1« IN I t «PKI llh<. OUANTirAIIVI DATA WtIM %f\% A lew mora »«•"»•' ntuai be defined to he able to go forth« in «mi Hudy, Wc need to talk j huk- iboul wtabfcl ind ihcdi «vjn-s Vati.ibli". and Mcisutemonl A variable Í8 a characteristic or quality that is observed, measured, ami recorded in a dttt tile l generally, in j single column). If you need 10 keep irock of (lie country of birth of ihe individuals in youi population, you will include in your study B win ahk- called Country ofbirlk. You may also »am io keep track of the nationality of the Individuals: you will itien Imvc another variable called NiiUiumtliy. I he two van ahlcs .ire ilislmcl. mi».1 some people nut) i.iih ihr nationality ni i touniiN ntliei than the one they weie born in Here are some examples ol variables utcd widely m MKÚd «letKCs Smlu-ilcmonfapble variables 5e* Religion level of education lliülicst degree obtained Marilal natal Country of birth Nauonaln) Mother tongue ľs>iholo«úal wniublcs I evel of anxiety Slumilus res|Hiiis,- ume Si on- obuineil m a personality test S,oie obtained hi .i" jptHude lest rioimmie variables Working status Income Value of individual assets Average number of hours, ol work per week Vm iuliles that refer lit units other than I In- iiiiliuilu.il Numlvr ol Inispilals in UCDUfHr) Percentage ol people who cen nud Percentage ol people who completed high school total population Birth raw li-mliiv rale Numlwr ol" teachers per I(HH) people Number nfdtKi.>ts per io.mi people Population L'tiiwth I'ledoininanl lelieioil You iii.iv have noUCCd that some til ibcSC v .»übles letci to i|iiahlies Im« h u iiioihei tongue) ami other* refer to quantities, such .is Ihe total population "i u countrj in ia.i l people m a boasehold, the Size Of a building, or ihe annual sales ol a producl Qualitative variables are Julk ten silts or qualities that are not numerical, sud) as mother tongue, or country Of Origin. The semes of the individuals Ol a popu I at inn on the various variables are called the values ol that variable Example Suppose you have the information shown in Table 1.2 aboul five students m vi'i.....Hece Table 1.2 EurnpUi of qualitative and quantitativ« variable« Nam« Wa Mai? Pd.-r Omim s ,1,1,- l"a 17 Itt ri Program of Study Sisul Soieoct Pure ami Appl»«l Sewnce < minima: Oflto SyuCTu Tttiaolog? tltuptiK Pwign________ Grad» Point Average 3.78 í.« M..-»VC» .1« llicie ne (hire vaitallies \y_, («|U itiliiaUVi] PrOgMBI "I Bind) (QU ifluUiví), .nul t.rade Point \vcrajje imwnlilativcl fhe VBhWli '»i sMues. taken by ihe individual« fOJ ihe vaiiahle Alte IIB 1 l, IH, L9(lwk0) Ud 10 fhlIVA! Mt I taten N'i ihe variable Cnntrani of Shuly arc Sucial Science, hm- and Applied Science, Cummer, i ' Ulice System« l iľ -l-'Ky. and Graphic Design. (Jualitaiive variables -tie sometimes referred to as categorical variables because (hey consist of categories ni which ihe population «in be classitied For instance, wc can classify all students in a college into categories according io the program of Study they are in. í arem) mention must iv ;:i.en to the wa> iibaofvatsDns penaining n a variable are "■'""'■>> We mu« find o system.....wording the data thai is very dear, and that can be uiierprewd without any ambiguii) CtMsider. i.« msumce. ihe roUowing char» lemtli i: age rank in the la.miy. und.....the, tongue ľne first characteristic ia a uuoniM) the sect....... a rank, and ihe third is......dii) i he 0 *wi.....d io record ""' <*wrvations al.......hesc chcracteristio. will he organiiwd into ihm levels ur measurement' 13 36 0933 12 INTÉBPflETING QUANTITATIVE DATA WITH SPSS • i measurement ai the nominal level; • measurement at the ordinal level; and • measurement at the numerical scale level. Each level of measurement allows us to perform certain statistical operations, and not others. The nominal level of measurement is used to measure qualitative variables. It is ihe simplest system lor writing down our observations: when we want to measure a Characteristic at the nominal level, we establish a number of categories in such a way-thai each observation falls into one and only one of these categories. For example, if you want to write down your observations about mother tongue in the Canadian context, you may have the following categories: • English. • French, • Native, and • Other. Depending on the subject of your research, you may have more categories to include other languages, or you may want to make a provision for those who have two mother tongues. It is important to note that when a variable is measured at the nominal level, the categories must be • exhaustive, and • mutually exclusive. The categories arc said to be exhaustive when ihey include the whole range of possible observations, that is. they exhaust all the possibilities. Thai means that every one of the observations can fit in one of the available categories. The categories arc said to be mutually exclusive if they are not overlapping: every observation fits in only one c-aiegory. These two properties ensure thai the system used lo write down the observations is clear and complete, and that there arc no ambiguities when recording the observations or when reading the data file. Tabic 1.3 displays examples of measurements made at the nominal level. Qualitative variables must be measured ai ihe nominal level. The ordinal level of measurement is used when the observations are organized in categories that arc ranked, or ordered. We can say lhal one category precedes another, but we cannot say by how much exactly lor if we can. we do not keep that information). I lere too the categories must be exhaustive anil mutually exclusive, but in addition you must be able to compare any iw« categories, and say which one precedes the other (or is bigger, or better, etc.). Table 1.4 displays examples of variables measured at the ordinal level. THE 8A51C LANGUAGE Of STATISTICS 13 Table 1.3 Examples of variables measured at the nominal level Variable Place of bi nh Woik M3IUS Categories used Male Female Ttie country where the survey is couitduiTetl Ahead Working full-iimc Working pan-time *l<*n>|»i4iilv oui of work Unemployed Retired Housekeeper Ohei Table 1.4 Examples of variables measured at the ordinal level Variable Rating of a rmuuani Rank arming siblinns Ranked Categories Excellent Ver, good Acceptable Poor Very I"** First child Second child eic. High Medium !■.■". The scale used to wriie down an ordinal variable is often referred to as a I.tkert scale. Ii usually has a limited number of ranked categories: anywhere from three to seven categories, sometimes more. For instance. If people arc asked m rate a service as: D Excellent Q Very good O Good O Poor □ Very poor, the proposed answers constitute a five-level Liken scale. 14 INTERPRETING QUANTITATIVE DATA WITH SPSS • Another example of n Liken scale, this (inn- with Ibui levels, i\ provided by the situations whewasliilcmcnl is given, and respondents arc asked to say whether they: Ü Totally agree □ Agree U Disagree G Totally disagree. A variable measured ai the ordinal level could he either qualitative or quantitative. In Tahle 1.4. the variable Income is quantitative, and the variable Kating of a Restaurant in qualitative, bill they arc ImuIi measured al llie ordinal level. Hor a variable measured at the ordinal level, we can say thai one value precedes another, but we cannot give an exact numerical value lor the difference between them. For instance, if we know lhal a respondent is the first child and the other is the second child in the same family, we do noi keep track of the age difference between them. It could Ik* one year1 in ime use and live years in another case, hut ihe values recorded under this variable do not give us this information: they only give us ihc rank. When recording information about categorical variables, the information is usually coded. Coding is die operation by which we determine the categories that will be recorded, and the codes used to refer to them. For instance, if the variable is Sex, and ihe two possible answers arc: Male Female. we usually code this variable as 1 Male 2 Female. The numbers 1 and ~ are the codes, and the categories Male and Female are the values of the variable. When coding a variable, a code must be given to the cases where no answer has been provided by thtf respondent, or when ihe respondent refuses to answer (if the answer is judged too personal or confidential, such as the exact income of a person). We refer to these answers as missing values and we give them different codes. Lab <> explains how to handle Ihcm in SPSS. Finally, some variables are measured hy a numerical stale Fvm observation is measured against the scale and assigned a numerical value, which measures a quantity, These variables are said to Ix* quantitative, fable 1.5 displays examples of numerical scale variables. ruf BASIC LANGUAGE Of STATISTICS table 1.5 Examples of variables measured at the numerical «ale level II Variable Numerical Scale Annual income in dollars, without decimals (n» cents) Age In years, wiih im fraction* Age In year*, with one decimal for fractions of a year Temperature In degrees Celsius Tin« In years. A siaumg pouti mu\i he specified Annual Income In dollar*, m ihe .-umicm thimsiind Notice thai the s;imť variable can be measured hy different scales, as shown in the examples above. So. when we use a numerical scale, we imisi determine the units used Ifor instance years or months!, and the number of decimals used. Numerical scales are sometimes subdivided into interval scales and ratio scuU\s. depending on whether there is an absolute zero to the scale or not. Thus, temperature ami time arc measured hy interval scales, whereas age and number of children are each measured by a ratio scale. However, this distinction will not be relevant for most of what we are doing in this course, and we will simply use the term numerical scale to talk about this level of measurement. The program SPSS thai we are going to use simply uses the term scale to refer to such variables. Most statistical software packages include more specific ways of writing down the observations pertaining to a numerical scale. For instance. SPSS will otter the possibility of specifying thai Ihe variable is a currency, or a dale. Moreover, it is also possible lo group the values of a quantitative variable into Classes. Thus, when observing (he variable OR«, we can write down the exacl age of a person in years, or we can simply write Ihe age group ihc person tails in. as is done in the following example: • 18 lo 30 years • 31 to 40 years • 41 to 50 years • 50 to 60 years • Over 60. When we group a variable such as age into a small number of categories as we have just done, wc musí code the eaiegories as we do tor categorical variables. For example, 1 would stand for ihe category IS to l<> years 2 would stand for ihe category 31 to 40 years eic. 15 IC INKRPRETING QUANTITATIV* DATA WITH SPSS 1 In such situations, we cannoi perform ihe same statistical operations tlmi we do when the values arc noi grouped. For instuncc. ihc mean, Or average of the variable (/i;<- is best calculated when (he ages arc not grouped. When we group ihe values, n is because we »'Hi 10 know (he relative importance (thai is. mc frequency, in percentage) or one group as compared lo the others. The information thai Ml'« <>l the population is under 20 years old in some developing countries is obtained by grouping the ages into 20 years old or less and more than 20 years old. When we collect the data, it is always better to collect it in actual years, since we can easily group it laier on in the data tile with the help of a statistical software package, [n this case, a new column is added to the data lile. and it contains the grouped data of the quantitative variable. For example, in the <;SS93 subset data tile that we use in the SPSS labs, you will find two variables for age: one is called age. and the other one is called agecat4. The latter is calculated from the former, by grouping individuals into lour age groups. In the column of agecaťl, the specific age of an individual is not recorded: only the age group of the individual is recorded. Finally, numerical scales can be either continuous or discrete. A scale is said to be continuous if the observations can theoretically lake any value over a certain range, including fractions of a unit. For instance, age. weight, length arc continuous variables bťcausc ihey are not limited to specific values, ami they can lake any value within B certain range. A variable is said to be discrete if H can take only a limited number of possible values, but noi values in between. Foi instance, the variable Number ofihildren is measured by a discrete scale because il can only be equal to a whole number: 0, I. 2. etc. Importance of the Level of Measurement The level of measurement used for a variable depends on whether it is qualitative or quantitative-Qualitative variables must be measured at the nominal or ordinal level. They cannot l>e measured ai the numerical scale level, even when their categories arc coded with numbers. For instance, as shown above, we usually code the variable Sex as follows: 1 Male 2 Female. In this t'QSCi »"* mimhers I and 2 have no numerical value. They arc simply codes. It is shorter to write I than male, and we could have assigned Ihe numbers differently. If you ask SPSS to compute ihc mean (or average) for a variable coded in this way. you will gel a numerical answer. Kul you must always keep in mind thai such a numerical answer is totally meaningless because the level of measurement ■ THE 8ASIC LANCUAfiE Of STATISTICS 17 'S of lhal variable is nominal. The numbers used to record the information are simply codes. (Juaiililalive variables are usually measured by a numerical scale, hut Ihey could be measured at Ihe ordinal level also, lor iiisiance. if you have the ;.....ual income o ľ an individual, you may treat it as a numerical scale, but you could also group ihe values into Low. Medium and High income and treat the variable at the ordinal level. When you perform a statistical analysis of data, it is very important to pay attention to the level of measurement of each variable. Some statistical computations are appropriate only to a given level of measurement, and should not be performed if the variable is measured at a different level. Concepts, Dimensions, and Indicators We oltcn want to observe social phenomena that are too abstract and complex to )>c expressed by a single variable. Suppose for instance that we want to observe and measure ihc degree of religious inclination (or (he tendency of a person towards religion) in a given social group. Religious inclination can be manifested in many ways: people may have or not have certain beließ about their religion: they may also perform or not certain rituals such as attending religious services, fasting, praying, etc.: ihey may also seek the advice «/ the religions leadership on important decisions, or ignore such leadership: finally, ihey may seek 10 look at everything from the poinl of view of religion, and apply Ihe teat'hing.x of their religion in their daily lives, or ignore them. All these aspects are not found all the time in all individuals. Some individuals may have strong beliefs, while avoiding ihe religious services. Other may attend all services while being skeptical about some of the religious dogma. The way to handle this complexity is to subdivide the concept of religious inclination t mo dimensions, which are themselves measured by several indicators, if we were to study religious inclination in the Catholic religion, we would gel a set or dimensions and indicators that would look as in Table 1.6 (we are simplifying the issues a little, of course). The items listed on the right-hand side of Table I .ň are indicators of the concept of religious inclination. None of them, taken alone, is a measure of religious inclination, but each of them constitutes one aspect of it Indicators that are seen as similar are grouped together to form one dimension of the concept. And finally the various dimensions, taken together, capture the «mcept as a whole. This way of breaking down a complex concept into dimensions and indicators is vailed the »pcr-atioiiali/ation of the concept. As an illustration, we may want lo see how economists ope rationalize ihe concept of cost of living- Ihey estimate the average cost of most of the standard expenses a family of four is expected to incur. The various expenses are divided into main dimensions such as food, housing, transportation, education, and leisure. Lach dimension is then subdivided into smaller dimensions; themselves subdivided further until indicators are reached. For instance food is 31 46 1» IN-ÍBPMTIHC QUANTITATIVE DATA WITH \P\% Table'1.6 Eximpl» of how a concept can bo brokun down into dimension» and indicator« Concept Dimensions Indicators. RELIGIOUS INCLINAflON I IkllstS H Rituals III (iuHlancc IV, Daily life Urlnf inti'xl Btltll m Ihc Holy Tnnily liclkl III (lie mum dogma i-1. AtWtfctMkC ot tei>icti IVrfunninrf player» H.i|iii'HiK i'hililicn •Mi i . ihc jwiesl J»»*i> CoMMilling (he offwul «puuons ol the chunk im ceiiain issue» such as binh coetrol ,i, HťiiiK lnul .mil geiKnui I» people Nul i «'.iinf (iihcis in LXMntncrcul trUUCHom M broken down as: meat, vegetables, milk producta, etc.. themselves subdivided Into specific item* such a» lonialocs.Ieilucc.clc. ľinnlly, ľ« each of diese indicators, ihc increase or oecnaM in ihc COM of living is measured against Ute corresponding cosi in some yew. culled the ruse year. By combining! (hew indicator,, economise .in-able 10 measure how ihc cost of living has changed, on the average, foi I i.imily of four. The way j omu-pl is broken down, or opcralioiializcd. into dimension» and iasScalori depends on the theoretical framework .atonicd for a study Rescauhci. may not agree on how to operalionalize a concept, and you will find in the Iitcraiutc different sluilie» lhal operalionalize concepts in completely ditfcrcni ways, because they rely on dillcieni theoretical fhuneworkl Summary Quantitative nothodl arc procedures and le.htuqtics for collecting, organizing, describing, analyzing, and interpreting »lala In ihisshapler we has e icamed ihc besk vocabulary mad to talk about quantitative ifMdxxh Dan is osgaarzed into elccironk dala tiles wilb the help ol statistical pack.ij.vs A data tile contains ÜK vuluc» taken by a number ol cases (which are ihc uniis ol ihc population under study] over some variables, hveiy row represents a case, while every column represent» a Variable The unil» in ihc il.itj lile usually form j »ample .stu»h i» it-Neil a Mih%et ol ihc whole population. Soiitcliincs. the data file refers 10 «lie whole population. Till SASIC LANGUAGE O' STATISTICS H The variables can be cilher qualitative or quantitative. The system used to record* the Information is called a meaiurotnant scale, There ait- uvea levels of measurement: nominal, ordinal and numerical (interval or ralio) The level ol measurement ol .i ■, .mahle will determine »hat statistical procedures can he pwfanitBJ, and what kind ol graphs must be used to illustrate the dala. When a concept is complex, it is not measured directly. It is usually broken down into dimensions and indicators, which arc then combined to provide a »ingle measure. 'Ihc statistical procedures themselves (all mto two hioud ttegori«»: descriptive »tad»IK» and inferential statistics IVwnpdse statistka) techniques ami at describing the data by summarizing it. while inferential statistical techniques aim ai generalizing m a whole population whm ha» been observed on n sample Keywords Siudeiiis should he able to deli in* and explain oil ihc following lerms Da» ii.n.i ni Case Unit Quantitative methods Vmahk variable label Sample ľ .i .i . i level nl measurement Value Value label Nominal level Ordinal level Variable type Numerical level interval s l_hiantitativc variable QualilMive variable Exhaustive Mutually exclusive categories Ltkeil scale Continuous numerical scales Disctele iiunienc.il scales Codes Coding Codebook Statistici (tne two meanings) Descriptlvo statistics Inferential statistics Dimension» <>i .,.....,.■-•■ Indicaiors of .1 concept OperattotuHmlon of s concept Suggestions for Further Reading BlalOCa li. Huben \l : -• _ i .•.u,pimil,ztiti|.i Saddle River. NJ: IVilllic Mall Rosenbaum, Soni* (1979) Quuimiauve Mtffteds ,„iä Statistku \ (kilét /». Seesai Research l(.....| HiHv SjSe PublkJth-nn. Itudcl. Kotvil and Antonius. R-h.nl > |WI > Methode* auamiiaii.e\ applique,% aux sciences humainrt Montreal: OK". 4249