hp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations

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1 HP 30S Statstcs Averages ad Stadard Devatos Average ad Stadard Devato Practce Fdg Averages ad Stadard Devatos

2 HP 30S Statstcs Averages ad Stadard Devatos Average ad stadard devato The HP 30S provdes several fuctos to calculate statstcs,.e. quattes that descrbe some propertes of a sample or of the whole populato ( ths last case, some authors prefer the term parameter), amely: Average or arthmetc mea (symbols: x, µ ).The average of quattes x 1, x,, x s defed as the sum of the quattes dvded by the umber of quattes: x x These quattes ca have frequeces f 1, f,, f so that f, whch case the average s ( fx ). A smlar cocept s that of the weghted average. The weghted average of quattes each havg weghts w 1, w x w. O the HP 30S averages ca be calculated by selectg the x ad y optos of w,, w s ( ) ( ) the STATVAR meu 1-VAR STAT mode (up to 40 data tems plus frequeces) or -VAR STAT mode (up to 40 pars of data tems). Eve though o specfc weghted average fucto s provded, t ca be easly calculated as show oe of the examples below. Sample ad populato stadard devatos (symbols: S ad σ, respectvely). The stadard devato s a measure of how dspersed the data values are about the average. The dfferece betwee the sample ad the populato stadard devato s that the former assumes the data s a samplg of a larger, complete set of data, whereas the latter assumes the data costtutes the complete set of data. They ca be calculated as follows: σ x ( x ) ( x ) ( x x) x ( x x) S 1 1 where s the umber of data pots. The sample stadard devato s calculated usg 1 as the dvsor. The HP 30S ca also calculate grouped stadard devato (whe data pots occur at gve frequeces). It ca be proved (Tchebycheff s equalty) that betwee the mea ad ±k σ are at least 100 ( 1 k )% of the data pots, regardless of the dstrbuto of the data. (Ths s also true for the sample stadard devato, because σ S 1 S > ). The stadard devato caot be egatve. Its square s kow as the varace., ( ) σ x y, x, y ad xy whch are useful calculatg other statstcs. Practce fdg averages ad stadard devatos Example 1: The sales prce of the last 10 homes sold the Parkdale commuty were: $198,000; $185,000; $05,00;$5,300; $06,700; $01,850; $00,000; $189,000; $19,100; $00,400. What s the average of these sales prces ad what s the sample stadard devato? Would a sales prce of $46,000 be cosdered uusual the same commuty? Soluto: Frst of all press 1select CLR-DATA y to be sure that o statstcal data remas from prevous calculatos. Now, let s put the data 1-VAR STAT mode: hp calculators - - HP 30S Statstcs Averages ad Stadard Devatos - Verso 1.0

3 HP 30S Statstcs Averages ad Stadard Devatos 1select 1-VAR ya 198u3??185u3??0500??53 00??06700??01850??u5??18 9u3??19100??00400? The average ad stadard devato are both show the STATVAR meu. Press to dsplay the average ad to dsplay the sample stadard devato. Aswer: The average of the sales prces s $00,355 ad the sample stadard devato s $11, Wth four stadard devatos o ether sde of ths average,.e. betwee $155, ad $45,111.18, at least ( 1 ) % 4 of all home sales prces wll fall. If a home were to sell for $46,000 ths area, t would be a uusual evet. Example : Below s a chart of daly hgh ad low temperatures for a week of July Bueos Ares, Argeta. What were the average hgh ad low temperatures for that week? Suday Moday Tuesday Wed. Thurs. Frday Sat. Hgh Low Soluto: We ll put the data -VAR mode, storg the hgh temperatures x ad the low oes y : 1select CLR-DATA y 1select -VAR ya 11??14?0?10?t1?8?t6?9?t5?8?t 4?7?t3? Press b@ to dsplay the average hgh temperature ( x ) ad to dsplay the average low temperature ( y ). Aswer: The average hgh ad low temperatures were 9.6 ad.6, respectvely. Example 3: Emma has bought gas ths week whle showg houses at four gasole statos as follows: Gallos Cost per gallo $1.56 $1.64 $1.70 $1.58 What s the average prce of the gasole purchased? Soluto: I ths case we have to calculate a weghted average. You wo t fd a fucto o the HP 30S STATVAR meu to calculate weghted averages; but, as oted o page, the weghted average calculato s 1 Remember that ths s true regardless of how the data s dstrbuted. Depedg o the dstrbuto, ths percetage ca actually crease. For example, f the data s ormally dstrbuted, 95.5% of the data pots wll fall wth µ±σ. hp calculators HP 30S Statstcs Averages ad Stadard Devatos - Verso 1.0

4 HP 30S Statstcs Averages ad Stadard Devatos mathematcally equvalet to the calculato of the average of grouped data (.e. data that occurs wth gve frequeces). Therefore, the average prce ca be calculated as follows: 1select CLR-DATA y 1select 1-VAR ya 1.56?15?1.64?7?1.7?10?1.58?17? Now, smply press b@ to dsplay the aswer. Alteratvely, sce the weghted average s defed as X w ( wx ) ( w ) follows: X w xy y, we ca calculate t as provded the umber of gallos purchased s stored, as above but ths tme -VAR mode because the xy ad y calculatos are preset the STATVAR meu oly -VAR mode: 1select -VAR yb<<</b<<<<yy Aswer: The average prce per gallo Emma has pad ths week whle showg houses s slghtly less tha $1.61. Example 4: Judgg by the coeffcet of varato, what ca we say for the followg data f t comes from the same populato? y Soluto: The rth momet about a value a s defed as: mr r (x a) x. If a 0 ad r the m. S The coeffcet of varato s defed as: CV. It s ofte gve as a percetage, that s: ( S x) 100. x Let s put the data 1-VAR mode remember to clear the prevous data frst, press 1select CLR- DATA y, ad the: 1select 1-VAR ya 1045??300??13??5??45??90??970??8? We ca ow fd the secod momet by pressg: b</byy I -VAR mode the umber of gallos purchased wll be cosdered a depedet varable, ad therefore the resultg average wll be wrogly calculated as ( ) / hp calculators HP 30S Statstcs Averages ad Stadard Devatos - Verso 1.0

5 HP 30S Statstcs Averages ad Stadard Devatos To fd the coeffcet of varato, press: Aswer: m Roudg to two decmal dgts, CV1.57, or 157%. The coeffcet of varato of postve data comg from a homogeeous populato s ormally less tha 100%. If t s greater tha 150%, the data probably comes from heterogeeous sources (e.g. from people of dfferet sex, age, etc.) hp calculators HP 30S Statstcs Averages ad Stadard Devatos - Verso 1.0

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