Inferential: methods using sample results to infer conclusions about a larger population.
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1 Chapter 1 Def : Statstcs: 1) are commoly kow as umercal facts ) s a feld of dscple or study Here, statstcs s about varato. 3 ma aspects of statstcs: 1) Desg ( Thk ): Plag how to obta data to aswer questos. ) Descrpto ( Show ): Summarzg the obtaed data. 3) Iferece ( Tell ): Makg decsos ad predctos based o data. Chapter - Data Def : A populato cossts of all elemets whose characterstcs are beg studed. Ex.1) A sample s a porto of the populato selected for study. Ex.) A parameter s a summary measure calculated for populato data. A statstc s a summary measure calculated for sample data. Types of statstcs: Descrptve: methods to vew a gve dataset. Iferetal: methods usg sample results to fer coclusos about a larger populato. Def : A varable s ay characterstc that s recorded for subjects a study. - Qualtatve (categorcal): caot assume a umercal value but classfable to or more o-umerc categores. - Quattatve (umercal): measured umercally. - Dscrete: oly certa values wth o termedate values. - Cotuous: ay umercal value over a certa terval or tervals. Chapter 3 Categorcal Data Graphs Def : A frequecy table (for qualtatve data) s a lstg of possble values for a varable, together wth the # of observatos for each value. Major Frequecy (f) Relatve frequecy Percetage (%) Scece Arts Busess Nursg Other Relatve frequecy = f f Percetage = Relatve frequecy 100%
2 Graphcal Summares Def : A bar chart s a graph of bars whose heghts represet the (relatve) frequeces of respectve categores. Ex3.1) (precedg table used class) Look for: frequetly ad frequetly occurrg categores. A pe chart s a crcle dvded to portos that represet (relatve) frequecy belogg to dfferet categores. Ex3.) (precedg table used class) Look for: categores that form large ad small proportos of the data set. A segmeted bar chart uses a rectagular bar dvded to segmets that represet frequecy or relatve freq. of dfferet categores. Ex3.3) (precedg table used class) Chapter 4 Numercal Varable Graphs Def : A stem-ad-leaf dsplay has each value dvded to two portos: a stem ad a leaf. The leaves for each stem are show separately. (Values should be raked.) Look for: - typcal values ad correspodg spread - gaps the data or outlers - presece of symmetry the dstrbuto - umber ad locato of peaks Ex4.1) Note: Dotplots also exst (see p. 5 textbook), but replace the values wth dots. Def : A hstogram, lke a bar graph, graphcally shows a frequecy dstrbuto. The data here, however, s quattatve. Look for: - cetral or typcal value ad correspodg spread - gaps the data or outlers - presece of symmetry the dstrbuto - umber ad locato of peaks The data dvde to tervals (ormally of equal wdth). Cumulatve Relatve Frequecy = (Cumul. freq. of a class) / (Total obs s dataset) Table 4X0 Total eargs as of Ja. 6/015 Worldwde Box Offce Number of moves ( mllos) f 00 to to to to to to to 3000 Relatve Frequecy Cumulatve rel. freq.
3 Ex4.) (draw class usg above data) NOTE: Dot ad S-ad-L plots are good for small data sets because data values are retaed. Hstograms are better for large data sets to codese the data. Hstogram shapes/trats: (correspodg fgures draw class) 1. Modes (umodal, bmodal, multmodal, uform). Skewess (symmetrc, left-skewed & rght-skewed) term refers to TAIL 3. Tal weght (ormal, heavy-taled, lght-taled) Def : A tmeplot s a graph of data collected over tme (or a tme seres). Look for: - a tred over tme, deotg a decrease or crease. - a patter repeatg at regular tervals (a cycle or seasoal varato) Ex4.3) (draw class) Chapters 4/5 Summary measures (ad oe more graph) Measures of Ceter Def : A outler s a obs that falls well above or below the overall bulk of the data. y y y 1 + y y Populato mea: µ = Sample mea: y = = N The meda s the value of the mdpot of a data set that has bee raked order, creasg or decreasg. If dataset has a eve # of observatos, use the average of the mddle values. Note: meda resstat to outlers, mea uses all observatos. Table 5X0 Estmated provcal populatos crca Jul. 011 ( mllos) ON PQ BC AB MB SK NS NB NL PEI Ex5.1) Avg. pop of all provces: Avg. pop from sample of 3 provces: y y µ = = = y = = = N 10 3 Meda pop of all provces: Outler effect? (remove Otaro & Quebec) y meda = = y = = = Comparg Mea ad Meda: (correspodg fgures draw class) 1. Symmetrc curve & hstogram - the are detcal, le at ceter of dstrbuto. Rght-skewed: Meda < Mea 3. Left-skewed: Mea < Meda
4 Def : The mode s the most frequet value a data set. Ex5.) Provces Moves Measures of Spread Def : Rage = largest value smallest value = max m Ex5.3) (from Table 5X0) rage = Devatos from the Mea: Ex5.4) 1,, 4, 3 y y y = = = = 0.5 ( y y) = 0 Note that ( y µ ) ad ( y), aka devato of x from the mea, both equal zero. y Varace ad Stadard Devato: The most commo measure of spread s stadard devato. Iformally terpreted as the sze of a typcal devato from the mea. Varace, however, must be calculated frst. The basc formulas for varace are: ( y µ ) ( y ) y σ = s = N 1 where σ s the populato varace ad s the sample varace. ( y ) Sce ( y y) = y, the varace formulas become y 1 ( ) y 1 ( ) σ = y ad s = y N N 1 Fdg the stadard devato oly requres takg the postve square root of the varace. Populato: σ = σ Sample: s = Ex5.5) 1,, 4, 3 s
5 Importat otes: 1. Stadard devato measures spread oly about the mea (.e. ot the meda).. Values of varace ad std. dev. are ever egatve. (Equals zero oly f o spread.) 3. The measuremet uts of varace are always the square of the uts of the orgal data. 4. Stadard devato, lke the mea, s ot resstat to outlers. 5. Cosder the sample varace s to have 1 degrees of freedom. There are observatos, ad devatos from the mea. Sce the total always sums to zero, 1 of these quattes determes the remag oe. Thus, oly 1 of the devatos, y y, are freely determed. (Degrees of freedom apply oly to samples.) Measures of Posto Def : The p th percetle s a value such that p percet of the observatos fall below or at that value. Three useful percetles are the quartles. The frst quartle has p = 5, the secod quartle (a.k.a. the meda) has p = 50, ad the thrd quartle has p = 75. The fve-umber summary cossts of the m, Q 1, meda, Q 3, ad the max. (vsual represetato of above draw class) Def : The terquartle rage (IQR) s the dfferece betwee the frst ad thrd quartles. IQR = Q 3 Q 1 (IQR s actually a measure of spread) Ex5.6) (examples regardg fdg quartles wth other # s of observatos dscussed class) Note: For exams, INCLUDE the meda each half whe calculatg Q 1 ad Q 3. Boxplots: Def : A boxplot shows the ceter, spread, ad skewess of a data set. To costruct t: Step 1: Rak the data creasg order ad fd the meda, Q 1, Q 3, ad IQR. Step : Fd the pots beyod the boudares: 1.5*IQR below Q 1 ad 1.5*IQR above Q 3, kow as the lower & upper er feces, respectvely. These pots are outlers. Ex5.7) 1.5*IQR = Lower.f. = Upper.f. = Step 3: Determe smallest & largest values wth the respectve er feces. small = large = Step 4: Draw lear scale cotag etre rage of data. Step 5: Draw perpedcular les to the scale to dcate Q 1 ad Q 3. Coect eds of both les. Box wdth = IQR Step 6: Draw aother le perpedcular to the scale to dcate the meda sde the box. Step 7: Draw two smaller les perpedcular to the scale for the values from Step 3. Jo ther ceters to the box to make whskers. (boxplot draw class)
6 What to do wth outlers? Cosder lower & upper outer feces at 3.0*IQR below Q 1 ad 3.0*IQR above Q 3. Ex5.8) 3.0*IQR = Lower o.f. = Upper o.f. = A (mld) outler s outsde a er fece but sde the outer fece. A far (or extreme) outler s outsde ether outer fece. All textbooks are dfferet for dstgushg outlers. Our textbook uses ope crcles for mld ad astersks, *, for far outlers. Whskers exted o each ed to the most extreme observatos that are ot outlers. Ex5.9) Lookg at ceter, spread, ad skewess: Approx. value of the ceter? Wdth of IQR? Symmetrc or skewed? Boxplot vs. Hstogram: Each graph hghlghts dfferet features of a data set (layers of skewess ad skewess/modalty, respectvely), so t s always better to costruct both. Chapter 6 Stadard Devato as a Ruler Emprcal Rule apples oly to a bell-shaped dstrbuto % of observatos le wth 1σ of the mea.. 95% of observatos le wth σ of the mea % of observatos le wth 3σ of the mea. Suppose we go further, say, 5σ. Software produces a value of %, whch meas far less chace for error (.e. the observatos beyod 5σ from the mea). Extra Measure of Posto/Potetal Outler Idetfer z-score = (observato mea) / (std. dev.) - z-score tells us how may stadard devatos the value s from the mea, postve OR egatve - more useful whe dstrbuto approxmately ormal. - a potetal outler s more tha 3σ from the mea. Ex6.1) µ = 31.6 σ = 6.4 y = 50
- Inferential: methods using sample results to infer conclusions about a larger pop n.
Chapter 6 Def : Statstcs: are commoly kow as umercal facts. s a feld of dscple or study. I ths class, statstcs s the scece of collectg, aalyzg, ad drawg coclusos from data. The methods help descrbe ad
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