Questions about the Assignment. Describing Data: Distributions and Relationships. Measures of Spread Standard Deviation. One Quantitative Variable

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Virgil Milo Knight
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Quetio about the Aigmet Read the quetio ad awer the quetio that are aked Experimet elimiate cofoudig variable Decribig Data: Ditributio ad Relatiohip GSS people attitude veru their characteritic ad

1 Quetio about the Aigmet Read the quetio ad awer the quetio that are aked Experimet elimiate cofoudig variable Decribig Data: Ditributio ad Relatiohip GSS people attitude veru their characteritic ad poue Share the relevat ample tatitic for oe categorical variable ad oe quatitative variable. Oe Quatitative Variable Whe decribig quatitative variable we are itereted i the ditributio of the value it hape, ceter, ad pread. Meaure of Spread Stadard Deviatio The ample tadard deviatio () meaure the pread of a ditributio. i1 x x 2 i 1 For each value i the ample, how much doe it deviate from the ample mea. The tadard deviatio provide the average amout the value i the ample (x i ) deviate from the ample mea ( ). The tadard deviatio i alway 0. Stadard Deviatio The more pread out the ditributio i, the larger the tadard deviatio will be. Frequecy Frequecy x 2 i x i = 1 The 95% Rule If a ditributio i ymmetric ad bell-haped, the approximately 95% of the data value will lie withi 2 tadard deviatio of the mea. = 0 =

2 The 95% Rule What i the ample mea for thi ditributio? = 64.5iche What i the tadard deviatio for thi ditributio? = 2.5iche What i your height if it i 2 tadard deviatio above the mea? 69.5 z-core: Number of Stadard Deviatio from the Mea A value z-core idicate how far it i from the mea (i.e., how may tadard deviatio away from the mea). x x z i Every value ha a correpodig z-core. It i a uit-free meaure of extremity of a data poit. z-core farther from 0 are more extreme. 95% of the z-core for a ditributio fall betwee -2 ad 2. z-core If you are 67 tall, what i the z-core for your height? z = i how may tadard deviatio below the mea? 2 What i your height if it z-core i 3? 72iche z x x i = 64.5 = 2.5 The tadard deviatio idicate the pread of the ditributio. z-core z x x i the poitio of a value o the ditributio The z core idicate Calculate the z-core for x 1 = 10 if the mea = 10 ad the tadard deviatio = 5 z-core = (10-10)/5 = 0 10 i 0 tadard deviatio from the mea Calculate the z-core for x 1 = 15 if the mea = 10 ad the tadard deviatio = 5 z-core = (15-10)/5 = 1 10 i 1 tadard deviatio above the mea Calculate the z-core for x 1 = 20 if the mea = 10 ad the tadard deviatio = 5 z-core = (20-10)/5 = 2 20 i 2 tadard deviatio = 4 above the mea Calculate the z-core for x 1 = 0 if the mea = 10 ad the tadard deviatio = 5 z-core = (0-10)/5 = -2 0 i 2 tadard deviatio below the mea z-core Which i better, a ACT core of 28 or a combied SAT core of 2100? ACT: mea = 21, d = 5 SAT: mea = 1500, d = 325 Aume ACT core ad SAT core are approximately ymmetric ad bell-haped Hit: Which tet core ha a higher z-core? Other Meaure of Locatio Maximum = larget data value Miimum = mallet data value Quartile: Q1 = media of the value below m. Q3 = media of the value above m. A. ACT core of 28 B. SAT core of 2100 ACT: (28 21)/5 = 1.4 SAT: ( )/325 = 1.85 => SAT core of 2100 ha the higher z-core 2

3 Mi Five Number Summary Q 1 m Q 3 25% 25% 25% 25% Max Percetile The P th percetile i the value of a quatitative variable that i greater tha P percet of the data. z-core ca determie whether a SAT core of 2100 or a ACT core of 28 i better. Percetile ca alo provide thi iformatio: ACT core of 28: 89 th percetile 28 i greater tha 89 percet of the core SAT core of 2100: 93 rd percetile 2100 i greater tha 93 percet of the core Mi Five Number Summary Q 1 m Q 3 25% 25% 25% 25% Max Meaure of Spread ad Outlier Rage = Mi Max Iterquartile Rage (IQR) = Q3 Q1 I the rage reitat to outlier? A. Ye The rage i determied by the mot extreme B. No value, o it i very affected by outlier. 0 th percetile 25 th percetile 50 th percetile 75 th percetile 100 th percetile I the IQR reitat to outlier? A. Ye The IQR i ot very affected by outlier, B. No becaue it igore the mot extreme data. Meaure of Spread Outlier ca be iformally idetified by lookig at a plot. Oe formal rule for idetifyig outlier i data value that are more tha 1.5 IQR beyod the quartile. A data value i a outlier if it i: Smaller tha Q1 1.5(IQR) or Larger tha Q (IQR) [ IQR = Q3 Q1] Outlier Q 3 Media Q 1 Boxplot Mi. Q1 Media Q3 Max Lie ( whiker ) exted from each quartile to the mot extreme value that i ot a outlier Q (IQR) ad Q1 1.5(IQR) }Middle 50% of data 3

Quatitative Statitic by Category The tatitic we calculate for a quatitative variable ca be looked at eparately for each level of a categorical variable.

4 Quatitative Statitic by Category The tatitic we calculate for a quatitative variable ca be looked at eparately for each level of a categorical variable. Relatiohip betwee Two Quatitative Variable Quatitative variable: # of hour pet tudyig per week Categorical variable: Academic major Mea # of hour pet tudyig per week by major: Art & Humaitie 16.2hr Social Sciece 17.5hr Natural Sciece 17.8hr Scatterplot A catterplot i a graph of the relatiohip betwee two quatitative variable. Each dot repreet oe cae. Do the dot form a clear tred? I the tred upward or dowward? Are there ay outlier poit? Directio of Aociatio A poitive aociatio mea that value of oe variable ted to be higher whe value of the other variable are higher. A egative aociatio mea that value of oe variable ted to be lower whe value of the other variable are higher. No aociatio exit betwee two variable if kowig the value of oe variable doe ot give you ay iformatio about the value of the other variable. Correlatio The ample correlatio (r) meaure the tregth ad directio of liear aociatio betwee two quatitative variable. z-core for x z-core for y 1 xi x yi y 1 i1 X Y r X = ample tadard deviatio of variable X Y = ample tadard deviatio of variable Y Correlatio z-core for x z-core for y 1 xi x yi y 1 i1 X Y r X = height ( = 64 ): x 1 = 67 [1.2] x 2 = 72 [1.8] x 3 = 60 [-1.3] Y = weight ( = 180lb): y 1 = 200[.9] y 2 = 220[1.3] y 3 = 135[-1.4] Poitive correlatio (r > 0) exit whe poitive z-core for X ted to occur with poitive z-core for Y ad vice vera. Negative correlatio (r < 0) exit whe poitive z-core for X ted to occur with egative z-core for Y ad vice vera. 4

5 Correlatio Propertie of Correlatio z-core for x z-core for y -1 r 1 1 xi x yi y Poitive aociatio: r > 0 1 Negative aociatio: r < 0 i1 X Y No liear aociatio:r 0 r X = height ( = 64 ): x 1 = 67 [1.2] x 2 = 72 [1.8] x 3 = 60 [-1.3] Y = age ( = 42yr): y 1 = 63 [1.8] y 2 = 34 [-1.1] y 3 = 57 [1.5] No correlatio (r 0) exit whe the ig of the z-core for X do ot coitetly correpod with the ig of the z-core for Y. The cloer r i to 1, the troger the liear aociatio r doe ot deped o the uit of meauremet The correlatio betwee X ad Y i the ame a the correlatio betwee Y ad X Correlatio Are correlatio reitat to outlier? Correlatio Same plot, but with Dolphi ad Raider (outlier) removed z-core for Pealty Yard NFL Team r =.43 z-core for Pealty Yard r = Fiercee Malevolece Ratig Ratig of Team of Uiform Macot Fiercee Malevolece Ratig Ratig of Team of Uiform Macot Correlatio Cautio Correlatio ca be heavily affected by outlier. Alway plot your data! r = 0 mea o liear aociatio; however the variable could have a o-liear aociatio. Alway plot your data! Correlatio doe ot imply cauatio! Aigmet Part I Graded Problem 2.108, 2.168, ad Additioal Practice Problem (ot to be tured i): 2.75, 2.87, 2.93, ad Part II See Next Slide 5

6 Aigmet Summary: Oe Quatitative Variable Part II: Goto Type up thi aigmet i a Word documet Provide the followig iformatio for 5 quatitative variable: Variable ame Quetio related to the variable Explai i your ow word what thi variable i meaurig The uit ued to meaure the variable (e.g., year, dollar, iche, etc.) Mi, Max, Mea, Media, Stadard Deviatio (Std Dev) Provide the followig iformatio for 5 categorical variable: Variable ame Quetio related to the variable Explai i your ow word what thi variable i meaurig The categorie for the variable (e.g., Race: White, Black or Other) The proportio for each category (You ca ue variable you have ued before.) Summary Statitic Ceter: mea, media Spread: tadard deviatio, rage, IQR Percetile 5 umber ummary Viualizatio Dotplot Hitogram Boxplot Other cocept Shape: ymmetric, kewed, bell-haped Outlier, reitace z-core Summary Statitic Correlatio Viualizatio Scatterplot Summary: Two Quatitative Variable Summary: Oe Quatitative ad Oe Categorical Summary Statitic Summary tatitic for quatitative variable ca be broke dow by each level of the categorical variable Viualizatio Side-by-ide boxplot Variable Viualizatio Summary Statitic Categorical Quatitative Categorical v Categorical Quatitative v Categorical Quatitative v Quatitative Decribig Data bar chart, pie chart dotplot, hitogram, boxplot ide-by-ide bar chart, egmeted bar chart ide-by-ide boxplot catterplot frequecy table, relative frequecy table, proportio mea, media, max, mi, tadard deviatio, rage, IQR, five umber ummary two-way table, differece i proportio tatitic by group correlatio 6

(# x) 2 n. (" x) 2 = 30 2 = 900. = sum. " x 2 = =174. " x. Chapter 12. Quick math overview. #(x " x ) 2 = # x 2 "

(# x) 2 n. ( x) 2 = 30 2 = 900. = sum. x 2 = =174. x. Chapter 12. Quick math overview. #(x x ) 2 = # x 2 Chapter 12 Describig Distributios with Numbers Chapter 12 1 Quick math overview = sum These expressios are algebraically equivalet #(x " x ) 2 = # x 2 " (# x) 2 Examples x :{ 2,3,5,6,6,8 } " x = 2 + 3+