Important Concepts not on the AP Statistics Formula Sheet

Transcription

1 Part I: IQR = Q 3 Q 1 Test for a outlier: 1.5(IQR) above Q 3 or below Q 1 The calculator will ru the test for you as log as you choose the boplot with the oulier o it i STATPLOT Importat Cocepts ot o the AP Statistics Formula Sheet Liear trasformatio: Additio: affects ceter NOT spread adds to, M, Q 1, Q 3, IQR ot σ Multiplicatio: affects both ceter ad spread multiplies, M, Q 1, Q 3, IQR, σ Whe describig data: describe ceter, spread, ad shape. Give a 5 umber summary or mea ad stadard deviatio whe ecessary. Histogram: fairly symmetrical uimodal skewed right Skewed left Ogive (cumulative frequecy) Boplot (with a outlier) Stem ad leaf Normal Probability Plot r: correlatio coefficiet, The stregth of the liear relatioship of data. Close to 1 or -1 is very close to liear residual = residual = observed predicted y = a+b Slope of LSRL(b): rate of chage i y for every uit The 80 th percetile meas that 80% of the data is below that observatio. Epoetial Model: y = ab take log of y Power Model: y = a b take log of ad y HOW MANY STANDARD DEVIATIONS AN OBSERVATION IS FROM THE MEAN Rule for Normality N(µ,σ) N(0,1) Stadard Normal Eplaatory variables eplai chages i respose variables. EV:, idepedet RV: y, depedet r : coefficiet of determiatio. How well the model fits the data. Close to 1 is a good fit. Percet of variatio i y described by the LSRL o Lurkig Variable: A variable that may ifluece the relatioship bewtee two variables. LV is ot amog the EV s y-itercept of LSRL(a): y whe = 0 Cofoudig: two variables are cofouded whe the effects of a RV caot be distiguished.

2 Part II: Desigig Eperimets ad Collectig Data: Samplig Methods: The Bad: Volutary sample. A volutary sample is made up of people who decide for themselves to be i the survey. Eample: Olie poll Coveiece sample. A coveiece sample is made up of people who are easy to reach. Eample: iterview people at the mall, or i the cafeteria because it is a easy place to reach people. The Good: Simple radom samplig. Simple radom samplig refers to a method i which all possible samples of objects are equally likely to occur. Eample: assig a umber to all members of a populatio of size 100. Oe umber is selected at a time from a list of radom digits or usig a radom umber geerator. The first 10 selected are the sample. Stratified samplig. With stratified samplig, the populatio is divided ito groups, based o some characteristic. The, withi each group, a SRS is take. I stratified samplig, the groups are called strata. Eample: For a atioal survey we divide the populatio ito groups or strata, based o geography - orth, east, south, ad west. The, withi each stratum, we might radomly select survey respodets. Cluster samplig. With cluster samplig, every member of the populatio is assiged to oe, ad oly oe, group. Each group is called a cluster. A sample of clusters is chose usig a SRS. Oly idividuals withi sampled clusters are surveyed. Eample: Radomly choose high schools i the coutry ad oly survey people i those schools. Differece betwee cluster samplig ad stratified samplig. With stratified samplig, the sample icludes subjects from each stratum. With cluster samplig the sample icludes subjects oly from sampled clusters. Multistage samplig. With multistage samplig, we select a sample by usig combiatios of differet samplig methods. Eample: Stage 1, use cluster samplig to choose clusters from a populatio. The, i Stage, we use simple radom samplig to select a subset of subjects from each chose cluster for the fial sample. Systematic radom samplig. With systematic radom samplig, we create a list of every member of the populatio. From the list, we radomly select the first sample elemet from the first k subjects o the populatio list. Thereafter, we select every kth subject o the list. Eample: Select every 5 th perso o a list of the populatio. Eperimetal Desig: A well-desiged eperimet icludes desig features that allow researchers to elimiate etraeous variables as a eplaatio for the observed relatioship betwee the idepedet variable(s) ad the depedet variable. Eperimetal Uit or Subject: The idividuals o which the eperimet is doe. If they are people the we call them subjects Factor: The eplaatory variables i the study Level: The degree or value of each factor. Treatmet: The coditio applied to the subjects. Whe there is oe factor, the treatmets ad the levels are the same. Cotrol. Cotrol refers to steps take to reduce the effects of other variables (i.e., variables other tha the idepedet variable ad the depedet variable). These variables are called lurkig variables. Cotrol ivolves makig the eperimet as similar as possible for subjects i each treatmet coditio. Three cotrol strategies are cotrol groups, placebos, ad blidig. Cotrol group. A cotrol group is a group that receives o treatmet Placebo. A fake or dummy treatmet. Blidig: Not tellig subjects whether they receive the placebo or the treatmet Double blidig: either the researchers or the subjects kow who gets the treatmet or placebo Radomizatio. Radomizatio refers to the practice of usig chace methods (radom umber tables, flippig a coi, etc.) to assig subjects to treatmets. Replicatio. Replicatio refers to the practice of assigig each treatmet to may eperimetal subjects. Bias: whe a method systematically favors oe outcome over aother. Types of desig: Completely radomized desig With this desig, subjects are radomly assiged to treatmets. Radomized block desig, the eperimeter divides subjects ito subgroups called blocks. The, subjects withi each block are radomly assiged to treatmet coditios. Because this desig reduces variability ad potetial cofoudig, it produces a better estimate of treatmet effects. Matched pairs desig is a special case of the radomized block desig. It is used whe the eperimet has oly two treatmet coditios; ad subjects ca be grouped ito pairs, based o some blockig variable. The, withi each pair, subjects are radomly assiged to differet treatmets.

3 Part II i Pictures: Samplig Methods Simple Radom Sample: Every group of objects has a equal chace of beig selected. Stratified Radom Samplig: Break populatio ito strata (groups) the take a SRS of each group. Cluster Samplig: Radomly select clusters the take all Members i the cluster as the sample. Systematic Radom Samplig: Select a sample usig a system, like selectig every third subject. Completely Radomized Desig: Eperimetal Desig: Radomized Block Desig: Matched Pairs Desig:

4 Part III: Probability ad Radom Variables: Coutig Priciple: Trial 1: a ways Trial : b ways Trial 3: c ways The there are a b c ways to do all three. A ad B are disjoit or mutually eclusive if they have o evets i commo. Roll two die: DISJOINT rollig a 9 rollig doubles Roll two die: ot disjoit rollig a 4 rollig doubles A ad B are idepedet if the outcome of oe does ot affect the other. Mutually Eclusive evets CANNOT BE Idepedet For Coditioal Probability use a TREE DIAGRAM: P(A) = 0.3 P(B) = 0.5 P(A B) = 0. P(A U B) = = 0.6 P(A B) = 0./0.5 = /5 P(B A) = 0. /0.3 = /3 For Biomial Probability: Look for out of trials 1. Success or failure. Fied 3. Idepedet observatios 4. p is the same for all observatios P(X=3) Eactly 3 use biompdf(,p,3) P(X 3) at most 3 use biomcdf(,p,3) (Does 3,,1,0) P(X 3) at least 3 is 1 - P(X ) use 1 - biomcdf(,p,) Normal Approimatio of Biomial: for p 10 ad (1-p) 10 the X is appro N(p, ) Discrete Radom Variable: has a coutable umber of possible evets (Heads or tails, each.5) Cotiuous Radom Variable: Takes all values i a iterval: (EX: ormal curve is cotiuous) Law of large umbers. As becomes very large Liear Combiatios: Geometric Probability: Look for # trial util first success 1. Success or Failure. X is trials util first success 3. Idepedet observatios 4. p is same for all observatios P(X=) = p(1-p) -1 µ is the epected umber of trails util the first success or P(X > ) = (1 p) = 1 - P(X )

5 Samplig distributio: The distributio of all values of the statistic i all possible samples of the same size from the populatio. Cetral Limit Theorem: As becomes very large the samplig distributio for is approimately NORMAL Use ( 30) for CLT Low Bias: Predicts the ceter well High Bias: Does ot predict ceter well Low Variability: Not spread out High Variability: Is very spread out High bias, High Variability Low Bias, Low Variability Low Bias, High Variability High Bias, Low Variability See other sheets for Part IV ART is my BFF Type I Error: Reject the ull hypothesis whe it is actually True Type II Error: Fail to reject the ull hypothesis whe it is False. ESTIMATE DO A CONFIDENCE INTERVAL Paired Procedures Must be from a matched pairs desig: Sample from oe populatio where each subject receives two treatmets, ad the observatios are subtracted. OR Subjects are matched i pairs because they are similar i some way, each subject receives oe of two treatmets ad the observatios are subtracted EVIDENCE - DO A TEST Two Sample Procedures Two idepedet samples from two differet populatios OR Two groups from a radomized eperimet (each group would receive a differet treatmet) Both groups may be from the same populatio i this case but will radomly receive a differet treatmet.

6 Major Cocepts i Probability For the epected value (mea,µ X ) ad the X or X of a probability distributio use the formula sheet Biomial Probability Simple Probability (ad, or, ot): Fied Number of Trials Probability of success is the same for all trials Trials are idepedet Fidig the probability of multiple simple evets. Additio Rule: P(A or B) = P(A) + P(B) P(A ad B) Multiplicatio Rule: P(A ad B) = P(A)P(B A) If X is B(,p) the (ON FORMULA SHEET) Mea p X Stadard Deviatio X p( 1 p) For Biomial probability use or: Eactly: P(X = ) = biompdf(, p, ) At Most: P(X ) = biomcdf(, p, ) At least: P(X ) = 1 - biomcdf(, p, -1) More tha: P(X > ) = 1 - biomcdf(, p, ) Less Tha: P(X < ) = biomcdf(, p, -1) or use: Mutually Eclusive evets CANNOT be idepedet A ad B are idepedet if the outcome of oe does ot affect the other. A ad B are disjoit or mutually eclusive if they have o evets i commo. Roll two die: DISJOINT rollig a 9 rollig doubles Roll two die: NOT disjoit rollig a 4 rollig doubles Idepedet: P(B) = P(B A) Mutually Eclusive: P(A ad B) = 0 You may use the ormal approimatio of the biomial distributio whe p 10 ad (1-p) 10. Use the mea ad stadard deviatio of the biomial situatio to fid the Z score. Geometric Probability You are iterested i the amout of trials it takes UNTIL you achieve a success. Probability of success is the same for each trial Trials are idepedet Use simple probability rules for Geometric Probabilities. P(X=) = p(1-p) -1 P(X > ) = (1 p) = 1 - P(X ) µ X is the epected umber of trails util the first success or Coditioal Probability Fidig the probability of a evet give that aother eve has already occurred. P( A B) Coditioal Probability: P ( B A) P( A) Use a two way table or a Tree Diagram for Coditioal Problems. Evets are Idepedet if P(B A) = P(B) Normal Probability For a sigle observatio from a ormal populatio P ( X ) P( z ) P ( X ) P( z ) For the mea of a radom sample of size from a populatio. Whe > 30 the samplig distributio of the sample mea is approimately Normal with: X To fid P( X y) Fid two Z scores ad subtract the probabilities (upper lower) Use the table to fid the probability or use ormalcdf(mi,ma,0,1) after fidig the z-score X If < 30 the the populatio should be Normally distributed to begi with to use the z-distributio. P ( X ) P( z ) P ( X ) P( z ) To fid P( X y) Fid two Z scores ad subtract the probabilities (upper lower) Use the table to fid the probability or use ormalcdf(mi,ma,0,1) after fidig the z-score

7 Mutually Eclusive vs. Idepedece You just heard that Da ad Aie who have bee a couple for three years broke up. This presets a problem, because you're havig a big party at your house this Friday ight ad you have ivited them both. Now you're afraid there might be a ugly scee if they both show up. Whe you see Aie, you talk to her about the issue, askig her if she remembers about your party. She assures you she's comig. You say that Da is ivited, too, ad you wait for her reactio. If she says, "That jerk! If he shows up I'm ot comig. I wat othig to do with him!", they're mutually eclusive. If she says, "Whatever. Let him come, or ot. He's othig to me ow.", they're idepedet. Mutually Eclusive ad Idepedece are two very differet ideas Mutually Eclusive (disjoit): P(A ad B) = 0 Evets A ad B are mutually eclusive if they have o outcomes i commo. That is A ad B caot happe at the same time. Eample of mutually eclusive (disjoit): A: roll a odd o a die B: roll a eve o a die Odd ad eve share o outcomes P(odd ad eve) = 0 Therefore, they are mutually eclusive. Eample of ot mutually eclusive (joit): A: draw a kig B: draw a face card Kig ad face card do share outcomes. All of the kigs are face cards. P(kig ad face card) = 4/5 Therefore, they are ot mutually eclusive. Idepedece: P(B) = P(B A) Evets A ad B are idepedet if kowig oe outcome does ot chage the probability of the other. That is kowig A does ot chage the probability of B. Eamples of idepedet evets: A: draw a ace B: draw a spade P(Spade) =13/5 = 1/4 P(Spade Ace) = 1/4 Kowig that the draw card is a ace does ot chage the probability of drawig a spade Eamples that are depedet (ot idepedet): A: roll a umber greater tha 3 B: roll a eve P(eve) = 3/6 = 1/ P(eve greater tha 3) = /3 Kowig the umber is greater tha three chages the probability of rollig a eve umber. Mutually Eclusive evets caot be idepedet Mutually eclusive ad depedet A: Roll a eve B: Roll a odd They share o outcomes ad kowig that it is odd chages the probability of it beig eve. Idepedet evets caot be Mutually Eclusive Idepedet ad ot mutually eclusive A: draw a black card B: draw a kig Kowig it is a black card does ot chage the probability of it beig a kig ad they do share outcomes. Depedet Evets may or may ot be mutually eclusive Depedet ad mutually eclusive A: draw a quee B: draw a kig Kowig it is a quee chages the probability of it beig a kig ad they do ot share outcomes. Depedet ad ot mutually eclusive A: Face Card B: Kig Kowig it is a face card chages the probability of it beig a kig ad they do share outcomes. If evets are mutually eclusive the: P(A or B) = P(A) + P(B) If evets are ot mutually eclusive use the geeral rule: P(A or B) = P(A) + P(B) P(A ad B) If evets are idepedet the: P(A ad B) = P(A)P(B) If evets are ot idepedet the use the geeral rule: P(A ad B) = P(A)P(B A)

8 Iterpretatios from Iferece Iterpretatio for a Cofidece Iterval: I am C% cofidet that the true parameter (mea or proportio p) lies betwee # ad #. INTERPRET IN CONTEXT!! Iterpretatio of C% Cofidet: Usig my method, If I sampled over ad over agai, C% of my itervals would cotai the true parameter (mea or proportio p). NOT: The parameter lies i my iterval C% of the time. It either does or does ot!! If p < I reject the ull hypothesis H 0 ad I have sufficiet/strog evidece to support the alterative hypothesis H a INTERPRET IN CONTEXT i terms of the alterative. If p > I fail to reject the ull hypothesis H 0 ad I have isufficiet/poor evidece to support the alterative hypothesis H a INTERPRET IN CONTEXT i terms of the alterative. Evidece Agaist H o P-Value "Some" 0.05 < P-Value < 0.10 "Moderate or Good" 0.01 < P-Value < 0.05 "Strog" P-Value < 0.01 Iterpretatio of a p-value: The probability, assumig the ull hypothesis is true, that a observed outcome would be as etreme or more etreme tha what was actually observed. Duality: Cofidece itervals ad sigificace tests. If the hypothesized parameter lies outside the C% cofidece iterval for the parameter I ca REJECT H 0 If the hypothesized parameter lies iside the C% cofidece iterval for the parameter I FAIL TO REJECT H 0 Power of test: The probability that at a fied level true. test will reject the ull hypothesis whe ad alterative value is

9 Oe Sample Z Iterval Use whe estimatig a sigle populatio mea ad σ is kow -SRS -Normality: CLT, stated, or plots -Idepedece ad N 10 Iterval: z CALCULATOR: 7: Z-Iterval Two Sample Z Iterval σ kow (RARELY USED) Use whe estimatig the differece betwee two populatio meas ad σ is kow -SRS for both populatios -Normality: CLT, stated, or plots for -both populatios -Idepedece ad N 10 for both populatios. Iterval: ( 1 ) z CALCULATOR: 9: -SampZIt * * 1 1 Cofidece Itervals Oe Sample t Iterval Use whe estimatig a sigle mea ad σ is NOT kow [Also used i a matched paired desig for the mea of the differece: PAIRED t PROCEDURE] -SRS -Normality: CLT, stated, or plots -Idepedece ad N 10 Iterval: df = -1 CALCULATOR: 8: T-Iterval t * Two Sample t Iterval σ ukow Use whe estimatig the differece betwee two populatio meas ad σ is NOT kow -SRS for both populatios -Normality: CLT, stated, or plots for both populatios -Idepedece ad N 10 for both populatios. Iterval: ( 1 ) 1 Use lower for df (df = -1) or use calculator CALCULATOR: 0: -SampTIt Cofidece iterval for Regressio Slope: Use whe estimatig the slope of the true regressio lie 1. Observatios are idepedet. Liear Relatioship (look at residual plot) 3. Stadard deviatio of y is the same(look at residual plot) 4. y varies ormally (look at histogram of residuals) Iterval: b t * SE b df = - CALCULATOR: t * s s 1 s Oe Proportio Z Iterval Use whe estimatig a sigle proportio -SRS -Normality: ˆ p 10, ( 1 pˆ ) 10 -Idepedece ad N 10 Iterval: ˆ * p z pˆ (1 pˆ ) CALCULATOR: A: 1-PropZIt Two Proportio Z Iterval Use whe estimatig the differece betwee two populatio proportios. -SRS for both populatios -Normality: -Idepedece ad N 10 for both populatios. Iterval: * pˆ 1(1 pˆ ˆ ˆ 1) p(1 p ( p ˆ ˆ 1 p) z CALCULATOR: B: -PropZIt LiRegTIt Use techology readout or calculator for this cofidece iterval. 1 )

10 Oe Sample Z Test Use whe testig a mea from a sigle sample ad σ is kow H o : µ = # H a : µ # -or- µ < # -or- µ > # -SRS -Normality: Stated, CLT, or plots -Idepedece ad N 10 Test statistic: z CALCULATOR: 1: Z-Test Two Sample Z test σ kow (RARELY USED) Use whe testig two meas from two samples ad σ is kow H o : µ 1 = µ H a : µ 1 µ -or- µ 1 < µ -or- µ 1 > µ -SRS for both populatios -Normality: Stated, CLT, or plots for both populatios -Idepedece ad N 10 for both populatios. Test statistic: 0 Hypothesis Testig Oe Sample t Test Use whe testig a mea from a sigle sample ad σ is NOT kow [Also used i a matched paired desig for the mea of the differece: PAIRED t PROCEDURE] H o : µ = # H a : µ # -or- µ < # -or- µ > # -SRS -Normality: Stated, CLT, or plots -Idepedece ad N 10 Test statistic: df = -1 0 t s CALCULATOR: : T-Test Two Sample t Test σ ukow Use whe testig two meas from two samples ad σ is NOT kow H o : µ 1 = µ H a : µ 1 µ -or- µ 1 < µ -or- µ 1 > µ -SRS for both populatios -Normality: Stated, CLT, or plots for both populatios -Idepedece ad N 10 for both populatios. Test statistic: use lower for df (df = -1) or use calculator. Oe Proportio Z test Use whe testig a proportio from a sigle sample H o : p = # H a : p # -or- p < # -or- p > # -SRS -Normality: p 0 10, (1-p 0 ) 10 -Idepedece ad N 10 Test statistic: CALCULATOR: 5: 1-PropZTest Two Proportio Z test Use whe testig two proportios from two samples H o : p 1 = p H a : p 1 p -or- p 1 < p -or- p 1 > p -SRS for both populatios -Normality: - Idepedece ad N 10 for both populatios.test statistic: CALCULATOR: 3: -SampZTest Χ Goodess of Fit L1 ad L Use for categorical variables to see if oe distributio is similar to aother H o : The distributio of two populatios are ot differet H a : The distributio of two populatios are differet -Idepedet Radom Sample -All epected couts 1 -No more tha 0% of Epected Couts <5 Test statistic: df = -1 CALCULATOR: 4: -SampTTest Χ Homogeeity of Populatios r c table Use for categorical variables to see if multiple distributios are similar to oe aother H o : The distributio of multiple populatios are ot differet H a : The distributio of multiple populatios are differet -Idepedet Radom Sample -All epected couts 1 -No more tha 0% of Epected Couts <5 Test statistic: df = (r-1)(c-1) CALCULATOR: 6: -PropZTest Χ Idepedece/Associatio r c table Use for categorical variables to see if two variables are related H o : Two variables have o associatio (idepedet) H a : Two variables have a associatio (depedet) -Idepedet Radom Sample -All epected couts 1 -No more tha 0% of Epected Couts <5 Test statistic: df = (r-1)(c-1) CALCULATOR:D: Χ GOF-Test(o 84) CALCULATOR: C: Χ -Test CALCULATOR: C: Χ -Test Sigificace Test for Regressio Slope: H o : β = 0 H a : β # -or- β < # -or- β > # 1. Observatios are idepedet. Liear Relatioship (look at residual plot) 3. Stadard deviatio of y is the same(look at residual plot) 4. y varies ormally (look at histogram of residuals) CALCULATOR: LiRegTTest Use techology readout or the calculator for this sigificace test b t SE b df = -

11 IQR Ier Quartile Rage Mea of a sample µ Mea of a populatio s Stadard deviatio of a sample σ Stadard deviatio of a populatio Sample proportio p s σ M Σ Q 1 Q 3 Z z * t t * N(µ, σ) r r Notatio Populatio proportio Variace of a sample Variace of a populatio Media Summatio First Quartile Third Quartile Stadardized value z test statistic Critical value for the stadard ormal distributio Test statistic for a t test Critical value for the t-distributio Notatio for the ormal distributio with mea ad stadard deviatio Correlatio coefficiet stregth of liear relatioship Coefficiet of determiatio measure of fit of the model to the data Equatio for the Least Squares Regressio Lie a y-itercept of the LSRL b Slope of the LSRL ( Poit the LSRL passes through y = a b Power model y = ab Epoetial model SRS Simple Radom Sample S Sample Space P(A) The probability of evet A A c A complemet P(B A) Probability of B give A Itersectio (Ad) U Uio (Or) X Radom Variable µ X Mea of a radom variable σ X Stadard deviatio of a radom variable Variace of a radom variable B(,p) pdf cdf N CLT Biomial Distributio with observatios ad probability of success Combiatio takig k Probability distributio fuctio Cumulative distributio fuctio Sample size Populatio size Cetral Limit Theorem Mea of a samplig distributio Stadard deviatio of a samplig distributio df Degrees of freedom SE Stadard error H 0 Null hypothesis-statemet of o chage H a Alterative hypothesis- statemet of chage p-value Probability (assumig H 0 is true) of observig a result as large or larger tha that observed α Sigificace level of a test. P(Type I) or the y-itercept of the true LSRL β P(Type II) or the true slope of the LSRL χ Chi-square test statistic

12 Regressio i a Nutshell Give a Set of Data: Eter Data ito L 1 ad L ad ru 8:Lireg(a+b) The regressio equatio is: predicted fat gai ( NEA) y-itercept: Fat gai is kilograms whe NEA is zero. slope: Fat gai decreases by for every uit icrease i NEA. r: correlatio coefficiet r = Moderate, egative correlatio betwee NEA ad fat gai. r : coefficiet of determiatio r = % of the variatio i fat gaied is eplaied by the Least Squares Regressio lie o NEA. The liear model is a moderate/reasoable fit to the data. It is ot strog. The residual plot shows that the model is a reasoable fit; there is ot a bed or curve, There is approimately the same amout of poits above ad below the lie. There is No fa shape to the plot. Predict the fat gai that correspods to a NEA of 600. predicted fat gai predicted fat gai (600 ) Would you be willig to predict the fat gai of a perso with NEA of 1000? No, this is etrapolatio, it is outside the rage of our data set. Residual: observed y - predicted y Fid the residual for a NEA of 473 First fid the predicted value of 473: predicted fat gai predicted fat gai (473) observed predicted = =

13 Take the log or l of y. The ew regressio equatio is: log(y) = a + b Trasformig Epoetial Data y = ab Take the log or l of ad y. The ew regressio equatio is: log(y) = a + b log() Trasformig Power Data y = a b Residual Plot eamples: Liear mode is a Good Fit Curved Model would be a good fit Fa shape loses accuracy as icreases Iferece with Regressio Output: Costruct a 95% Cofidece iterval for the slope of the LSRL of IQ o cry cout for the 0 babies i the study. Formula: df = = 0 = 18 * b t SE b (.101)(0.4870) (0.4697,.5161) Fid the t-test statistic ad p-value for the effect cry cout has o IQ. From the regressio aalysis t = 3.07 ad p = Or t b SE b s = This is the stadard deviatio of the residuals ad is a measure of the average spread of the deviatios from the LSRL.

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