Stat 104 Lecture 2. Variables and their distributions. DJIA: monthly % change, 2000 to Finding the center of a distribution. Median.

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1 Stat 04 Lecture Statistics 04 Lecture (IPS. &.) Outlie for today Variables ad their distributios Fidig the ceter Measurig the spread Effects of a liear trasformatio Variables ad their distributios Variable: Ay characteristic of a idividual - takes o differet values for differet idividuals Distributio: Describes what values a variable takes ad how frequetly these values occur The distributio of a variable ca be described graphically ad umerically i terms of shape, ceter ad spread Shape: A picture is worth a thousad words DJIA: mothly % chage, 000 to 007 Listig % % % % % % % N = 9 Careful exploratio ad visual display of data ofte reveal patters of regularity, variability ad outliers 3 Fidig the ceter of a distributio Patters difficult to see with raw umbers Simple tools to describe the distributio of a set of umbers (data) Commo umerical summaries of the ceter of a distributio Mea Media Mode Mea The mea is the average value If there are observatios x, x,, x, the the mea is x+ x + x3 + + x x = xi = i= For example if the data are: 3,, 3, 6, the their mea (or average) is ( )/5 = 3.0 Media The media is the midpoit 50% of observatios are smaller tha the media ad 50% are larger tha the media If is odd the the media is the ceter observatio i the ordered list If is eve the the media is the mea of the two ceter observatios i the ordered list For example if the data are: 3,, 3, 6,, we ca order them,, 3, 3, 6 ad see that the media is 3 5 6

2 Stat 04 Lecture Mode The mode is the observatio that occurs most frequetly The mode may ot be uique; there may be more tha oe mode For example if the data are: 3,, 3, 6,, the mode is 3 because it occurs most frequetly Mea versus media Whe the shape of the distributio is symmetric, the mea ad media are very similar Whe the distributio is skewed, the mea is further out i the tail tha the media the mea follows the tail The media is much less sesitive to extreme observatios (ofte referred to as outliers ) The mea makes efficiet use of the data - especially if the data is moud shaped Example: Harvard Salary Survey I 998, Harvard coducted salary survey of eterig class of 973 They were iterested i determiig a typical salary 5 years after first eterig Harvard They determied Mea salary: $750,000 Media salary: $75,000 Why such a large discrepacy?

3 Stat 04 Lecture Outliers DJIA: mothly % chage, 000 to 007 Outliers (extreme values) usually demad ivestigatio Ofte they are errors i the data (e.g. due to istrumet failure or errors i recordig) But they also may be very importat (e.g. a ew scietific observatio) If there is o reaso to suspect they have bee wrogly recorded, may wat to use summaries that are resistat to their ifluece (e.g., medias rather tha meas) Outliers should ot be discarded without good reaso Mea % chage: 0. Media % chage: Measurig the spread of a distributio "Should we scare the oppositio by aoucig our mea height or lull them by aoucig our media height?" A measure of spread coveys iformatio regardig variability how dispersed the distributio is Commo umerical summaries of spread Variace (s ) Stadard Deviatio (SD & s) Note: SD = Variace Rage (largest mius smallest observatio) Iterquartile Rage (IQR) 5 The cocept of variace The ceter of a group of observatios ca be measured by the mea The variability of a sigle observatio ( x i ) ca be measured by its distace from the ceter (e.g. mea) ( xi x) Sice we wat this to always be a positive umber we cosider the square of the above If we cosider the sum of such squared deviatios from the mea as a measure of variability - we realize that we eed to take its average A problem the uits of variace are squared uits 7 Variace The variace is the average (almost) of squared deviatios from the mea If there are observatios x, x,, x, the the variace is s x x ( x x) + ( x x) + + ( x x) = ( i ) = i= 8

4 Stat 04 Lecture Stadard Deviatio The stadard deviatio (SD) is the square root of the variace s s x x = = ( i ) i= Note: The SD is i the origial uits of measuremet The variace is i the (origial uits) 9 Example: variace & stadard deviatio Cosider the height of five suflower plats Their heights (i feet) are: 3,, 3, 6, ad Their mea is x = xi = = 3.0 feet i = 5 Ad their variace is (3 3) + ( 3) + (3 3) + (6 3) + ( 3) s x x = ( i ) = i= s = = 3.5 feet 4 s So their stadard deviatio is = = 3.5 =.87 feet s 0 Gettig a feel for stadard deviatio Cosider the distributio of Iowa Test vocabulary scores for Gary, Idiaa 7 th graders or ay moud-shaped distributio Quartiles ad the Iterquartile Rage The first quartile Q is the media of the observatios i the ordered list to the left of the overall media (5% are smaller tha Q ad 75% are larger) The third quartile Q 3 is the media of the observatios i the ordered list to the right of the overall media (75% are smaller tha Q 3 ad 5% are larger) Iterquartile Rage, IQR = Q 3 -Q, is a measure of variability of the distributio (IQR cotais middle 50% of the observatios) Example: For the observatios,, 3, 4, 5 Q =.5, Media = 3, Q 3 = 4.5, ad IQR = Roughly 4-6 stadard deviatios Variace versus IQR We saw earlier, the mea is sesitive to skew ad outliers; the media is ot Similarly The variace (stadard deviatio) is sesitive to skew ad outliers, the IQR is ot A outlier has o affect o the computatio of IQR because it does t chage Q or Q 3 The variace (stadard deviatio) makes efficiet use of the data especially if it is moud shaped 3 DJIA: mothly % chage, 000 to 007 Guess the SD & IQR Mea: 0. SD: 4.03 Media: 0.33 IQR: 4.53 Stadard deviatios (like meas) ca be sesitive to extreme observatios (outliers) e.g. SD of (, 3, 5) = versus SD of (, 3, 0) = 0.4 4

5 Stat 04 Lecture Outlie for today Variables ad their distributios Fidig the ceter Measurig the spread Effects of a liear trasformatio 5 Effects of a liear trasformatio Cosider a liear trasformatio from X to Y Ay liear trasformatio ca be put i the form y i = a + bx i The role of the a i the equatio above is to shift the locatio of all the values If a > 0, the shift is to the right If a < 0, the shift is to the left The role of the b i the equatio above is to chage the scale of all the values If b >, the values will be more spread out If b <, the values will be closer together 6 Effects of a liear trasformatio y i = a + bx i Why worry about effects of liear trasformatios o data? Addig a costat (a i the equatio above) or chagig the scale (multiplyig by b above) is very commo Example: How cold is Bosto i Jauary? Movig betwee Fahreheit ad Celsius temperature scales Effect o measures of ceter ad measures of spread Mea (Y) = a + b mea (X) Media (Y) = a + b media (X) Variace (Y) = b variace (X) SD (Y) = b SD (X) IQR (Y) = b IQR (X) Note: a shift of locatio (a) does t have y i = a + bx i Movig from Fahreheit to Celsius Cel = 5/9 * (Fahr - 3) = /9 * Fahr Movig from Celsius to Fahreheit Fahr = 3 + 9/5*Cel ay ifluece o measures of spread 7 8 Example: How cold is Bosto i Jauary? Jauary 004 miimum daily temperatures (3 days) were measured i Fahreheit ad showed a mea of 0.5 ad a stadard deviatio (SD) of.9 Covert the mea ad SD from Fahr to Celsius usig a liear trasformatio From last slide, Cel = /9 * Fahr (So Cel (Y) = a + b Fahr (X), a = -7.8 ad b = 5/9) So the mea i Celsius (Y) = a + b mea (X) = /9 (0.5) = -.0 Ad the SD i Celsius (Y) = b SD (X) = 5/9 *.9 = The last word Survey questioaire We wish to collect some survey data from you to be used i class as examples Please complete the survey form (o ames) ad deposit i box As i all surveys, participatio is volutary Before ext class Look over IPS. &. A certai statistics professor left Harvard to go to Yale, thereby improvig the average quality of both departmets. - A old joke 30