Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean


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1 1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class. Stadard deviatio ad variace for grouped data See text, sectio 5.9.2, pp Whe workig with data that are grouped ito categories or itervals, the variace ad stadard deviatio are agai obtaied usig the deviatios about the mea ad the squared value of these. But i this case, the sum of squares of differeces1 about the mea are weighted by the umber of times each occurs. The defiitios are as follows. Defiitio. A variable X that takes o values X 1, X 2, X 3,...X k with respective frequecies f 1, f 2, f 3,...f k has mea X Σ(fX)/ where Σf. From Module 3, this is the formula for the mea usig grouped data. The variace is Variace s 2 Σf(X X) 2 1 ad the stadard deviatio is the square root of the variace, that is, Stadard deviatio s s 2. Begiig with data that are grouped, agai the mea is obtaied first, the the differece of each value of X from the mea is calculated. These deviatios about the mea are squared ad the multiplied, or weighted, by the respective frequecies of occurrece (f). While the above method ca be used, it results i awkward ad timecosumig calculatios. A more straightforward procedure is to use the followig alteratie formula. See text pp
2 SOST 201 October 13, Stadard deviatio for grouped data 2 Alterative formula. A alterative formula for the variace ad stadard deviatio for grouped data, givig exactly the same result, is Variace s 2 1 [ ΣfX 2 (ΣfX)2. 1 The stadard deviatio is the square root of the variace Stadard deviatio s s 2. The formula give i the defitio is computatioally iefficiet, so i this class we geerally use the alterative formula for the variace. I the examples ad exercises that follow, oly the alterative formula is used. See text pp for proof of equivalece of formulae ad a example. Steps used i tabular format. As with ugrouped data, a tabular format is ordiarily used to obtai the variace ad stadard deviatio. Employig the alterative formula, the procedure for calculatig the variace ad stadard deviatio is as follows: Create a table with the values of X i the first colum ad the frequecies of occurrece f i a secod colum. Create a third colum fx with the products of the f (from secod colum) ad the X (from first colum). Sum the f etries i the secod colum to determie the sample size, that is, Σf. Sum the products, fx, i the third colum to obtai the colum total ΣfX. Divide this sum by to determie the mea of X ΣfX/. To this step, the procedures are idetical to those used i Module 3 for calculatig the mea of grouped data. Create a fourth colum with values of fx 2, that is the f multiplied by the square of the X. The square of the X values multiplied by the frequecies f are etered ito the fourth colum. Also, this is equivalet to multiplyig the fx of the third colum by aother X (the value i the first colum). That is, the fourth colum is the etry
3 SOST 201 October 13, Stadard deviatio for grouped data 3 i the third colum, multiplied by the etry i the first colum. This produces the values of fx 2. Sum the values i the fourth colum to obtai ΣfX 2. The sums of the fourth colum (ΣfX 2 ) ad the third colum (ΣfX) are etered ito the formula s 2 1 [ ΣfX 2 (ΣfX)2 1 ad this is the variace. The stadard deviatio s is the square root of the variace of the last step, that is, s s 2. I terms of procedures, it is geerally preferable to proceed row by row. Begi by eterig all the X values i the first colum ad all the frequecies f i the secod colum. The proceed row by row to obtai the etries i the third ad fourth colums. That is, for the first row, calculate fx ad eter it i the third colum; the obtai fx 2 ad eter it i the fourth colum. The go to the secod row ad do the same, ad so o, util all the etries i each row have bee obtaied. Fially, sum all the etries i each colum ad eter the sums ito the appropriate place i the formula for the variace ad stadard deviatio. The followig example illustrates the use of the alterative formula.
4 SOST 201 October 13, Stadard deviatio for grouped data 4 Example 4.8 Variatio i alcohol cosumptio by icome level. The data i Table 1 is adapted from Statistics Caada, 1991 Geeral Social Survey  Cycle 6: Health. Table 1: Distributio of Saskatchewa adults by umber of alcoholic driks cosumed per week ad by persoal icome Number Number of respodets of driks with persoal icome per week < $20,000 $20,000 plus Noe Total Use these data to compute the mea ad stadard deviatio of the umber of alcoholic driks cosumed per week for adults i each of the two icome groups. Usig these statistics ad the data i Table 1, write a short ote comparig the two distributios. Aswer. The variable i this questio is umber of alcoholic driks cosumed per week. Assumig that ay drik is equal i alcoholic cotet to ay other drik, this variable has a ratio scale. That is, the uit is oe drik ad a value of zero meas o alcoholic cosumptio. Low icome group. The table for the calculatios of the mea ad stadard deviatio of the umber of alcoholic driks cosumed per week for Saskatchewa adults at the lower icome level is Table 2. This table is costructed for the alterative formula, that is, with colums for the values of f, X, fx, ad fx 2. The first step is to obtai the X values associated with each category ito which the data are grouped these are the midpoits of
5 SOST 201 October 13, Stadard deviatio for grouped data 5 Table 2: Calculatios for measures of variatio of umber of alcoholic driks cosumed per week icome of less tha $20,000 Number of driks per week X f fx fx 2 Noe , , ,192 Total ,696 each iterval (0, 3, 8, 13, ad 32). The frequecies of occurrece for each iterval, f, are copied from Table 1. The ext colum is the fx colum ad the fial colum the fx 2 colum. For the first row, multiply the f value of 178 by the X value of 0. This gives a value of fx 0 ad fx 2 0. For the secod row, f 76 ad X 3, so the etry i the fx colum is Multiplyig this fx by aother X produces This is the value etered ito the fial colum. Note also that fx 2 f X X However, it is more efficiet to obtai this etry for the fial colum by multiplyig the etry i the fx colum by aother X. For the third row, f 28 ad X 8, that is, there are 28 respodets who drik a average of 8 driks per week. The etry i the fx colum is Multiplyig this fx by aother X produces , 792 ad this is the value etered ito the fx 2 colum.
6 SOST 201 October 13, Stadard deviatio for grouped data 6 For the fourth row, f 12, X 13, so fx Multiplyig this by aother X 13 gives fx , 028. Fially, the last row has f 8 ad X 32, so fx ad fx 2 fx X , 192. The sum of the f colum is 302, the sum of the etries i the fx colum is ΣfX 864, ad the sum of the etries i the last colum is ΣfX 2 12, 696. Usig these values i the formulae for the mea ad variace gives X ΣfX or a mea of 2.9 driks per week. The variace is [ s 2 1 ΣfX 2 (ΣfX)2 1 1 [ 12, [ 746, , , 696 2, , The stadard deviatio is the square root of the variace, or s s Rouded to the earest teth of a drik, the stadard deviatio for lower icome idividuals is 5.8 driks per week. Higher icome group. For the higher icome group,the tabular format for the calculatios is provided i Table 3. The format ad procedures are the same as for the lower icome group. From Table 3, the sum of the f colum is 235, the sum of the
7 SOST 201 October 13, Stadard deviatio for grouped data 7 Table 3: Calculatios for measures of variatio of umber of alcoholic driks cosumed per week icome of $20,000 plus Number of driks per week X f fx fx 2 Noe , , ,432 Total 235 1,304 24,466 etries i the fx colum is ΣfX 1, 304 ad the sum of the etries i the last colum is ΣfX 2 24, 466. From the sums i Table 3, the mea is X ΣfX 1, or 5.5 driks per week. The variace is [ s 2 1 ΣfX 2 (ΣfX)2 1 1 [ 1, , [ 1, 700, , , 466 7, , ad the stadard deviatio is s s or 8.6 driks per week.
8 SOST 201 October 13, Stadard deviatio for grouped data 8 These statistics are summarized i Table 4. From these statistics, higher icome idividuals both cosume more alcohol, ad are more varied i their alcohol cosumptio patter, tha lower icome idividuals. Mea alcohol cosumptio for those with higher icomes (5.5 driks per week) is almost double that for those with lower icomes (2.8 driks per week). I additio, the stadard deviatio for higher icome idividuals is 8.6 driks per week, over half agai as great as for the lower icome idividuals (s 5.8 driks per week). Table 4: Summary Statistics for Alcoholic Driks Cosumed Low High Measure Icome Icome Mea X Variace s s.d s
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