Descriptive Statistics

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Harold Kennedy
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1 Page Techcal Math II Descrptve Statstcs Descrptve Statstcs Descrptve statstcs s the body of methods used to represet ad summarze sets of data. A descrpto of how a set of measuremets (for eample, people s weghts measured pouds) s dstrbuted volves two dfferet kds of summary statstcs. These are called measures of ceter ad measures of spread. Measures of ceter volve a typcal score about whch all the other scores are dstrbuted. The most commoly used measures of ceter are the mea, the meda ad the mode. The mea or arthmetc average s just the sum of all scores dvded by the umber of scores. The meda s the mddle score wth as may scores above t as below. The mode s the score that occurs most ofte (has the hghest frequecy). The mode s oly really meagful for large sets of data where t s lkely for the same score to be measured more tha oce. For sgle-peaked symmetrc dstrbutos the mea, meda ad mode are all the same. Therefore the dffereces betwee these three measures are a dcato of skewess or departure from a sgle-peaked symmetrc dstrbuto. The most commoly used measures of spread are the stadard devato, the rage ad the terquartle rage. The stadard devato measures a typcal dstace of scores the dstrbuto from the mea. The rage s just the dstace from the lowest score (the mmum) to the hghest score (the mamum). The ter-quartle rage s the dstace over the mddle half of the data from the score that marks the frst quarter pot to the score that marks the thrd quarter pot. For the famous bell-shaped curve (what s geerally called a ormal or Gaussa dstrbuto) 68% of all scores are foud wth oe stadard devato of the mea ad 9.% of all scores are wth two stadard devatos of the mea. More geerally, for ay dstrbuto of scores, at least 7% of all scores must be wth two stadard devatos of the mea.

2 Page Techcal Math II Descrptve Statstcs To make the calculato of these statstcs easer the followg otato wll be used. The measuremets whch make up the data set (the sample) wll be desgated as the subscrpted varable, where the de rus from to. Furthermore, t wll be assumed that the scores have bee sorted from smallest to largest (a creasg sort). Thus, s the mmum ad s the mamum. The formulas for the mea ad stadard devato are most ofte stated usg summato or sgma otato. Here the symbol Σ, whch s the Greek letter upper case sgma,

3 Page 3 Techcal Math II Descrptve Statstcs stads for the mathematcal operato of summato ad the sum s wrtte cocsely as. The varable s called a dummy summato de ad s just used to sgfy that varous values of are beg summed. The desgato beeath the sgma dcates where the sum begs ad the above the sgma dcates where the sum eds. I statstcs t s early always the case that sums beg at ad ed at, so the followg abbrevated symbols are ofte employed The mea s usually dcated by (called bar) ad ts calculato s dcated by the formula. A secod symbol for the mea (called the populato mea, used whe all possble scores have bee measured) s the Greek letter lower case mu, μ. The stadard devato s ( ) calculated by the followg formula s. Here the symbol s s used for the sample stadard devato. If all possble scores have bee measured the we have the slghtly dfferet formula (whch gves a slghtly smaller result) for the populato stadard devato ( μ) σ. The Greek letter lower case sgma s used to desgate that ths s for a populato of scores as opposed to just a sample. Both of these formulas ca be mapulated usg algebra to gve the computatoally more effcet formulas show below. ( ) ( ) s ( ) μ ; σ. Here order of operatos s crucal. meas that each s frst squared ad the summed (.e., ths s the sum of squares). I cotrast, ( ) ( ) meas that the s are frst summed ad ths aswer s the squared (.e., ths s the square of the sum). For eample, f wth, 3, 3 4, 4, ad 7, the , whle ( ) ( ) Regardg these scores as a sample gves the results ; s.36, whle regardg these scores as a populato (the set of all possble scores for the problem of terest) gves the results μ 4 ; σ 4. These formulas may seem etremely complcated. Fortuately, they are bult to may calculators. It s strogly recommeded that you lear.

4 Page 4 Techcal Math II Descrptve Statstcs how to properly use the statstcal fuctos o your calculator so as to avod havg to perform the eplct summatos. O the Caso f-300ms frst press the MODE key (the very top, secod butto from the rght) ad from the lst of optos preseted choose (SD for Stadard Devato ). To eter the data keystroke the value ad the press the DT (data butto) whch SD mode s the M+ butto (foud above the AC butto). For eample, the data set of fve scores used above s etered as M+ 3 M+ 4 M+ M+ 7 M+. To get the sample mea,, eter SHIFT S-VAR (S-VAR s the key). To get the sample stadard devato, σ, eter. To get the populato stadard devato, σ, eter SHIFT S-VAR. The values for, ad ca be accessed wth the meu actvated by the commad sequece SHIFT S-VAR 3 S-SUM SHIFT (S-SUM s the key). Before eterg a ew data set t s ecessary to remove old data from the calculator s memory. Ths s accomplshed by accessg the CLR meu wth SHIFT CLR (CLR s the same butto as MODE) ad the pressg to actvate Scl whch clears the statstcal memory. O the TI-30Xa to eter the data keystroke the value ad the press + (the data butto, located the frst colum, fourth row). For eample, the data set of fve scores used above s etered as After each score s etered the curret value of s dsplayed To get the sample mea eter d X (whch s d ), to get the sample stadard devato eter d σ - (whch s d ), ad to get the populato stadard devato eter d σ (whch s d ). The values for, ad are dsplayed by usg d ), (, ad EE respectvely. To clear a data set from the calculator s memory press the OFF butto.

5 Page Techcal Math II Descrptve Statstcs The meda of a set of scores ca be calculated by the followg procedure. From the frst score ( ) to the last score ( ), there s a chage of posto of magtude. The meda s the score halfway through ths chage. Thus, the poter or de to the meda s gve by + ( ). Now, f s odd (say ), ths de pots to a actual score ( 8 ) whch has as may scores above t as below. O the other had, f s eve (say 6), ths de pots to a o-estet score ( 8. ). Ths s to be terpreted as the value that s half-way betwee 8 ad 9. Ths s epressed eplctly the followg formula: for 6, M d ( 9 8 ). Ths procedure ca be geeralzed to calculate the quartles Q ad Q 3 (Q M d ), whch are the scores oe quarter ad three quarters of the way through the data respectvely. The de of Q (called the frst quartle) s gve by + ( ), ad the de of Q 3 (called the thrd quartle) s 4 3 gve by + ( ). For eample, f 6, the de to Q s 4.7 ad the de to Q 3 s 4., so that Q ( 4 ) ad Q ( 3 ). The terquartle rage or IQR s the dstace from Q to Q 3,.e., IQR Q 3 Q. The values of Q, M d, ad Q 3 are the bass of represetg the data what s called a bo plot or a bo ad whskers plot. I ts smplest form ths s a scaled dagram wth a straght le (whsker) draw from the mmum score ( ) to Q. From Q to Q 3 a bo s draw wth the posto of M d marked. A secod whsker the draw from Q 3 to the mamum score ( ). Ths costructo geerates a graphcal represetato of the dstrbuto of the data set. For large data sets where the same scores occurr more tha oce t s coveet to orgaze the data to a frequecy dstrbuto. The frequecy, f, s the umber of tmes a gve score occurs. Thus, the total umber of scores s just the sum of the frequeces,.e., f. I computg the mea ad stadard devatos we eed the sum of all scores, whch meas that each score should be multpled by how may tmes that score happeed. Thus the formulas preseted earler are ofte rewrtte as show below. f f ; f s ( ) f f ( f ) ; σ ( μ) ( f ) The mode of a frequecy dstrbuto s the just the score wth the largest frequecy. To locate scores t s coveet to troduce the cumulatve frequecy, whch s just the umber of occurrg scores less tha or equal to the score questo. Ths s a rug sum of the frequeces. The output of a spreadsheet s dsplayed o page. It shows the results for a frequecy dstrbuto of 0 scores. To llustrate the use of the cumulatve frequeces, cosder f f

6 Page 6 Techcal Math II Descrptve Statstcs the scores 8 ad 9 ths table. Sce the cumulatve frequecy of the score 8 s ad the cumulatve frequecy of the score 9 s 67, we kow that through 67 are all a score of 9. The followg procedure o the Caso f-300ms allows you to easly eter a frequecy dstrbuto:. Eter a gve score.. Eter ; (.e., SHIFT, ). 3. Eter the value of that score s frequecy. 4. Press the M+ key. The dsplay wll the show the curret value of, the umber of scores etered so far. If the scores are raked from lowest to hghest ths correspods to the cumulatve frequecy of the score.. Repeat ths process utl all scores have bee etered. The correspodg procedure o the TI-30Xa s as follows:. Eter a gve score.. Press d FRQ (.e., d / ). 3. Eter the value of that score s frequecy. 4. Press the + key.. Repeat ths process utl all scores have bee etered. To perform comparable statstcal calculatos o the ewer le of TI calculators, such as the TI-30X IIB, perform the followg:. Press d STAT (.e., d DATA ).. Press to select -VAR statstcs. 3. Press DATA. 4. Eter the frst score after.. Press the dow arrow butto to advace to FRQ. 6. Eter the frequecy of the frst score.

7 Page 7 Techcal Math II Descrptve Statstcs 7. By repeated use of the dow arrow butto eter all of the remag scores ad ther assocated frequeces. 8. After the frequecy of the last score has bee etered, press the STATVAR key. 9. Use the rght arrow to access the computed values of the sample mea,, the sample stadard devato, s, ad the populato stadard devato, σ. A bar chart of frequecy versus the values of the scores s called a frequecy hstogram ad gves aother vsual represetato of the dstrbuto of scores. Eample of a Frequecy Dstrbuto for 0 Scores The mea or average of the scores s desgated by the Greek letter : (mu). X (Score) f (Frequecy) Cumulatve f f*x f*x f*(x-:) f*(x-:) Sum : E(f*X)/ 9.97 Mode 9 Rage -3 ( f * X) f * X f * ( X X) s ( f * X) f * X f * ( X X) σ 3.00 Ide(Q ) + 0.*(-) 30.7 Q X *(X 3 -X 30 ) 7+0.7*(7-7) 7 Ide(M d ) + 0.*(-) 60. M d X *(X 6 -X 60 ) *(9-9) 9 Ide(Q 3 ) + 0.7*(-) 90. Q 3 X *(X 9 -X 90 ) + 0.*(-).

8 Page 8 Techcal Math II Descrptve Statstcs Hstogram of Data Frequecy f Scores (X)

is the score of the 1 st student, x

is the score of the 1 st student, x 8 Chapter Collectg, Dsplayg, ad Aalyzg your Data. Descrptve Statstcs Sectos explaed how to choose a sample, how to collect ad orgaze data from the sample, ad how to dsplay your data. I ths secto, you wll