= 1. UCLA STAT 13 Introduction to Statistical Methods for the Life and Health Sciences. Parameters and Statistics. Measures of Centrality

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1 UCLA STAT Itroducto to Statstcal Methods for the Lfe ad Health Sceces Istructor: Ivo Dov, Asst. Prof. of Statstcs ad Neurolog Teachg Assstats: Brad Shaata & Tffa Head Uverst of Calfora, Los Ageles, Fall 7 Parameters ad Statstcs Varables ca be summarzed usg statstcs. Defto: A statstc s a umercal measure that descrbes a characterstc of the sample. Defto: A parameter s a umercal measure that descrbes a characterstc of the populato. We use statstcs to estmate parameters Slde Slde Measures of Cetralt Measures of Ceter Recall that ceter s # of the BIG three. Measures of ceter clude: the mea the meda the mode (the value wth the hghest frequec) These measures all descrbe the ceter of a dstrbuto a slghtl dfferet wa Slde The Mea aka the average ca be thought of as the balacg pot of a dstrbuto Slde Measures of Ceter Measures of Ceter Example: I a expermet wth some statstcs studets, 8 male studets were radoml selected ad asked to perform the stadg log jump. I realt ever studet partcpated, but for the ease of calculatos below we wll focus o these eght studets. The log jumps were as follows: log jump (.) Slde 7.7ches The Meda ca be thought of as the pot that dvdes a dstrbuto half (/) Steps to fd the meda: ( + ). Arrage the data ascedg order observato a. If s odd, the meda s the mddle value b. If s eve, the meda s the average of the mddle two values the average of observatos ad + Slde 6

2 Measures of Ceter Resstace Example: Log Jump (cot ) Because s eve, the meda wll be the average of the mddle two values log jump (.) log jump (.) meda 7ches Defto: A statstc s sad to be resstat f the value of the statstc s relatvel uchaged b chages a small porto of the data Referecg the formulas for the meda ad the mea, whch statstc seems to be more resstat? Example: Log Jump (cot ) Let's remove the studet wth the log jump dstace of 6 ad recalculate the meda ad mea. Descrptve Statstcs: dstace Varable N N* Mea SE Mea StDev Mmum Q Meda Q Maxmum dstace Slde 7 Slde 8 Ceter vs. Shape We ca also use the mea ad meda to help terpret the shape of a dstrbuto I a umodal dstrbuto: mea meda mea > meda mea < meda (d) Smmetrc Postvel skewed (log upper tal) Negatvel skewed (log lower tal) Mea, Meda, Mode, Quartles, # summar The sample mea s the average of all umerc obs s. The sample meda s the obs. at the dex (+)/ (ote take avg of the obs s the mddle for fractos lke.), of the observatos ordered b sze (small-to-large)? The sample meda usuall preferred to the sample mea for skewed data? mea Uder what crcumstaces ma quotg a sgle ceter (be t mea or meda) ot make sese?(mult-modal) What ca we sa about the sample mea of a qualtatve varable? (meagless) Slde 9 Slde A addtoal graphcal dspla for the data that utlzes some of these measures of ceter s called a boxplot. slghtl paful to costruct b had we wll rel o the computer, but we wll stll dscuss the formulas of the mportat aspects of the plot The fve umber summar: mmum: the smallest observato maxmum: the largest observato meda: splts the data to / quartles: splt the data to quarters Q s the lower quartle ad Q s the upper quartle A boxplot s a vsual represetato of the fve umber summar Slde Slde

3 There are four addtoal features of a boxplot Iterquartle rage (IQR): Q Q, the spread of the mddle % of the data whskers exted from Q ad Q to the smallest* ad largest* observatos wth the *feces *feces used to detf extreme observatos lower fece (LF): Q.(IQR) upper fece (UF): Q +.(IQR) outlers extreme observatos that fall outsde the feces Example (cot ): Usg the log jump data a boxplot of dstace would be: 6 7 Boxplot of dstace 8 dstace 9 Slde Slde Recall that spread s # of the BIG three. Measures of spread clude: the rage the varace the stadard devato The rage easest measure of spread to calculate ot the best measure of spread rage max - m Example: Log Jump (cot ) Calculate the rage for the log jump data Descrptve Statstcs: dstace Varable N N* Mea SE Mea StDev Mmum Q Meda Q Maxmum dstace Rage Slde Slde 6 A & B Dot plot The rage (cot ) Wh s the rage ot the best measure of spread? Suppose we have the followg data sets, dotplots below. Itutvel whch plot (A or B) seems to have more spread (e. less cluster)? A B Data 6 7 The stadard devato The logc behd the stadard devato s to measure the dfferece (e. devato) betwee each observato ad the mea A devato s What seems lke a reasoable wa to fd a average devato? Bg problem, wh? ( ) How could we solve ths problem? Slde 7 Slde 8

4 The varace s ( ) The stadard devato (sd) ( ) s Wh use the sd ad ot the varace? Example (cot ): Calculate the sd ( ) s ( ).79 ches + ( ) 8 Descrptve Statstcs: dstace ( 76 log jump (.) ) Varable N N* Mea SE Mea StDev Mmum Q Meda Q Maxmum dstace Slde 9 Slde Below we have four relatve frequec hstograms ad four portos of output. Match the graph to the approprate output. Mea Meda StDev A B...6 C D...7 Percet Percet Percet Percet The Emprcal Rule The emprcal rule s useful whe talkg about a dstrbuto, usg the stadard devato terms of t s dstace from the mea. I geeral, for smmetrc dstrbutos: ± s 68% ± s 9% ± s > 99% NOTE: If the dstrbuto s ot umodal smmetrc the emprcal rule ma ot hold Slde Slde The Emprcal Rule The Emprcal Rule Example (hotdogs cot ): From the hotdog data we have the followg output: Descrptve Statstcs: Calores Varable N N* Mea SE Mea StDev Mmum Q Meda Q Calores Varable Maxmum Rage Calores ± s. ± 9.8 (6.6,7.8) ± s. ± (9.8) (86.68,.) ± s. ± (9.8) (7.,.8) Slde Example (hotdogs cot ): From the hotdog data we have the followg tervals: ± s. ± 9.8 (6.6,7.8) ± s. ± (9.8) (86.68,.) ± s. ± (9.8) (7.,.8) / % s ths close to 68%? Slde Character Stem-ad-Leaf Dspla Stem-ad-leaf of Calores N Leaf Ut ()

5 The Goal The Goal Defto: A statstcal ferece s the process of drawg coclusos about a populato based o observatos a sample. To make a statstcal ferece we wat the sample to be represetatve of the populato. How could we esure ths? Defto: Radom meas that each subject of the populato must have a equal chace of beg selected. Wh does ths seem mportat for statstcs? How ca we esure radom selecto? Slde Slde 6 More Notato More Notato Both samples ad populatos have umerc quattes of terest, such as: mea (the average) stadard devato (the spread) proporto (percet) For what tpe of varable(s) would each of these umerc quattes be approprate? Recall: A characterstc of the populato s called a parameter ad a characterstc of a sample s called a statstc. Mea Stadard Devato Proporto Populato μ σ P Sample x s pˆ Uder what crcumstaces would we kow μ? What seems lke a good estmate of μ? Slde 7 Slde 8 More Notato Recall: Statstcs estmate parameters. Populato μ Sample The bg questo s: how good of a estmate are these values? Slde 9