Statistics fundamental, Transcriptions of Statistics

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—— | o*. « INTRODUCTION In the modern world of information and communication technology, the ad ed «* 9 ane = o © importance of statistics is very well recognised by all the disciplines. Statistics -z-., rd . 3 has originated as .a science of statehood and found applications slowly and ;%-— ‘ bias eae in Agriculture, Economics, Commerce, Biology, Medicine, Industry, 3 bg ie ae {ane of today, there is no other human waite of life; whe e 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 in oration avout a given- - scbiect Statistic is a science of understanding data and m aking decisions in the. face of randomriess. | _ J ' The, study of statistics is therefore. escontial for SO und reasoning, precise EEN and objective decision in the face o° up- to- date accurate and reliabie “ata Thus. smany researchers, ‘educationalists, business men and sovenment | “agencies: ‘at the national, ‘state or. local levels rely on data to answer operations . ‘ ‘Stat osn and programs. § tistics is usual mutually elution namely: Descriptive statistics-and inferential statistics. ally divided into two categories, which is not "DESCRIPTIVE STATIS TICS This is the act of summarizing and given a descriptive accoun t of numerical oo info: ‘mation in form of reports, charts and diagrams. ‘She goal of descriptive -. statistics is to gain information from collected data. It begins with collection of data by either counting or measurement i in an inquiry. It involves the summary of specific "aspect of the data, such | as averages values and measure of dispersion “7 at = - (spread). suit DLE | graphs, diagram 1S and. charts are then used. to gain | | ‘ " | << "s understanding an and clear interpretation of the phenomenon under investigation | . | | i ee eee ee _ eee Qn err ge 8 ye OF See * , ~--* ra ' ‘ se iv * 4 ‘ ‘ - . keeping firmly in mind where the data comes from. Normally, a descriptive statistics should: i.’ - be single - valued — \~4ir- -***bealgebraically tractable iii. | consider every observed value. . INFERENT PIAL STATISTICS - This is the act of making deductive statement about a population — 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 : ‘poametes and hypothesis testing. directly from deidionalineit by personal | interview, measurements or observation. Statistical data can be obtained from: questionnaire, (i) ‘ Census ~ complete enumeration of all the unit of the population a). | Surveys — the study of representative part of 2 p2pulation : i pe i ft “1 Experimentation. - - observation from experiment carrieds out in “Jaboratories and research center. ea (iv). Administrative process e.g. Record of births and deaths. > *) ADVANTAGES | | . he v- Comprises of sctial data needed Capen = --~ ¥ -Itis more ‘reliable with clarity ov Comprises a more detail information | as ‘ "DISADVANTAGES — Sa beck. ia # tee VUE > Cost of data collection shih at 7 _ > Time consuming je i7 eee > There may larger range non response 7, - ee et | ae er sciences through laboratory works than in social sciences. But this is very useful studying small communities and institutions, ro. , INTERVIEWING | ! 44 mathad +h wm nmnnn nnllarntinn than Aataiioc eallad tha + TATETAM RAC EN Ael In this mecmnOG, tne PETSON COueCung tié Gata iS CauEG wie iiveTViEWer EVES LO adn the ‘person (interviewee) direct questions. The interviewer has to go to the interviewees personally to collect the information required verbally. This makes ~i different from the next method called questionnaire method. | QUESTIONNAIRE | i ia ee -Aset 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 f questionnaire. Human beings usually are required to respond to the questions or | statements on the. questionnaire. Copies .of the questionnaire can be tong tig dministéred personally by its 1 user or sent to people by post. Both interviewing | and questionnaire methods are used in the social sciences “where human - oi "ot . 4. . “ | ~ . . . * . . ; ° “population is mostly involved. Gi be» Fe SS fe Mh 4 i oa; - is ° . ’ 3 e* (4 pty, wee ey t- 9 wom Sameer ® es + @-stee ’ . . ’ “ “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. Va at Other methods includes: Telephone method, Document/Report method, Mail or Postal questionnaire, On-line interview method, etc. -— — ee © oe /} ENTATION OF DATA When raw data are collected, they are organized numericatly by nos distributing them into classes or categories in order to det ermine the nun ber OF ‘individuals belonging to each class. Most cases, itis necessary to present Cate in ‘tables, charts and diagrams in order to have a clear understanding of the data, and to illustrate the relati onship existing between the variables being.examined. ‘FREQUENCY BABU This is a tabular arrangement of data into various classes together with ‘sheix corresponding frequencies. , ; . : “Procedure for forming frequency distribution a _ Given a set of observation: xy; Mp) Xgy vec Xr for a single variable. “1. Determine the range (R) =L- Swhere L = rane observation in the raw -data; and $= sthallest observation i in the Taw data. - | 2 Determine the appropriate number of classes or sroups ®. The choice of eal XK is arbitrary but. as a general rule, ‘it should be a Thumber (integer) between 5 and 20 depending on the size of. the data given. There are several suggested guide linés aimed at helping one decided on how many | - ; ; class intervals to employ. Two of such, methods are: c (a K=1 43 kya 2 (log 07). | LA | | 6) K= oa , where n= = ‘number of observations. | es eae tl o° * OS R , 38. Determine the width fw) of the class interval. Ttis dete ratined as £2 4 Determine the numbers of observations falling into sich class interval 1: e. - find the class frequencies. _ 4 eee ~ oN OTE: With advent of computers, all these steps can be accomplishes easily. aos , ‘> lo a ee eee ee a oe ee _-——— © < -. _———. — . \ a Si ‘etc. In case of continuous variables, the exclusive method should be usea. The inclusive method should be used in case of discrete variable. % Open end cla asses: ‘ fir di 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 hich values or few ver 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” - £iass bounds; fes:. 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 95-195. - a i of a. ‘ . . ; . € ° ° . ‘ ’ 4 , . « Cumulative frequency: This is the sum 7G) : frequency of the —_ class to ~ the frequencies of the class before it. , Exanjple 1: The following are the marks of 50 students in STS 102: | 48 70 60 47 51.55 59 63 68 63 a7 53 72 53, 67.62 64 70 57 48° 51 58 62 65 62 49 64 359 63 50 61 67. 72 56 64 66.49 | ied be 52 62 71.5853 63 69 59 64 73 56. mC) Construct a frequency table for the above data. ce 0) Answer the following questions using the table obtained: a ie Speed - me how many students scored between 5 51 and 62? ) how many students scored: above e507 A Se otater cates oS) pone eg A ee Cit what i is the probability that a meres selected at random from the class a Ee ad will score eles than 63? it : a -* ‘ . ee . ** ° e ~ e ~ . : e' ® ie . . . ae $ 3 ; +8. t< e ‘ * 4e ee e . ” - e . ° . * >. = * J e 2 > e e ° ,. n : ” > ° . * *,'°* . "> “> ° e . i . “8. © P ° ° '_7 ° Oia . - ee . ° ‘ . . ° . . . . ‘se 6. § ° é = ’ - . ; ue . - ° o . A “ r ‘CG uscee * . » os ‘ . m . . -_ - ~ 5 ‘ ‘ese “+ - . ‘ es. ~ . = © i . ’ . we . » ® , - are P - » ® Ld . . ® * . a ae . * . - it -— Oe * = ~— a. o-@ - = . . . le — ¢ 2 ‘ : ogee. lta a : mons Oe . o , ’ . - ° * "oe ee Ad - — - . e_* e 8 oa ‘ ” ne? _ * ° . - ade ‘ se ¢ x . - . ® “+ . ’ . 4 . ® ae ‘+ ad es ‘ . _ ~teg * ‘ . s . . * . . t. ® 5 : ms ‘ e . . ’ " : a Solution: - wats . of gcteet hy of ae Cob Ne Aree Uae ane) nh Ss “ Range (R= 69— 2 67 2a on eee * se eee us bt a a0 se Solution: (a) Range (R) = 73- 47 = 26 No of classes. (k) =Vn =V¥50=7.07 = | 7 Class size (w) ) = 26/5 = 3.7 & 4 | . Frequency Table | a Mark | Tally Ifrequency | "4 +47-50 | tr || 7 51. 54 |e ||| 7 55- a8 | 7 , 59-2 ue ABBE: etl) | 67-70 91 atte | 71-74 uF - Rxample 2: The following data represent the ages fin years) of f people living i ina : housin g estate in “Abeoktuta. - | : | “sg 31 30 6 16 17 18 43°28 32 33° 9 18 33 19: 21 13°13 14 Soo 4a 6 52> 45° 61 23 26. 15 14 15 14 27 36 12 37. it 19 4 ie Ye S99" 12. 39° 20° 40 69 63 29 64 Oh: 15° 28. ty | oe Present the above data i in a frequency’ table showing the Gioia columns; class | vy { SF ‘interval, class boundary, class mark (mid-point, tally, frequency: and cumulative * ee : oe ; frequency i in that order. | | | . | ; "No of classes () = Vi V50 = 7.07 & “700, PSE ek aR eae gee Class width (w) = = = 61/y 295 10} oon es “* ——— —— ee ceme -_— —_ eee —— ’ ‘ = ae ¢ ‘ . ou=7 - = . — _§ epuae@=s 4 7 . Ye’. 2 j ‘ -e —— . a 6 ee Cee 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 Ey 3 rT. © 3 =F os a [2 (D =% 4 7 °D 2 J cD aa aca ...,Xn, the arithmetic mean denoted by X is defined by yu7 + X2 + Xo Fo +X _ nr i=1 Example . a2 “The ages: of ~~ ten students in STS 102 are 16,20,19,21,18,20,17,22,20,17, determine the mean age. ee Solution: X = }%4= Th 164+20+19+214+18+204+17422+20417 . _— ° e— —— — ee 10 - bg . — = 190 a Lovears. “ If the numbers %4,%2,_,X, occur f; f2 fs, -..f, times respectively, the | X ‘nome fixytfex2ttfartn _ _ Liz1 fix: (or 2§% gy. short.) fitfatetfn | Bier ht | _ nt =a. et tl - * * Rearearoner [3 Solution: Waiting (inumin) INo of custom SSMACK UC Viale ye ae 15-19 3 17 | 20-24 — 10 2.2 2 25-29 - 18 27 8.6 3.0-3.4 10, 3.20 82 35-39 | 7 37 9 4.0-44 : 42 4 , py =50 IS fx = 142 R= of | =e 2ns | han Use of Assume mean Calculation of mean from grouped data If the items of a frequency distribution are classified In Intervals, we mak: the assumption that every item in an interval has the mid-values of the Interval and we use this midpoint for x. Example 3: The table below shows the distribution of the waiting items for some customers In a certain petrol station in Abeokuta. 3.0-3.4 13.5-3.9 | 40-44 eee Find the average waiting time of the customers. Waiting time(in|] 15-19 | 2.0-2.4 mins) ef-s One ee er eee —=— 28 &-— ..- -_--— - —_ —— —_— — ns 7s ——— —_—-—- —_— af ci rder to make our computation ¢ easier, we may assume one of the values as the x . mean. Thiss{A=assumed mean,and d=deviation of x from A, i, e. d= de a A ‘y , * re : a7) he ‘= ometimes, large values of the variable are involve in calculation of mean, via esis tee ' Feat Weoed oe od ¥*" Example 5: Consider the data in example 3, using a suitable assume mean, compute the mean. Solution: Waiting time x 15-139 3 {1.7 -1 2.0 - 2.4 10 {2.2 -0.5 2.5-2.9 18 |2.7A |0 3.0 - 3.4 10 |3.2 0.5 3.5-3.9 7 | i 1 AN-44 2 4.2 had aie yaa+2t2 LJ =2.7+—. = 2.7+0.14 = 2.84 | — —_— «= d=x-A dl “3 “5 0 5 7 — NOTE: It is always easier to select the class mark with the longest frequency as the assumed mean. . 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. ®, « a *@ ee —— —_ —— a —— —— — es Game DISAVANTAGE OF MEAN re oe ee ee ee eee 7 ome a -- Its value is greatly affected by the extremely too large or too small observation. >= LLL TC Ce P " py TTT sat e . . ° = 4 THE HARMONIC MEAN (H.M) The H.M of a set of numbers x,, 3, ...,X, is the reciprocal of the arithmetic mean of the reciprocals of the numbers. It is used when dealing with the rates of the type x per d (such as kilometers per hour, Naira per liter). The formula is expressed thus: 1 1M = yr = —r n “i= =1x, 4i=1y, If x has frequency/, then HM =s7 Example: Find the harmonic mean of 2,4,8,11,4. . Solution: = 5 HiM = raya =4—— = 4,112, 24a teal r eo Note: a _ (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. | =| a oP) tml Cc —! Tl ttl | = = = — > = os The G.M is an analytical 1 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 X41 X2. «Xp, is the nth root of the product of the number. Thus | | i GM= VG@q Xx Xu x Xn) —_ EE ep ‘ ‘If f, is the frequency of x,, then —— 7 2 Ge eee G.M = NACSUE! x Male HK sos xi fi) See fae eae reer eee af Vee | Example: The rate. of inflation in fir€ successive year in a country was — ee re me rr eee oe ee qe eens o “Oe -——enE ee «w+ = ow so 5%, 8%, 12%, 25% and 34%, What was the average rate of inflation per year? eg OO (6) The median, ¥ =X es ° =Xq en = § Example 2: The value Of a random variable x are given as 15,15, 17,19, 21, 22, 25, and 28. Find the median. Solution: mnisodd ever Median, X¥ = Xn +X Calculation of Median froma grouped data tes - The formula for calculating the median from grouped data is defined as - LAE 9 Letina. KL + (=) w Sean ' Where: L, = Lower class boundary of the median class. __ ate : N =)' f =Total frequency Cf, = Cumulative frequency before the median class _ Sm = Frequency of the median class. ~ w=Class size or width. Example3: The table ‘below shows the height of 70 men randomly selected at Sango Ota. re - —_ —— — Compute the median, =~ ~ Solution Height Freauencv Cumulative frequency 118 - 126 3) 127-135 18 136 - 144 Wd 145 - 153 50 154 - 162 99 163 - i71 66 172 - 180 70 = 2S 4 =e ~-34. 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 L, = 144.5, n= 70,cf, = 32,w=9 n | | a (z-h \ | i | ne xX =L, + - W \ =] | x9 =: = 1445.4 (=). a = 144.5+1.5 = 136. ADVANTAGE OF THE MEDIAN (i) Its value is not affected by extreme values; thus it is a resistant: measure of central tendency. (iy 1t iS a £0 OU MeaSUTE O -- - - —_ ADVANTAGE OF THE MODE 1) Itis easy to calculate. DISADVANTAGE OF THE MODE (i) Itis not a unique measure of location. (ii) lt presents a misleading picture of the distribution. (iii) It does not take into account all the available data. Exercise 2 1. Find the mean, median and mode of the following observations: 5, “an eat 737 424 £ ah £ 6,1U,10,22,10,0,1U,0. 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 Calculate the (i)Mean (ii)Mode (iii) Median | tal _—— ee ee cc oe ee ee —+A es ore eee ee e ae