Measures of Dsperson Defenton Range Interquartle Range Varance and Standard Devaton Defnton Measures of dsperson are descrptve statstcs that descrbe how smlar a set of scores are to each other The more smlar the scores are to each other, the lower the measure of dsperson wll be The less smlar the scores are to each other, the hgher the measure of dsperson wll be In general, the more spread out a dstrbuton s, the larger the measure of dsperson wll be Defnton Measures of Dsperson Measure of dsperson estmate the spread or varablty of a dstrbuton around the centre Dsperson s a key concept n statstcal thnkng. The basc queston beng asked s how much do the scores devate around the Mean? The more bunched up around the mean the better your ablty to make accurate predctons. Whch of the dstrbutons of scores has the larger dsperson? The upper dstrbuton has more dsperson because the scores are more spread out That s, they are less smlar to each other 5 00 5 5 0 5 00 5 5 0 4 5 9 0 4 5 9 0 4 Measures of Dsperson If all values are the same, then they all equal the mean. There s no varablty. Varablty ests when some values are dfferent from (above or below) the mean. We wll dscuss the followng measures of spread: range, quartles, varance, and standard devaton The Range The range s defned as the dfference between the largest score n the set of data and the smallest score n the set of data, X L - X S What s the range of the followng data: 4 9 9 The largest score (X L ) s 9; the smallest score (X S ) s ; the range s X L - X S 9 -
When To Use the Range The range s used when you have ordnal data or you are presentng your results to people wth lttle or no knowledge of statstcs The range s rarely used n scentfc work as t s farly nsenstve It depends on only two scores n the set of data, X L and X S Two very dfferent sets of data can have the same range: 9 vs 5 9 Quartles Three numbers whch dvde the ordered data nto four equal szed groups. Q has 5% of the data below t. Q has % of the data below t. (Medan) Q has 5% of the data below t. Quartles Unform Dstrbuton Obtanng the Quartles Order the data. For Q, just fnd the medan. For Q, look at the lower half of the data values, those to the left of the medan locaton; fnd the medan of ths lower half. st Qtr Q nd Qtr Q rd Qtr Q 4th Qtr 9 0 Obtanng the Quartles For Q, look at the upper half of the data values, those to the rght of the medan locaton; fnd the medan of ths upper half. Obtanng the Quartles Poston of -th Quartle: poston of pont Q ((n+))/4 Eg. n Ordered Array: Poston of Q ((9+))/4 0/4.5 Q.5
The nterquartle range Poston of Q ((9+))/4 0/45 Q Poston of Q ((9+))/4 0/4.5 Q9.5 s the dstance between the lower quartle and the upper quartle. 5 0 5 The nterquartle range gves the spread of the mddle % of the data often the bt you are most nterested n. Dfference Between Thrd & Frst Quartles: Interquartle Range Q-Q 9.5-.5 Not Affected by Etreme Values It s not affected by any etreme values. A pet shop owner weghs hs mce every week to check ther health. The weghts of the 0 mce are shown below: weght (g) (f) 0 < w 0 0 < w 0 5 0 < w 0 5 0 < w 40 9 40 < w < w 0 5 0 < w 0 4 0 < w 0 0 < w 90 90 < w 00 4 4 0 0 (f) 0 < w 0 0 < w 0 5 0 < w 0 5 0 < w 40 9 40 < w < w 0 5 0 < w 0 4 0 < w 0 0 < w 90 90 < w 00 4 Cumulatve 4 0 0 Cumulatve frequency 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 90 00
From ths graph we can now fnd estmates of the medan, and upper and lower quartles The lower quartle s the 0 th pece of data ¼ of the total peces of data The upper quartle s the 0 th pece of data ¾ of the total peces of data Medan poston Lower quartle Cumulatve frequency Lower quartle s g Medan weght s 54g Upper quartle s g Upper quartle 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 90 00 Varance and Standard Devaton varablty ests when some values are dfferent from (above or below) the mean. Each data value has an assocated devaton from the mean: Devatons what s a typcal devaton from the mean? (standard devaton) small values of ths typcal devaton ndcate small varablty n the data large values of ths typcal devaton ndcate large varablty n the data Varance Varance Formula of Sample Fnd the mean Fnd the devaton of each value from the mean Square the devatons Sum the squared devatons Dvde the sum by n- s (n ) n ( ) (gves typcal squared devaton from mean) 4 4
Standard Devaton Standard Devaton Formula typcal devaton from the mean Standard devaton varance Varance standard devaton s ( n ) n ( ) [ standard devaton square root of the varance ] 5 BPS - 5th Ed. Chapter Varance and Standard Devaton Eample from Tet Metabolc rates of men (cal./4hr.) : 9 4 40 49 9 + + + 4 + 40 + + 49,00 00 Varance and Standard Devaton Eample from Tet Observatons Devatons Squared devatons ( ) 9 9 00 9 (9),4 00 () 4,5 00 - (-) 5,44 4 4 00 4 (4) 9 40 40 00-40 (-40) 9,00 00 (),9 49 49 00 - (-) 5,9 sum 0 sum 4,0 Varance and Standard Devaton Eample from Tet s s 4,0 5,. 5,. 9.4 calores BPS - 5th Ed. Chapter 9 Varance Formula of a Populaton When calculatng varance, t s often easer to use a computatonal formula whch s algebracally equvalent to the defntonal formula: ( X ) X N X µ σ N N ( ) σ s the populaton varance, X s a score, µ s the populaton mean, and N s the number of scores 0 5
Varance Formula of a Populaton Computatonal Formula Eample X X X-µ (X-µ) 9 4 4-5 5-4 4 - Σ 4 Σ 0 Σ 0 Σ σ X 4 0 0 94 ( X ) N N σ ( X ) µ N Range for grouped data Interquartle Range Grouped n - F M e d a n L m + f m Where: n the total frequency F the cumulatve frequency before class medan Fm the frequency of the class medan f m L m the class wdth the lower boundary of the class medan Quartles Usng the same method of calculaton as n the Medan, we can get Q and Q equaton as follows: n - F 4 Q L Q + fq Tme to travel to work 0 0 0 40 4 n - F Q L Q + 4 fq Eample: Based on the grouped data below, fnd the Interquartle Range 4 9
Soluton: st Step: Construct the cumulatve frequency dstrbuton Tme to travel to work 0 0 0 40 4 4 9 n Class Q. 5 4 4 Class Q s the nd class Therefore, Cumulatve 4 4 n - F 4 Q LQ + fq. 5-0. 5 + 0 4. 4 Q ((n+))/4 ( ) n Class Q. 5 4 4 n - F 4 Q L Q + fq. 5-4 0. 5 + 0 9 4. 9 Interquartle Range IQR Q Q IQR Q Q calculate the IQ IQR Q Q 4.9.4 0.4 A pet shop owner weghs hs mce every week to check ther health. The weghts of the 0 mce are shown below: weght (g) (f) 0 < w 0 0 < w 0 5 0 < w 0 5 0 < w 40 9 40 < w < w 0 5 0 < w 0 4 0 < w 0 0 < w 90 90 < w 00 4 4 0 0 (f) 0 < w 0 0 < w 0 5 0 < w 0 5 0 < w 40 9 40 < w < w 0 5 0 < w 0 4 0 < w 0 0 < w 90 90 < w 00 4 Cumulatve 4 0 0 Cumulatve frequency 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 90 00
From ths graph we can now fnd estmates of the medan, and upper and lower quartles The lower quartle s the 0 th pece of data ¼ of the total peces of data The upper quartle s the 0 th pece of data ¾ of the total peces of data Medan poston Lower quartle Cumulatve frequency Lower quartle s g Medan weght s 54g Upper quartle s g Upper quartle 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 90 00 Standard devaton for Grouped Varance and SD Standard devaton for Grouped Standard devaton for Grouped
Standard devaton for Grouped Standard devaton for Grouped Standard devaton for Grouped Alternatve formula 9
Standard devaton for Grouped eercse 0