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## PSTAT 109 Lecture and Section Notes

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# PSTAT 109 Lecture and Section Notes

Marketplace > University of California Santa Barbara > PSTAT > PSTAT 109 Lecture and Section Notes
Ryan Notetaker
UCSB

Professor Mangalam

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Here are lecture and section notes for PSTAT 109. This bundle of notes will be added to over the course of the quarter and will eventually include 10 weeks of lecture and section notes. So far it ...
COURSE
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Professor Mangalam
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## Reviews for PSTAT 109 Lecture and Section Notes

So much better than office hours. Needed something I could understand, and I got it. Will be turning back to StudySoup in the future

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Date Created: 04/04/14
PSTAT 109 Lecture 1 statistical data classified into 4 kinds Nominal Ordinal Interval and Ratio Scale Nomina Scale When data are only qualitative not quantitative we use nominal scale o ie color gender names of people nonnumerical Ordinal Scale if variable takes a sort of order o ie Agree Disagree No opinion Interval in this scale you can compare the difference between 2 values w the difference between 2 other o ie temperature 2030 degrees is the same as 4050 degrees with this type of scale 20 degrees is not twice as hot as 10 degrees though no meaningful zero Ratio If there is a meaningful zero the ratio scale is most appropriate o ie height weight speed and acce Kelvin Quantitative and Qualitative o Qualitative nominal o Quantitative numerical can be in ordinal interval or ratio but mostly in the latter 2 o Location species name and color are qualitative Continuous Variabescan take any value in a given range Discrete Variables assume only a specified quotcountablequot set of values In set theory quotcountablequot means the set can be put in a onetoone correspondence with a subset of integers Frequency Polygons 0 6466 4 0 6668 7 0 6870 9 o Midpoints 65 gt4 67gt7 69gt9 75gt1 77gt0 Stem and Leaf Plot 13 11 16 18 15 14 27 25 29 32 34 38 36 41 45 Leaf unit first number 1 1 3 4 5 6 8 2 5 7 9 Number of Mistakes Number of Students frequency midpoints of the blocks in the histogram are obtained by taking midpoints of intervals Number of visits to the zooexampe of a frequency distribution Cumulative frequency adding all of the frequencies above or below the value Sturge39s Rule observations n Choose of class intervals k such that n2quotk1 o If n 16 then choose k5 162quot4 1 Use between 6 and 15 classes The exact number will depend on how many observations there are 2 Make sure each item goes into exactly one class 3 Try to make the classes have equal width midpoints of classes are called class marks 51 56 61 are lower class limits 1st in class 55 60 65 are upper class limits 2nd number in class mean median and mode descending order of importance mean sum of values divided by the number of values denoted by x quotbarquot median is the middle value Arrange data in ascending order and pick the middle number mode most frequently occurring observation o If data are in the form of class intervals pick the interval w the highest frequency PSTAT 109 Lecture 2 Notes Computing median is more tricky if given class intervals 0 Look at where the sum of frequencies passes the halfway point o If the sum of frequencies is 80 the halfway point is 40 0 There are 27 observations in the classes below the median class o n80 n12 27135 frequencies inside the median class o Frequency of median class is 25 o Median should be 13525 way into the class 0 171352541916 o MLn12Ffc o L is lower boundary M is median n is the sum of frequencies F is the number of observations below the median class f is the frequency of the median class Mode is easy look at the interval with the highest frequency Mean sum of the class marks x times frequencies f divided by the sum of the frequencies f Median is the haf way mark Similarly for any p in O1 we can find the point in the data that Is p fraction into the data The median is the 5th quartile or the 50th percentile 30th percentile is the 3rd decile 25th 50th and 75th percentiles are also the 1st 2nd and 3rd quartiles Variation Values in any data set differ Their variation the extent to which the are different is of great importance in statistics and words like dispersion variability and spread are used to describe this Range difference between largest and smallest values in data Interquartile range difference between the third and first quartiles IQR Mean Deviation avg of deviations of the data points from the mean 0 MD1n sum of xi xbar 0 The variance of a whole population is given by oquot21N sum of yi uquot2 0 Sample variance is sxquot21n 1 sum of xi xbarquot2 0 Mew u instead of xbar creates a bias This bias is accounted for with n1 instead of n Population standard deviation o is the sqaure root of the pop Var and the std dev Sample variance 1n 1sum xiquot2 nxbarquot2 Population variance 1n instead of 1n1 Skewness is a measure of the asymmetry of the distribution of the data A negative skew skew to the left indicates that the tail on the left side of the distribution has the bulk of Positive skew means the mean is further to the right than the median Negative skew means the mean is further to the left than the median Skewness of 0 means the values are evenly distributed on both sides of the mean Skewness of 0 does not mean the distribution is symmetric Boxplot also known as box and whisker plot is another pictorial representation of data Outliers are observations that lie within 15QR of the quartiles 0 Marked with asterisks 1 b L66 4rE 339 ENIOVV W ajf 0s 7 of P men will 9 wfLm 1 L 3 sfan qr E4 V l 0DS of Ykg H 5 is H 19Q12 35 Lsxa C00 WC ZFroLqLgtHI I quot rom am elt9amprmen is one w age out 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7nf 5 f f 33 m jgnjgaa VKS uSiv Z O 393g mm 4B v r7 303 j rd A 339 39 39 v A UBgte Ajffa L CAaBgt6 57 8 S C3 uI qU uncoun a L x P 70 L ingtC7rgti ex qW Co397 H55 wICamp K H 3 quot 0 36339 d7ComampS1 FT 4 oh ouhomes 7a r leaml one heals W34 1 Ki c f WI f2 xx p5 2e eltafc P 6 Ms J gt31rJ3 iL41 5 quot9 M8 F n8 Ea m 0 ma f 5 PAn8 gF916 A pM t FM3 1u psvrr um Leo3xgs M or M oj US015 z Earn wlscci CJFSOF ran L9 i1oLaEIe a x quot qMcanAom u9 3quJaon1 v q w quot 1 36 dfamp c31buJ5PO0 lt1 lt c1s9FQT6c9nltJam fa 1 Z mm InaJS r4 H 0Q ro a or p of m iTQrB I I gt 1 zquotL amp 6 w 1 W 0 1 LJgt1 rLfimLax4L zM14aa C mi B 3oC g 43 T T W igigcghii HS FrA1c vcre4quot 1e4v zrgt lnc cm L 39an3a43 slew of 1gt JCq1scSam ymfq o 39M19 0tCQ50fquotamp F namecl B j llt1ro gtlt aFwJsvA F emJI t op P p vtz liy I5 61 u Wquot mZ 1 L5quotm9 Lo g W runglo x7 F39 eme Cpa1v zf PSTAT Lecture 6 April 11 2014 Examples are included on last pages Definition 23 Let X be a discrete random variable with probability mass function px then the epectation of x is given by EX quotsum functionquot xp x Where the summation is taken over all possible values x of X EY is not equal to the square root of EYquot2 VY EY EYquot2 expected value can be negative Definition 26 If X is a random variable then the variance of X is defined by oquot2xsubscript VX Exquot2 EXquot2 Its standard deviation is defined by oxsubscipt St Dev X square root of VX o sigma u mew and y quotskewness variablequot fX has a mean u mew and st dev o the skewness ofx is defined as y1 Exuquot3oquot3 Proposition 6 Let X be any random variable continuous or discrete For any 2 real numbers a and b 1 EaXb aEXb 2 VaXb aquot2VX 3 SDaXb aSDX For Exa and EX Vx does not change Vax aquot2Vx SDaX aquot2quot12VXquot12 Some commonly occurring distributions have special names Discrete Uniform Bernoulli Binomial Hypergeometric Geometric Poisson If a random variable X takes n possible values a1 a2 an all with probability 1n then X is said to have discrete uniform distribution on a1 an Example 28 Roll a die and let X be the number that came up If the die is well balanced then X has discrete uniform on 1 2 3 4 5 6 o A Bernoulli random variable is one that takes values 1 and O with probabilities p and 1 p L c f M053 1 E 39 L 0Ln 3905 J IL 00532 8 E Ogf 2735 I e PRc 9ulL and j 45quot a Let b s P A f hm Xh 1 on u or 39rur1 squot5 I 0 695 i ql 45L 0 139 u L 0 AA 19 e gs cw 19 f 39 A Z nnp J1 I39lt if 3 L gQmz399 f 93 V2 L n F0 lt5 0739ln J I s L 4 W 9oltIltgt v 9 quot whore 4gs FAHL F mgg 39 C R F F39P 5 FPS FFSS 1 i 0 5 11 bo 0 Imace P heqA339n Q X 3 jj 39 o1J C l qQ inn 393 T m 55 b 0 A A 393 ii 95 39 V A 7 N H 397 quot 39 39 V e gtOgtAH 5 an cc gtjp 39 eSaI ELM quot 9 9 5 ckgggg VJ e39p A 0 we Bab m llesqgy a i If 397 quotSquotlt Of L S 4 M 15 ID MAJ 5 01 p J J 3 Q Qc nnt La 1 g T ema B I3 4 390 g2s m 5 3e V4 t rm Qgrnow c n aS i 395 LT ass rs a 7 9 V X 1391 Q1 i or Cquotwf 5 Ms 3 0m Arquot quot 3 lr quot 5 F5 FPS FFF 539 3 FFFF s 0 Pl 79QzCC i 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