Measures of Central Tendency


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1 Measures of Cetral Tedecy A studet s grade will be determied by exam grades ( each exam couts twice ad there are three exams, HW average (couts oce, fial exam ( couts three times. Fid the average if the studet has the followig grades. Exam: 7, 85, 8 HW: 90 Fial Exam: 85 You wat to fid the average height of a perso i this classroom. Process: You wat to fid the average height of a college studet at Agelo State Uiversity. Process: Populatio vs Sample Now You have recetly graduated with a college degree. You get a job offer from a compay that claims to have a average startig salary of $35, excludig emotioal attachmet, weather climate uimportat stuff like this, umerically speakig what else should you look for? For example: Cosider these two optios a average of $35, 000: small compay with salaries 0 K, 0K, 0K, 0K, 0K, 0K, 35K, 35K, 45K, 5K b average of $35,000: 5K, 5K, 30K, 30K, 35K, 35K, 40K, 40K, 55K, 55K Which oe oe be the best fit for you ad why?
2 Defie differet ways of measurig data Uless otherwise stated, we will assume that we have the etire populatio. Def. Give values x, x, x3 we defie the arithmetic average, x = x + x + x3 3 or i geeral x = x, by x + x + x x ex. Fid the average umber of abseces i a class with te studets.,, 0,,,,, 3, 4, = x i I the evet that the data is large ad ca be grouped together i differet classes, the we ca use the followig formula. ex. A large class of 00 studets has met for 5 five days. Here is a descriptio of the umber of times a studet has bee abset. What is the average umber of days that a studet has missed. 0 abseces 4 studets absece 3 studets abseces 35 studets 3 abseces 7 studets 4 abseces studets Aother useful measure Mode: value that appears with the largest frequecy. ex. The umber of times that a driver has bee pulled over durig the last five years 0,, 0, 3, 0,,,,,, ex. A class of 40 studets is asked the umber of times they ate out durig a five day period. 0 times 0 time 5 times 3 3 times 4 times
3 Media: the middle value of a give set of data. If o middle value exists, the we choose the average of the two middle values. ex. Salary withi a sixmember departmet i terms of thousads of dollars ex. The umber of miles that a perso walks per moth Rage: The differece of the largest ad smallest value. Review of some of the measures of cetral tedecy A study is doe of the eterig freshme studets at ASU. They list the umber of schools that they seriously cosidered attedig before comig to ASU. 000 had choice 560 had choices 300 had 3 choices 00 had 4 choices 40 had 5 Nobody had more tha five. Fid the arithmetic mea: Fid the mode: Fid the media: Fid the rage: A class of 8 eight studets are asked the umber of times that they ate out durig this past week. Here are their resposes: 0,, 0, 3,,,, 4, 3, 5 What is the arithmetic mea? What is the mode? What is the media? What is the rage?
4 Defie aother term. Fid a average of the distaces from the arithmetic mea average deviatio Aother example: A teacher marks dow the umber of times that a studet aswers a questio icorrectly i class. 0, 0, 0, 0, 0 Calculate the average deviatio The ext day the process is doe, 0, 0, 3, average deviatio Problem? Redefie: Average Squared Deviatio: A sd = ( x i x ad if the data is i terms of frequecies ( x i x f i This will elimiate the problem we had above
5 This gives two more ways to look at the distributio of data values. Variace: ( xi x fi Stadard deviatio: var iace We use the formulas above if the etire populatio is kow. I most cases we do ot kow the etire populatio so we use sample variace ad sample stadard deviatio. Variace: ( x i x f i sample stadard deviatio = s = sample variace A study is doe to determie the umber of accidets that a studet has bee ivolved i. A sample of 50 studets is doe with the results that follow 6 have bee i zero accidets 9 have bee i accidet have bee i accidets 3 have bee i 3 accidets Noe have bee i more tha 3. Fid the sample stadard deviatio.
6 Normal Distributios Normal Curves   mea, stadard deviatio, iflectio poits, area uder a curve, If the ormal curve has µ = 0 ad a variace = stadard deviatio =, we call it a stadard ormal curve. We use tables to fid area uder a curve. Notice that half of the area is to the right of the mea, half to the left (symmetric. We have a fuctio that expresses the curve ad there are ways of fidig the area uder a curve.( see page f(x = ( x µ σ e σ π ex. f(x = 4. Fid the area uder the curve betwee x = ad ex. f(x = x. Fid the area uder the curve betwee x = 0 ad 4 ex. f(x = x. Fid the area uder the curve betwee x =  ad x =
7 It is ot as easy to fid the area uder a ormal curve. Cosider the followig fuctios: (see page 60 f(x = ( x µ σ σ e π This is the fuctio that we would try to work with whe fidig the area uder a a ormal curve. You ca see the problem that we would have. A table is costructed for a stadard ormal curve. We use the followig formula ad this table to fid areas uder a ormal curve. x  µ z = σ Table z : : :. ex. Fid the area to the left of  uder a stadard ormal curve. ex. Fid the area to the left of 6 uder a ormal curve with µ = 0 ad variace = 6.
8 ex. Fid the area to the right of 90 uder a ormal curve with µ = 00 ad variace = 8. ex. Fid the area betwee 0 ad 30 uder a ormal curve with mea = 8 ad variace = 5
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