information and communication technology, the importance of statistics is very well recognised by all the disciplines. Statistics has originated as a science of statehood and found applications slowly and steadily in Agriculture, Economics, Commerce, Biology, Medicine, Industry, planning, education and so on. As of today, there is no other human walk of life, where statistics cannot be applied. Statistics is concerned with the scientific method of collecting, organizing, summarizing, presenting and analyzing statistical information (data) as well as drawing valid conclusion on the basis of such analysis. It could be simply defined as the “science of data”. Thus, statistics uses facts or numerical data, assembled, classified and tabulated so as to present significant information about a given subject. Statistic is a science of understanding data and making decisions in the face of randomness. The study of statistics is therefore essential for sound reasoning, precise judgment and objective decision in the face of up- to- date accurate and reliable data. Thus many researchers, educationalists, business men and government agencies at the national, state or local levels rely on data to answer operations and programs. Statistics is usually divided into two categories, which is not mutually elution namely: Descriptive statistics and inferential statistics. DESCRIPTIVE STATISTICS This is the act of summarizing and given a descriptive account of numerical information in form of reports, charts and diagrams. The goal of descriptive statistics is to gain information from collected data. It begins with collection of data by either counting or measurement in an inquiry. It involves the summary of specific aspect of the data, such as averages values and measure of dispersion (spread). Suitable graphs, diagrams and charts are then used to gain understanding and clear 3
- 4. www.crescent-university.edu.ng interpretation of the phenomenon under investigation keeping firmly in mind where the data comes from. Normally, a descriptive statistics should: i. be single – valued ii. be algebraically tractable iii. consider every observed value. INFERENTIAL STATISTICS This is the act of making deductive statement about a population from the quantities computed from its representative sample. It is a process of making inference or generalizing about the population under certain conditions and assumptions. Statistical inference involves the processes of estimation of parameters and hypothesis testing. However, this concept is not in the context of this course. USES OF STATISTICS Statistics can be used among others for: 1) Planning and decision making by individuals, states, business organizations, research institution etc. 2) Forecasting and prediction for the future based on a good model provided that its basic assumptions are not violated. 3) Project implementation and control. This is especially useful in on-going projects such as network analysis, construction of roads and bridges and implementation of government programs and policies. 4) The assessment of the reliability and validity of measurements and general points significance tests including power and sample size determination. 4
- 5. www.crescent-university.edu.ng STATISTICAL DATA Data can be described as a mass of unprocessed information obtained from measurement of counting of a characteristics or phenomenon. They are raw facts that have to be processed in numerical form they are called quantitative data. For instance the collection of ages of students offering STS 102 in a particular session is an example of this data. But when data are not presented in numerical form, they are called qualitative data. E.g.: status, sex, religion, etc. Definition: Statistical data are data obtained through objective measurement or enumeration of characteristics using the state of the art equipment that is precise and unbiased. Such data when subjected to statistical analysis produce results with high precision. SOURCES OF STATISTICAL DATA 1. Primary data: These are data generated by first hand or data obtained directly from respondents by personal interview, questionnaire, measurements or observation. Statistical data can be obtained from: (i) Census – complete enumeration of all the unit of the population (ii) Surveys – the study of representative part of a population (iii) Experimentation – observation from experiment carried out in laboratories and research center. (iv) Administrative process e.g. Record of births and deaths. ADVANTAGES Comprises of actual data needed It is more reliable with clarity Comprises a more detail information 5
- 6. www.crescent-university.edu.ng DISADVANTAGES Cost of data collection is high Time consuming There may larger range of non response 2. Secondary data: These are data obtained from publication, newspapers, and annual reports. They are usually summarized data used for purpose other than the intended one. These could be obtain from the following: (i) Publication e.g. extract from publications (ii) Research/Media organization (iii) Educational institutions ADVANTAGES The outcome is timely The information gathered more quickly It is less expensive to gather. DISADVANTAGES Most time information are suppressed when working with secondary data The information may not be reliable METHODS OF COLLECTION OF DATA There are various methods we can use to collect data. The method used depends on the problem and type of data to be collected. Some of these methods include: 1. Direct observation 6
- 7. www.crescent-university.edu.ng 2. Interviewing 3. Questionnaire 4. Abstraction from published statistics. DIRECT OBSERVATION Observational methods are used mostly in scientific enquiry where data are observed directly from controlled experiment. It is used more in the natural sciences through laboratory works than in social sciences. But this is very useful studying small communities and institutions. INTERVIEWING In this method, the person collecting the data is called the interviewer goes to ask the person (interviewee) direct questions. The interviewer has to go to the interviewees personally to collect the information required verbally. This makes it different from the next method called questionnaire method. QUESTIONNAIRE A set of questions or statement is assembled to get information on a variable (or a set of variable). The entire package of questions or statement is called a questionnaire. Human beings usually are required to respond to the questions or statements on the questionnaire. Copies of the questionnaire can be administered personally by its user or sent to people by post. Both interviewing and questionnaire methods are used in the social sciences where human population is mostly involved. 7
- 8. www.crescent-university.edu.ng ABSTRACTIONS FROM THE PUPLISHED STATISTICS These are pieces of data (information) found in published materials such as figures related to population or accident figures. This method of collecting data could be useful as preliminary to other methods. Other methods includes: Telephone method, Document/Report method, Mail or Postal questionnaire, On-line interview method, etc. PRESENTATION OF DATA When raw data are collected, they are organized numerically by distributing them into classes or categories in order to determine the number of individuals belonging to each class. Most cases, it is necessary to present data in tables, charts and diagrams in order to have a clear understanding of the data, and to illustrate the relationship existing between the variables being examined. FREQUENCY TABLE This is a tabular arrangement of data into various classes together with their corresponding frequencies. Procedure for forming frequency distribution Given a set of observation , for a single variable. 1. Determine the range (R) = L – S where L = largest observation in the raw data; and S = smallest observation in the raw data. 2. Determine the appropriate number of classes or groups (K). The choice of K is arbitrary but as a general rule, it should be a number (integer) between 5 and 20 depending on the size of the data given. There are several suggested guide lines aimed at helping one decided on how many class intervals to employ. Two of such methods are: 8
- 9. www.crescent-university.edu.ng (a) K = 1 +3.322 (b) K = 3. Determine the width where = number of observations. of the class interval. It is determined as 4. Determine the numbers of observations falling into each class interval i.e. find the class frequencies. NOTE: With advent of computers, all these steps can be accomplishes easily. SOME BASIC DEFINITIONS Variable: This is a characteristic of a population which can take different values. Basically, we have two types, namely: continuous variable and discrete variable. A continuous variable is a variable which may take all values within a given range. Its values are obtained by measurements e.g. height, volume, time, exam score etc. A discrete variable is one whose value change by steps. Its value may be obtained by counting. It normally takes integer values e.g. number of cars, number of chairs. Class interval: This is a sub-division of the total range of values which a (continuous) variable may take. It is a symbol defining a class E.g. 0-9, 10-19 etc. there are three types of class interval, namely: Exclusive, inclusive and open-end classes method. Exclusive method: When the class intervals are so fixed that the upper limit of one class is the lower limit of the next class; it is known as the exclusive method of classification. E.g. Let some expenditures of some families be as follows: 0 – 1000, 1000 – 2000, etc. It is clear that the exclusive method ensures continuity of data as much as the upper limit of one class is the lower limit of the next class. In the above example, there are so families whose expenditure is between 0 and 999.99. A family whose expenditure is 1000 would be included in the class interval 1000-2000. 9
- 10. www.crescent-university.edu.ng Inclusive method: In this method, the overlapping of the class intervals is avoided. Both the lower and upper limits are included in the class interval. This type of classification may be used for a grouped frequency distribution for discrete variable like members in a family, number of workers in a factory etc., where the variable may take only integral values. It cannot be used with fractional values like age, height, weight etc. In case of continuous variables, the exclusive method should be used. The inclusive method should be used in case of discrete variable. Open end classes: A class limit is missing either at the lower end of the first class interval or at the upper end of the last class interval or both are not specified. The necessity of open end classes arises in a number of practical situations, particularly relating to economic and medical data when there are few very high values or few very low values which are far apart from the majority of observations. Class limit: it represents the end points of a class interval. {Lower class limit & Upper class limit}. A class interval which has neither upper class limit nor lower class limit indicated is called an open class interval e.g. “less than 25”, ’25 and above” Class boundaries: The point of demarcation between a class interval and the next class interval is called boundary. For example, the class boundary of 10-19 is 9.5 – 19.5 Cumulative frequency: This is the sum of a frequency of the particular class to the frequencies of the class before it. 10
- 11. www.crescent-university.edu.ng Example 1: The following are the marks of 50 students in STS 102: 48 70 60 47 51 55 59 63 68 63 47 53 72 53 67 62 64 70 57 56 48 51 58 63 65 62 49 64 53 59 63 50 61 67 72 56 64 66 49 52 62 71 58 53 63 69 59 64 73 56. (a) Construct a frequency table for the above data. (b) Answer the following questions using the table obtained: (i) how many students scored between 51 and 62? (ii) how many students scored above 50? (iii) what is the probability that a student selected at random from the class will score less than 63? Solution: (a) Range (R) = No of classes Class size Frequency Table Mark Tally frequency 7 51- 54 7 55 - 58 7 59 – 62 8 63 – 66 11 67 – 70 6 71 – 74 4 (b) (i) 22 (ii) 43 (iii) 0.58 11
- 12. www.crescent-university.edu.ng Example 2: The following data represent the ages (in years) of people living in a housing estate in Abeokuta. 18 31 30 6 16 17 18 43 2 8 32 33 9 18 33 19 21 13 13 14 14 6 52 45 61 23 26 15 14 15 14 27 36 19 37 11 12 11 20 12 39 20 40 69 63 29 64 27 15 28. Present the above data in a frequency table showing the following columns; class interval, class boundary, class mark (mid-point), tally, frequency and cumulative frequency in that order. Solution: Range (R) No of classes Class width Class interval Class boundary Class mark Tally Frequency Cum.freq 2 – 11 1.5 – 11.5 6.5 7 7 12 – 21 11.5 – 21.5 16.5 21 28 22 – 31 21.5 – 31.5 26.5 8 36 32 – 41 31.5 – 41.5 36.6 7 43 42 – 51 41.5 – 51.5 46.5 2 45 52 – 61 51.5 – 61.5 56.5 2 47 62 – 71 61.5 – 71.5 66.5 3 50 Observation from the Table The data have been summarized and we now have a clearer picture of the distribution of the ages of inhabitants of the Estate. 12
- 13. www.crescent-university.edu.ng Exercise 1 Below are the data of weights of 40students women randomly selected in Ogun state. Prepare a table showing the following columns; class interval, frequency, class boundary, class mark, and cumulative frequency. 96 84 75 80 64 105 87 62 105 103 76 93 75 110 88 97 69 96 73 91 101 106 110 64 105 117 84 96 91 82 81 94 108 117 99 114 88 60 98 77 Use your table to answer the following question i. How many women weight between 71 and 90? ii. How many women weight more than 80? iii. What is the probability that a woman selected at random from Ogun state would weight more than 90? GRAPHICAL PRESENTATION OF DATA It is not enough to represent data in a tabular form. The most attractive way of representing data is through charts or graphs. PICTOGRAM Pictograms or pictographs are representations in form of pictures. They convey broad meanings and relationships among data. Also they are the simplest way of presenting information. Pictograms are popularly used in newspapers by journalists and advertisers. EXAMPLE 1: In a certain secondary school, there are 4 Eng teachers, 3 math teachers, 2 Biology teachers, 2 government teachers and 1 Physics teacher. Draw a pictogram to represent this information. 13
- 14. www.crescent-university.edu.ng SOLUTION: Let us use to represent a teacher The different subject Teachers in a certain school English Teachers Maths teachers Biology teachers Government teachers Physics teachers PIE CHART A pie chart is a circular graph in which numerical data are represented by sectors of a circle. The angles of the sectors are proportional to the frequencies of the items they represent EXAMPLE 2: In ADAS international school, the lesson periods for each week are given below. English 7, Maths 10, Biology 3, Physics 4, Chemistry 3, others 9. Draw a pie chart to illustrate this information. 14
- 15. www.crescent-university.edu.ng Solution: Total no of periods in a week Subject No of period English 7 Maths 10 Biology 3 Physics 4 Chemistry 3 Others 9 Total Angle of sector 36 The pie chart showing the lesson period in ADAS international school English Mathematics Biology Physics Chemistry Other 15
- 16. www.crescent-university.edu.ng BAR CHART A bar chart is a statistical graph in which bars (rectangular bars) are drawn such that their lengths or heights are proportional to the quantities or item they represent. Each bar is separated by equal gaps. Example 3: The allotment of time in minutes per week for some of the university courses in second semester is; Courses Minutes GNS 104 60 MTS 102 120 STS 102 180 ECO 102 120 BFN 108 120 PHS 192 140 Construct a bar chart to represent the above data. Solution: Bar chart showing the allotment of time in minutes 200 180 160 140 120 100 Series1 80 60 40 20 0 GNS 104 MTS 102 STS 102 ECO 102 BFN 108 16 PHS 192
- 17. www.crescent-university.edu.ng HISTOGRAM A histogram is a graphical representation of a frequency distribution. It consists of a number of rectangles. The area of each rectangular bar of a histogram is proportional to the corresponding frequency. Unlike bar charts, the rectangles are joined together for histogram. Example 4: Draw a histogram to represent the data in the table below. Height (cm) Frequency 6 9 15 5 2 Solution: 174.5 164.5 154.5 144.5 134.5 124.5 Frequency Histogram showing the heights Height CUMMULATIVE FREQUENCY CURVE (Ogive) This curve is the plotting of cumulative freq. against upper class bounding of the class intervals. The points, when joined, result usually in a smooth curve which is the form of an elongated S. 17
- 18. www.crescent-university.edu.ng Example 5: Using the data below: Weight (kg) Frequency Cumulative frequency 20 – 24 5 5 25 – 29 4 9 30 – 34 3 12 35 – 39 5 17 40 – 44 5 22 45 – 49 5 27 50 – 54 18 45 55 – 59 12 57 60 – 64 3 60 The cumulative frequency or Ogive will appears as follows: Stem plots (Stem – and – leaf plots) A stem-and-leaf plot is used to visualize data. To set up a stem-and-leaf plot, we follow some simple steps. First we have a set of data. Then we identify the lowest and greatest number in the data set and then we draw a vertical line. On the left hand side of the line we write the numbers that correspond to the tens and on the right hand side of the line, we will write the numbers that corresponds with the units. The digits should be arranged in order. Example : Draw the stem plot for the following data (a) 13, 24, 22, 15, 33, 32, 12, 31, 26, 28, 14, 19, 20, 22, 31, 15. (b) 46 59 35 41 46 21 24 33 40 45 49 53 48 54 61 36 70 58 47 12 18
- 19. www.crescent-university.edu.ng Solution: (a) Stem Leaf 1 2 0 2 2 4 6 8 3 (b) 2 3 4 5 5 9 1 1 2 3 Stem Leaf 1 2 2 1 4 3 3 5 6 4 0 1 5 6 6 7 8 9 5 3 4 8 9 6 1 7 1 Exercise 2: 1. The population by classes of Adas international school is as shown below Class I II III IV V VI No of pupils 30 25 27 26 22 26 Draw a pictogram, pie chart and a bar chart to represent the information. 2. The age distribution (in years) of a group of 100 individual is given below. Ages(x) 2–4 5–9 10 – 15 – 50 – 25 – 30 14 No. of 8 11 19 24 29 34 21 20 17 10 10 – 35 – 39 individuals(f) Draw a histogram and a cumulative frequency curve for the data. 19 3
- 20. www.crescent-university.edu.ng ERRORS AND APPROXIMATION Rounding of statistical data is the method of approximating a number such that the last digits affected are changed to zero and the number become clearer. Since in reality most measurement is not exact, hence offence is not much committed when data are rounded up. Methods of rounding are rounding to specific units. E.g. 4,234,120 to the nearest million is 4,000,000. Also rounding to specific significant figures e.g. to 5 significant figures will give 4,235,100. Again when an amount, say N750.3 is used to denote the average salary per person in a month, the figure in this wise is rounded to one place of decimal and the actual figure is between the ranges N750.25 to less than N 750.35. This implies rounded figures have errors. Error: For any given figure the correct figure lies in a certain range. Half of this range is the Error. Suppose correct figure; Absolute error: Percentage Error rounded figure e = error; . Relative error . i.e. percentage Error Example: The length of a pole is measured as 10meters to the nearest meter. What is the range of its actual length? Calculate the percentage error. Solution The actual length of the pole will be between 9.5 and 10.5 Error Percentage Error 20
- 21. www.crescent-university.edu.ng MEASURES OF LOCATION These are measures of the centre of a distribution. They are single values that give a description of the data. They are also referred to as measure of central tendency. Some of them are arithmetic mean, geometric mean, harmonic mean, mode, and median. THE ARITHMETIC MEAN (A.M) The arithmetic mean (average) of set of observation is the sum of the observation divided by the number of observation. Given a set of a numbers , the arithmetic mean denoted by is defined by Example 1: The ages of ten students in STS 102 are determine the mean age. Solution: . If the numbers times respectively, the ( for short.) Example 2: Find the mean for the table below Scores (x) 2 5 6 8 Frequency (f) 1 3 4 2 Solution 21
- 22. www.crescent-university.edu.ng . Calculation of mean from grouped data If the items of a frequency distribution are classified in intervals, we make the assumption that every item in an interval has the mid-values of the interval and we use this midpoint for . Example 3: The table below shows the distribution of the waiting items for some customers in a certain petrol station in Abeokuta. Waiting time(in 1.5 – 1.9 2.0 – 2.4 2.5 – 2.9 3.0 – 3.4 3.5 – 3.9 4.0 – 4.4 10 18 10 7 2 mins) No. of customers 3 d the average waiting time of the customers. Solution: Waiting (in min) No of customers Class mark mid-value(X) 1.5 – 1.9 3 1.7 5.1 2.0 – 2.4 10 2.2 22 2.5 – 2.9 18 2.7 48.6 3.0 – 3.4 10 3.2 32 3.5 – 3.9 7 3.7 25.9 4.0 – 4.4 2 4.2 8.4 = 2.84 22 Fin
- 23. www.crescent-university.edu.ng Use of Assume mean Sometimes, large values of the variable are involve in calculation of mean, in order to make our computation easier, we may assume one of the values as the mean. This if A= assumed mean, and d= deviation of from A, i.e. Therefore, since . If a constant factor C is used then . Example 4: The exact pension allowance paid (in Nigeria) to 25 workers of a company is given in the table below. Pension in N 25 30 35 40 45 No of person 7 5 6 4 3 Calculate the mean using an assumed mean 35. Solution Pension in N frequency 25 7 25 – 35 = -10 - 70 30 5 30 – 35 = -5 - 25 35 A 6 35 – 35 = 0 0 40 4 40 – 35 = 5 20 45 3 45 – 35 = 10 30 25 - 45 23
- 24. www.crescent-university.edu.ng Example 5: Consider the data in example 3, using a suitable assume mean, compute the mean. Solution: Waiting time 1.5 – 1.9 3 1.7 -1 -3 2.0 – 2.4 10 2.2 -0.5 -5 2.5 – 2.9 18 2.7 A 0 0 3.0 – 3.4 10 3.2 0.5 5 3.5 – 3.9 7 3.7 1 7 4.0 – 4.4 2 4.2 1.5 3 50 7 NOTE: It is always easier to select the class mark with the longest frequency as the assumed mean. 24
- 25. www.crescent-university.edu.ng ADVANTAGE OF MEAN The mean is an average that considers all the observations in the data set. It is single and easy to compute and it is the most widely used average. DISAVANTAGE OF MEAN Its value is greatly affected by the extremely too large or too small observation. THE HARMONIC MEAN (H.M) The H.M of a set of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers. It is used when dealing with the rates of the type per (such as kilometers per hour, Naira per liter). The formula is expressed thus: H.M If has frequency , then H.M Example: Find the harmonic mean of . Solution: H.M . Note: (i) Calculation takes into account every value (ii) Extreme values have least effect (iii) The formula breaks down when “o” is one of the observations. 25
- 26. www.crescent-university.edu.ng THE GEOMETRIC MEAN(G.M) The G.M is an analytical method of finding the average rate of growth or decline in the values of an item over a particular period of time. The geometric mean of a set of number is the root of the product of the number. Thus G.M If is the frequency of , then G.M Example: The rate of inflation in fire successive year in a country was . What was the average rate of inflation per year? Solution: G.M Average rate of inflation is Note: (1) Calculate takes into account every value. (2) It cannot be computed when “0” is on of the observation. Relation between Arithmetic mean, Geometric and Harmonic In general, the geometric mean for a set of data is always less than or equal to the corresponding arithmetic mean but greater than or equal to the harmonic mean. That is, H.M G.M A.M The equality signs hold only if all the observations are identical. THE MEDIAN This is the value of the variable that divides a distribution into two equal parts when the values are arranged in order of magnitude. If there are 26 (odd)
- 27. www.crescent-university.edu.ng observation, the median location of the median is But if is even, the median is the center of observation in the ordered list. The th item. is the average of the two middle observations in the ordered list. i.e. Example 1: The values of a random variable are given as Find the median. Solution: In an array: . is odd, therefore The median, Example 2: The value 0f a random variable and are given as . Find the median. Solution: is odd. Median, Calculation of Median from a grouped data The formula for calculating the median from grouped data is defined as Where: Lower class boundary of the median class 27
- 28. www.crescent-university.edu.ng Total frequency Cumulative frequency before the median class Frequency of the median class. Class size or width. Example3: The table below shows the height of 70 men randomly selected at Sango Ota. Height 118-126 127-135 136-144 145-153 154-162 163-171 172-180 No of rods 8 10 9 7 4 14 18 Compute the median. Solution Height Frequency Cumulative frequency 118 – 126 8 8 127 – 135 10 18 136 – 144 14 32 145 – 153 18 50 154 – 162 9 59 163 – 171 7 66 172 – 180 4 70 70 . The sum of first three classes frequency is 32 which therefore means that the median lies in the fourth class and this is the median class. Then 28
- 29. www.crescent-university.edu.ng . ADVANTAGE OF THE MEDIAN (i) Its value is not affected by extreme values; thus it is a resistant measure of central tendency. (ii) It is a good measure of location in a skewed distribution DISAVANTAGE OF THE MEDIAN 1) It does not take into consideration all the value of the variable. THE MODE The mode is the value of the data which occurs most frequently. A set of data may have no, one, two or more modes. A distribution is said to be uni-model, bimodal and multimodal if it has one, two and more than two modes respectively. E .g: The mode of scores 2, 5, 2, 6, 7 is 2. Calculation of mode from grouped data From a grouped frequency distribution, the mode can be obtained from the formula. Mode, Where: lower class boundary of the modal class Difference between the frequency of the modal class and the class before it. Difference between the frequency of the modal class and the class after it. Class size. Example: For the table below, find the mode. 29
- 30. www.crescent-university.edu.ng Class 11 – 20 frequency 6 21 – 30 31 – 40 41 - 50 51 – 60 61 – 70 20 12 10 9 9 Calculate the modal age. Solution: Mode, ADVANTAGE OF THE MODE 1) It is easy to calculate. DISADVANTAGE OF THE MODE (i) It is not a unique measure of location. (ii) It presents a misleading picture of the distribution. (iii) It does not take into account all the available data. Exercise 3 1. Find the mean, median and mode of the following observations: 5, 6,10,15,22,16,6,10,6. 2. The six numbers 4, 9,8,7,4 and Y, have mean of 7. Find the value of Y. 3. From the data below 30
- 31. www.crescent-university.edu.ng Class 21 – 23 Frequency 2 24 – 26 27 – 29 30 – 32 33 – 35 36 – 38 37 – 41 5 8 9 7 3 1 Calculate the (i)Mean (ii)Mode (iii) Median MEASURES OF PARTITION From the previous section, we’ve seen that the median is an average that divides a distribution into two equal parts. So also these are other quantity that divides a set of data (in an array) into different equal parts. Such data must have been arranged in order of magnitude. Some of the partition values are: the quartile, deciles and percentiles. THE QUARTILES Quartiles divide a set of data in an array into four equal parts. For ungrouped data, the distribution is first arranged in ascending order of magnitude. Then First Quartiles: Second Quartile: Third Quartile: For a grouped data Where The quality in reference 31
- 32. www.crescent-university.edu.ng Lower class boundary of the class counting the quartile Total frequency Cumulative frequency before the The frequency of the Class size of the class class class. DECILES The values of the variable that divide the frequency of the distribution into ten equal parts are known as deciles and are denoted by the fifth deciles is the median. For ungrouped data, the distribution is first arranged in ascending order of magnitude. Then For a grouped data Where 32
- 33. www.crescent-university.edu.ng PERCENTILE The values of the variable that divide the frequency of the distribution into hundred equal parts are known as percentiles and are generally denoted by The fiftieth percentile is the median. For ungrouped data, the distribution is first arranged in ascending order of magnitude. Then For a grouped data, Where Example: For the table below, find by calculation (using appropriate expression) (i) Lower quartile, (ii) Upper Quartile, (iii) 6th Deciles, 33
- 34. www.crescent-university.edu.ng (iv) 45th percentile of the following distribution Mark 20 – 29 30 – 39 40 – 49 50 – 59 60 – 69 70 – 79 80 – 89 90 – 99 Frequency 8 10 14 26 Solution Marks frequency cumulative frequency 20 – 29 8 8 30 – 39 10 18 40 – 49 14 32 50 – 59 26 32 60 – 69 20 58 70 – 79 16 78 80 – 89 4 98 90 – 99 2 100 100 (i) Lower quartile, w (ii) Upper Quartile, 34 20 16 4 2
- 35. www.crescent-university.edu.ng (iii) (iv) MEASURES OF DISPERSION Dispersion or variation is degree of scatter or variation of individual value of a variable about the central value such as the median or the mean. These include range, mean deviation, semi-interquartile range, variance, standard deviation and coefficient of variation. THE RANGE This is the simplest method of measuring dispersions. It is the difference between the largest and the smallest value in a set of data. It is commonly used in statistical quality control. However, the range may fail to discriminate if the distributions are of different types. Coefficient of Range 35
- 36. www.crescent-university.edu.ng SEMI – INTERQUARTILE RANGE This is the half of the difference between the first (lower) and third quartiles (upper). It is good measure of spread for midrange and the quartiles. THE MEAN/ABSOLUTE DEVIATION Mean deviation is the mean absolute deviation from the centre. A measure of the center could be the arithmetic mean or median. Given a set of the mean deviation from the arithmetic mean is defined by: In a grouped data Example1: Below is the average of 6 heads of household randomly selected from a country. 47, 45, 56, 60, 41, 54 .Find the (i) Range (ii) Mean (iii) Mean deviation from the mean (iv) Mean deviation from the median. Solution: (i) Range (ii) Mean ( ) 36
- 37. www.crescent-university.edu.ng (iii) Mean Deviation (iv) In array: 41, 45, 47, 54, 56, 60 Median Example2: The table below shown the frequency distribution of the scores of 42 students in MTS 101 Scores 0–9 No of student 2 10 – 19 20 – 29 30 – 39 40 – 49 50 – 59 60 – 69 5 8 12 9 5 1 Find the mean deviation from the mean for the data. 37
- 38. www.crescent-university.edu.ng Solution: Classes midpoint 0–9 4.5 10 – 19 14.5 5 72.5 -19.52 19.52 97.60 20 – 29 24.5 8 196 -9.52 9.52 76.16 30 – 39 34.5 12 414 0.48 0.48 5.76 40 – 49 44.5 9 400.5 10.48 10.48 94.32 50 – 59 54.5 5 272.5 20.48 20.48 102.4 60 – 69 64.5 1 64.5 30.48 30.48 30.48 42 1429 2 9 -29.52 29.52 59.04 465.76 THE STANDARD DEVIATION The standard deviation, usually denoted by the Greek alphabet (small signal) (for population) is defined as the “positive square root of the arithmetic mean of the squares of the deviation of the given observation from their arithmetic mean”. Thus, given as a set of observations, then the standard deviation is given by: (Alternatively, . ) 38
- 39. www.crescent-university.edu.ng For a grouped data The standard deviation is computed using the formula If A= assume mean and Note: We use is deviation from the assumed mean, then when using sample instead of the population to obtain the standard deviation. MERIT (i) It is well defined and uses all observations in the distribution. (ii) It has wider application in other statistical technique like skewness, correlation, and quality control e.t.c DEMERIT (i) It cannot be used for computing the dispersion of two or more distributions given in different unit. THE VARIANCE The variance of a set of observations is defined as the square of the standard deviation and is thus given by 39
- 40. www.crescent-university.edu.ng COEFFICIENT OF VARIATION/DISPERSION This is a dimension less quantity that measures the relative variation between two servers observed in different units. The coefficients of variation are obtained by dividing the standard deviation by the mean and multiply it by 100. Symbolically The distribution with smaller C.V is said to be better. EXAMPLE3: Given the data 5, 6, 9, 10, 12. Compute the variance, standard deviation and coefficient of variation SOLUTION Hence C.V EXAMPLE4: Given the following data. Compute the (i) Mean (ii) Standard deviation (iii) Coefficient variation. 40
- 41. www.crescent-university.edu.ng Ages(in years) 50 – 54 55 – 59 60 – 64 65 – 69 70 – 74 75 – 79 80 – 84 Frequency 1 2 12 18 25 9 10 SOLUTION Classes 50 – 54 52 1 52 -20.06 55 – 59 57 2 114 -15.06 60 – 64 62 10 620 -10.06 1012.04 65 – 69 67 12 804 -5.06 307.24 70 – 74 72 18 1296 -0.06 75 – 79 77 25 1925 4.94 610.09 80 – 84 82 9.94 889.23 9 738 77 5549 402.40 453.61 0.07 3674.68 C.V 41
- 42. www.crescent-university.edu.ng Exercise 4 The data below represents the scores by 150 applicants in an achievement text for the post of Botanist in a large company: Scores 10 – 19 20 – 29 30 – 39 40 – 49 50 – 59 60 – 69 70 – 79 80 – 89 90 – 99 Frequency 1 6 9 31 42 32 17 10 Estimate (i) The mean score (ii) The median score (iii) The modal score (iv) Standard deviation (v) Semi – interquartile range (vi) D4 (vii) P26 (viii) coefficient of variation SKEWNESS AND KURTOSIS Skewness means “lack of symmetry”. We study skewness to have an idea about the shape of the curve which we can draw with the help of the given data. If in a distribution mean in median mode, then that distribution is known as symmetrical distribution. If in a distribution, mean median mode, then it is not a symmetrical distribution and it is called a skewed distribution and such a distribution could either be positively skewed or negative skewed. (a) Symmetrical distribution: mean median 42 mode 2
- 43. www.crescent-university.edu.ng Mean, median and mode coincide and the spread of the frequencies is the same on both sides of the centre point of the curve. (b) Positively Skewed distribution: mode median mean In a positive skewed distribution, the value of the mean is the maximum and that of the mode is the least, and the median lies in between the two. The frequencies are spread out over a grater range of values on the right hand side than they are on the left hand side. (c) Negatively Skewed distribution: mean median mode In a negatively skewed distribution, the value of the mode is the maximum and that of the mean is the least. The median lies in between the two. The frequencies are spread out over a greater range of values on the left hand side than they are on the right hand side. Measures of Skewness The important measures of skewness are: (i) Karl-Pearson’s coefficient of skewness (ii) Bowley’s coefficient of skewness 43
- 44. www.crescent-university.edu.ng (iii) Measures of skewness based on moments. Karl-Pearson’s coefficient of skewness This is given by: Karl-Pearson’s Coefficient Skewness . In case of mode is ill-defined, the coefficient can be determined by the formula: Coefficient of Skewness . Bowley’s coefficient of skewness This is given by: Bowley’s Coefficient of Skewness . Measure of skewness based on moment First, we note the following moments: 1st moment about mean: ; . 2nd moment about mean: ; . 3rd moment about mean: ; . 4th moment about mean: ; . Then the measure of skewness based on moments denoted by is given by: . KURTOSIS The expression ‘Kurtosis’ is used to describe the peakedness of a normal curve. The three measures – central tendency, dispersion and skewness, describe the characteristics of frequency distributions. But these studies will not give us a 44
- 45. www.crescent-university.edu.ng clear picture of the characteristics of a distribution. Measures of kurtosis tell us the extent to which a distribution is more picked or more flat topped than the normal curve, which is symmetrical end bell – shaped, is designated as Mesokurtic. If a curve is relatively more narrow and peaked at the top, it is designated as Leptokurtic. If the frequency curve is more flat than normal curve, it is designated as Platykurtic. L M P Measures of Kurtosis The measure of kurtosis of a frequency distribution based on moment is denoted by and is given by: . If , the distribution is said to be normal and the curve is Mesokurtic. If , the distribution is said to be more peaked and the curve is Leptokurtic. If , the distribution is said to be flat peaked and the curve is Platykurtic. Example 1: Calculate Karl – Pearson’s coefficient of skewness for the following data: 25, 15, 23, 40, 27, 25, 23, 25, 20. 45
- 46. www.crescent-university.edu.ng Solution: Computation of mean and standard deviation using assume mean method: Size Deviation from A 25 25 0 0 15 -10 100 23 -2 4 40 15 225 27 2 4 25 0 0 23 -2 4 25 0 0 20 -5 25 Mean Mode , as this size of item repeats three times. Karl – Pearson’s coefficient of skewness 46
- 47. www.crescent-university.edu.ng Example 2: Find Karl – Pearson’s coefficient of skewness for the given distribution: Class 0 – 5 6 – 11 12 – 17 18 – 23 24 – 29 30 – 35 36 – 41 42 – 47 F 5 7 13 21 16 8 3 2 Solution: Mode lies in 24 – 29 group which contains the maximum frequency. Mode Computation of mean and standard deviation: Class Mid Point ( ) 0–5 2.5 2 5 -23.28 541.96 1083.92 6 – 11 8.5 5 42.5 -17.28 298.59 1492.95 12 – 17 14.5 7 101.5 -11.28 127.24 890.68 18 – 23 20.5 13 266.5 -5.28 27.88 362.44 24 – 29 26.5 21 556.5 0.72 0.52 10.92 30 – 35 32.5 16 520 6.72 45.16 722.56 36 – 41 38.5 8 308 12.72 161.79 1294.32 42 – 47 44.5 3 133.5 18.72 350.44 1051.32 Mean Standard deviation 47
- 48. www.crescent-university.edu.ng . Therefore, Karl – Pearson’s coefficient of skewness Example 3: Find the Bowley’s coefficient of skewness for the following series: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22. Solution: The given data in order: 2, 4, 6, 10, 12, 14, 16, 18, 20, and 22. size of size of th item th item size of 3rd item size of th item size of th item size of 9th item Median size of th item size of th item size of 6th item Bowley’ coefficient of skewness 48
- 49. www.crescent-university.edu.ng Since skewness , the given series is a symmetrical data. Example 4: Calculate (measure of skewness based on moment) and (measure of kurtosis based on moment) for the following data. X 0 1 2 3 4 5 6 7 8 F 5 10 15 20 25 20 15 10 5 Solution: First Moment: where Second Moment: Third Moment: Fourth Moment: . That is, symmetrical curve. . The value of , hence the curve is Platykurtic. Example 5: For the data below, determine the and Mark 30 – 33 Frequency 2 . 34 – 37 38 – 41 42 – 45 46 – 49 50 – 53 4 26 47 15 6 Solution: 49
- 50. www.crescent-university.edu.ng Mark Mid Point ( ) 30 – 33 31.5 2 63 -11.48 131.79 263.58 34 – 37 35.5 4 142 -7.48 55.95 223.80 38 – 41 39.5 26 1027 -3.48 12.11 314.87 42 – 45 43.5 47 2044.5 0.52 0.27 12.71 46 – 49 47.5 15 712.5 4.52 20.43 306.46 50 – 53 51.5 6 309 8.52 72.59 435.54 100 4298 1556.96 -1512.95 -3025.91 17368.71 34737.42 -18.51 -1674.04 3130.45 12521.79 -42.14 -1095.64 146.66 3813.16 0.14 6.58 0.07 3.44 92.35 1385.25 417.40 6261.02 618.47 3710.82 5269.37 31616.22 -692.94 88952.90 . Since , the curve is Leptokurtic. 50
- 51. www.crescent-university.edu.ng Exercise 5 Calculate the and for the data below and interpret your result. Class 10 – 14 15 – 19 20 – 24 25 – 29 30 – 34 35 – 39 40 – 44 45 – 49 f 4 8 19 1 35 20 7 5 Rates, Proportion and Index Numbers Proportion Proportion is the ratio of a number of items with certain characteristics (X) and the total number of items exposed to such characteristics (N). It is defined as . The above expresses the chance of occurrences of such characteristics (i.e. Probability of event X). Example: If the voting age population (people 18 years and above) in Ifo ward I consists of 550 males and 600 females. What is the proportion of males? Solution: , Total population Proportion of males, . . Rates When proportion refers to the number of events or cases occurring during certain period of time, it becomes a rate and is usually expressed as so many per 1000. Thus we refer to birth rate as the number of birth per 1000 population in a year. So also we have death rate, migration rate, marriage rate etc. 51
- 52. www.crescent-university.edu.ng INDEX NUMBERS There are various types of index numbers, but in brief, we shall discuss three kinds, namely: (a) Price Index (b) Quality Index (c) Value Index. (a) Price Index For measuring the value of money, in general, price index is used. It is an index number which compares the prices for a group of commodities at a certain time as at a place with prices of a base period. There are two price index numbers such as whole sale price index numbers and retail price index numbers. The wholesale price index reveals the changes into general price level of a country, but the retail price reveals the changes in the retail price of commodities such as consumption of goods, bank deposit etc. (b) Quantity Index This is the changes in the volume of goods produced or consumed. They are useful and helpful to study the output in an economy. (c) Value Index Value index numbers compare the total value of a certain period with total value in the base period. Here a total vale is equal to the price of commodity multiplied by the quantity consumed. NOTATION: For any index number, two time periods are needed for comparison. These are called the Base period and the Current year. The period of the year which is used as a basis year and the other is the current year. The various notations used are as given below: 52
- 53. www.crescent-university.edu.ng Price of current year Price of base year Quantity of current year Quantity of base year Definition (The weight): Different weights are used in different part of the country for a particular commodity. For instance, “congo” – western part of Nigeria, “mudu” – northern part etc. Problems in the construction of Index numbers No index number is an all purpose index number. Hence, there are many problems involved in the construction of index numbers, which are to be tackled by an economist or statistician. They are: 1. Purpose of the index numbers 2. Selection of base period 3. Selection of items 4. Selection of source of data 5. Collection of data 6. Selection of average 7. System of weighting METHOD OF CONSTRUCTION OF INDEX NUMBERS Index numbers may be constructed by various methods as shown below: INDEX NUMBERS Un weighted Simple aggregate index numbers Weighted Simple average of price relative 53 Weighted aggregate index number Weighted average of price relative
- 54. www.crescent-university.edu.ng Simple Aggregate Index Number This is the simplest method of construction of index numbers. The prices of the different commodities of the current year are added and the sum is divided by the sum of the prices of those commodities by 100. Symbolically, S.A.P.I, Where total prices for the current year total prices for the base year And we noted that Price Relative Index . Example 1: Calculate index numbers from the following data by simple aggregate method taking prices of 2007 as base. Commodity Price per unit (in N) 2007 2013 A 80 95 B 50 60 C 90 100 D 65 97 Solution: Commodity Price per unit (in N) 2007 2013 A 80 95 B 50 60 C 90 100 D 65 97 285 352 Total 54
- 55. www.crescent-university.edu.ng S.A.P.I, Simple Average Price Relative Index In this method, first calculate the price relative for the various commodities and then average of these relative is obtained by using arithmetic mean and geometric mean. When arithmetic mean is used for average of price relative, the formula for computing index is S.A. of P. R. by arithmetic mean where n is the number of items. When geometric mean is used, the following formula is adopted: Example 2: From the following data, construct an index for 2012 taking 2011 as base by the average of price relative using (a) arithmetic mean (b) geometric mean. Commodity Price in 2011 Price in 2012 Beans 50 70 Elubo 40 60 Rice 80 100 Garri 20 30 Solution: 55
- 56. www.crescent-university.edu.ng Commodity Price in 2011 ( ) Price in 2012( ) Beans 50 70 140 2.1461 Elubo 40 60 150 2.1761 Rice 80 100 125 2.0969 Garri 20 30 150 2.1761 Total 565 8.5952 (a) Simple average of price relative index by arithmetic mean (b) Price relative index number using geometric mean . Weighted Aggregate Index Number In order to attribute appropriate importance to each of the items used in an aggregate index number, some reasonable weights must be used. There are various methods of assigning weights and consequently index numbers have been devised of which some of the most important ones are: 1. Laspeyre’s method 56
- 57. www.crescent-university.edu.ng 2. Paasche’s method 3. Fisher’s Ideal method 4. Bowley’s method 5. Marshall – Edgeworth method 6. Kelly’s method 1. Laspeyre’s method This is a weighted aggregate price index, where the weights are determined by quantity in the based period and is given by: Laspeyre’s price index . 2. Paasche’s method It is a weighted aggregate price index in which the weight are determined by the quantities in the current year. The formula for constructing the index is given by: Paasche’s price index . 3. Fisher’s Ideal method This is the geometric mean of the Laspeyre and Paasche indices. Symbolically, Fisher’s Ideal index method . 4. Bowley’s method This is the arithmetic mean of Laspeyre’s and Paasche’s method. Symbolically, Bowley’s price index number . 57
- 58. www.crescent-university.edu.ng 5. Marshall – Edgeworth method In this method, the current year as well as base year prices and quantities are considered. The formula in using to get this is given as follows: Marshall – Edgeworth price index . 6. Kelly’s method Kelly has suggested the following formula for constructing the index number: Kelly’s price index number where . Here the average of the quantities of two years is used as weights. Example 3: Construct price index number from the following data by applying (a) Laspeyre’s method (b) Paasche’s method and (c) Fisher’s ideal method. Commodity 2000 Price Quantity 2012 Price Quantity A 2 8 4 5 B 5 12 6 10 C 4 15 5 12 D 2 18 4 20 Solution: 58
- 59. www.crescent-university.edu.ng Commodity A 2 8 4 5 16 10 32 20 B 5 12 6 10 60 50 72 60 C 4 15 5 12 60 48 75 60 D 2 18 4 20 36 40 72 80 172 148 251 220 L.A.P.I.N . P.R.I.N . F.I.I.N Interpretation: The results can be interpreted as follows. If N100 were used in the base year to buy the given commodities, we have to use N145.93k in the current year to buy the same amount of the commodities as per the Laspeyre’s formula. Other values give similar meaning. Example 4: Calculate the index number from the following data by applying (a) Bowley’s method (b) Marshall – Edgeworth price index. 59
- 60. www.crescent-university.edu.ng Commodity Base year Current year Quantity Price Quantity Price A 10 3 8 4 B 20 15 15 20 C 2 25 3 30 Solution: Commodity A 10 3 8 4 30 24 40 32 B 20 15 15 20 300 225 400 300 C 2 25 3 30 50 75 60 90 500 422 380 324 (a) B.P.I.N . (b) M.E.P.I.N . Example 5: Calculate a suitable price index from the following data. 60
- 61. www.crescent-university.edu.ng Commodity Quantity Price 1999 2000 X 20 2 4 Y 15 5 6 Z 8 3 2 Solution: Here the quantities are given in common and so we can use Kelly’s index price number. K.P.I.N . Remark: 1. For Quantity or Volume index number, interchange and in the formula for Laspeyre, Paasche and Fisher’s ideal. 2. For Consumer Price index number: (a) Aggregate Expenditure method: Consumer Price Index Number (based on Laspeyre) (b) Family Budget method or Method of Weighted Relative Consumer Price Index Number weight i.e. where and value . Exercise 6 1. Five feed components are to be used in the construction of an animal feedstuff index number. From the figures given in the following table, calculate 61
- 62. www.crescent-university.edu.ng (i) simple average of price relative index by arithmetic mean and by geometric mean. (ii) Laspeyre’s price index number (iii) Paasche’ price index number (iv) Fisher’s ideal index number (v) Bowley’s price index number (vi) Marshall – Edgeworth price index number (vii) Kelly’s price index number Component 1914 Price per ton 2014 Consumption(tons) Price per ton Consumption A 40 3600 41 2750 B 39 2750 53 1500 C 38 2050 35 2350 D 37 500 30 750 E 36 1475 24 2850 2. From the following data, compute quantity indices by (i) Laspeyre’s method (ii) Paasche’s method (iii) Fisher’s method. 2008 Commodity Price 2010 Total value Price Total value X 10 100 12 180 Y 12 240 15 450 Z 15 225 17 340 Hint: Quantity 62
- 63. www.crescent-university.edu.ng 3. Construct the consumer price index number for 2012 on the basis of 2002 from the following data using Aggregate expenditure method. Price in Commodity Quantity consumed 2002 2012 A 100 8 12 B 25 6 7 C 10 5 8 D 20 15 18 4. Calculate consumer price index by using Family Budget method for year 2013 with 2012 as base year from the following data. Price in Items Weights 2012 (in N) 2013 (in N) Food 35 150 140 Rent 20 75 90 Clothing 10 25 30 Fuel 15 50 60 Miscellaneous 20 60 80 63

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