I. Measures of Central Tendency: -Allow us to summarize an entire data set with a single value (the midpoint).

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I. Meaure of Cetral Tedecy: -Allow u to ummarize a etire data et with a igle value (the midpoit.. Mode : The value (core that occur mot ofte i a data et. -Mo x Sample mode -Mo Populatio mode.

1 I. Meaure of Cetral Tedecy: -Allow u to ummarize a etire data et with a igle value (the midpoit.. Mode : The value (core that occur mot ofte i a data et. -Mo x Sample mode -Mo Populatio mode. Media : the poit (core which divide the data et i ½ : e.g. ½ of the ubject are above the media ad ½ are below the media. -Md x Sample Media -Md Populatio Media 3. Mea: the arithmetic average: Directly coider every core i a ditributio. - Sample Mea - µ Populatio Mea - II. Skewed Ditributio & the 3M -Skewe refer to the hape of the ditributio which ca be iflueced by extreme core. - Skewe i alo a etimate of the deviatio of the Mea, Media, ad Mode. -Symmetrical Dit. Mea, Media, Mode are all i the ame locatio i the dit. - Skewed Right (Poitively Skewed Mode i peak of dit.(left of ceter, Media i ceter of ditributio, Mea i right tail of ditributio Skewed Left (egatively Skewed Mode i peak of dit (right of ceter, Media i ceter of ditributio, Mea i left tail of ditributio

2 I. Meaure of Variability (Diperio -Like meaure of cetral tedecy, meaure of variability allow u to ummarize our data et with a igle value. -Whe cetral tedecy ad variability are coidered together we get a more accurate picture of our data et. -The 3 mai meaure of variability: Rage, Variace, ad Stadard Deviatio. -thee formula are the root formula for may of the tatitical tet that will be covered later -Thee meaure tell u how much obervatio i a data et vary or how they are dipered withi the ditributio. -Although thi iformatio i idirectly cotaied withi meaure of cetral tedecy, they do't tell u much about the variace withi our data Example. umber of mile traveled before travelig compaio appear huma Zoo Pegui E 4 South Pole Pegui E 4 Mea, Mode, Media for both data et (They do ot differ -all zoo pegui halluciate after travelig mile, while there i much more variability i the ditace traveled by South Pole Pegui. -I order to draw accurate cocluio about our data both cetral tedecy ad variability mut be coidered. II. Rage : The umerical ditace betwee the larget ( maximum ad mallet value ( miimum, tell u omethig about the variatio i core we have i our data, or it tell u the width of our data et. Rage maximum - miimum - Rage for Zoo pegui - ;

3 - Rage for South Pole P' - 6 -Problem with Rage, Agai thi i a ummary meaure that doe ot directly coider every value i the data et ( here oly the two extreme umber; larget ad mallet therefore we do ot kow if mot of the core occur at the extreme of the ditributio or toward the ceter. It i a very crude meaure of variability. For example: Zoo Pegui E 4 Rage South Pole Pegui E 4 Rage 6 orth Pole Pegui E 4 Rage 6 III. Variace mathematically ae the total amout of variability i a data et by directly coiderig every obervatio. -To do thi require a poit from which each obervatio ca be compared to ae the amout they fluctuate. The Mea ca be ued for thi (ice it coider every obervatio i it calculatio. Mea Deviatio E( - Zoo Pegui - Mea South Pole Pegui - Mea orth Pole Pegui - Mea E 4 E(-Mea E 4 E(-Mea E 4 E(-Mea Mea Mea Mea Rage Rage 6 Rage 6 - The um of the mea deviatio for ay data et i alway. Thi limit the uefule of the mea deviatio for ummarizig differet data et with a igle poit. -If we quare each deviatio value the the egative value cacel out ad we are left with a more

4 meaigful value. Zoo Pegui - Mea ( - Mea South Pole Pegui - Mea ( - Mea E 4 E(-Mea E(-Mea E 4 E(-Mea E(-Mea Mea Mea Rage Rage 6 orth Pole Pegui - Mea ( - Mea E 4 E(-Mea E(-Mea Mea Rage 6 -If we um thee value we o loger get, but a umber that reflect the total variace for thi data et, if we divide that umber by or we get the average variace for thi data et. Defiitioal Populatio Formula F E ( - Defiitioal Sample Formula E ( - -ote ample variace ue - rather tha becaue it i a etimate of the populatio variace, becaue of thi reduced deomiator the ample variace will alway be lightly larger tha the populatio variace. Zoo Pegui σ ( (

5 South Pole Pegui σ ( ( 3. 4 orth Pole Pegui σ ( (.. 4 -give a good idea of how we get variace but it i time coumig for large data et, o we have developed mathematically idetical (algebraic equivalet formula that are a little eaier to calculate Computatioal Formula Populatio Variace F E - (E / Sample Variace E - (E / - -ote ample variace ue - rather tha becaue it i a etimate of the populatio variace, becaue of thi reduced deomiator the ample variace will alway be lightly larger tha the populatio variace. Zoo Pegui South Pole Pegui orth Pole Pegui E E Zoo Pegui: σ ( ( ( 4 ( 4 6 6

6 South Pole Pegui: σ ( ( ( 4 ( orth Pole Pegui: σ ( ( ( 4 ( Problem: Thi formula i the bae for may other tatitical formula, however a a igle ummary meaure it ha little umerical meaig util it i coverted to a tadardized core. - Right ow it repreet the average ditace each pegui i from the mea, i quared mile uit. 3. Stadard Deviatio The quare root of a variace. The tadardized variace value. It provide u with a umerically meaigful meaure of variace. Thi value (whe combied with other tat method allow u to ifer what percetage of our obervatio are a certai ditace from the mea. Stadard Deviatio (baed o computatioal formula of variace Populatio St. Dev. : Sample St. Dev. σ σ Σ σ Σ -The larger the value of variace or tadard deviatio, relative to the umerical value of the obervatio, the greater the amout of variability that i preet i the data et.

7 Zoo Pegui : σ Σ σ σ Σ South Pole Pegui : Σ σ σ σ 3.. Σ 4 orth Pole Pegui : Σ σ σ σ.. Σ

Data Analysis and Statistical Methods Statistics 651

Data Analysis and Statistical Methods Statistics 651 Data Aalyi ad Statitical Method Statitic 65 http://www.tat.tamu.edu/~uhaii/teachig.html Lecture 9 Suhaii Subba Rao Tetig o far We have looked at oe ample hypothei tet of the form H 0 : µ = µ 0 agait H