Descriptive Statistics: Measures of Center

Transcription

1 Secto 2.3 Descrptve Statstcs: Measures of Ceter Frequec dstrbutos are helpful provdg formato about categorcal data, but wth umercal data we ma wat more formato. Statstc: s a umercal measure calculated from data Descrptve statstcs: are statstcs that descrbe a set of data. A deas of some descrptve statstcs? Mea: s the most wdel kow measure of ceter of a data set. It s also kow as the average. It s calculated b summg all of the observatos ad the b dvdg the umber of observatos. Example 1 The average legth of sta b patets a short term hosptal A radom sample of e patets elded the followg data o the legth of sta das 4, 12, 18, 9, 12, 6, 7, 3, 55 What s the mea of these data pots?

2 Summato Notato the varable uses dces 1 s the frst data pot, 2 ad to dcate each data pot s the secod data pot, s the last data pot the data set. s the Greek letter Sgma whch represets the S for sum (to add) 1 s read as the summato of - values from where 1 to where s the sample sze. letter These smbols are used to defe the equato fdg the sample mea. Sample Mea For a varable Y, the mea of the observatos for a sample s called a sample mea ad s deoted b Where s the sample sze.. Smbolcall,

3 Hosptal Sta Data das Devato: the dfferece betwee a data pot a data set ad the mea of the data set. devato The mea s the ceter of the data set ad has the propert that the sum of the devatos of the data set s 0 0

4 Meda: s the value a ordered data set that les most earl the mddle of the sample. It s the umber that dvdes the bottom 50% of the data from the top 50%. Procedure 1 Meda of a Data Set Arrage the data creasg order If the umber of observatos s odd, the the meda s the observato exactl the mddle of the ordered lst. If the umber of observatos s eve, the the meda s the mea of the two mddle observatos the ordered lst. I both cases, f we let deote the umber of observatos, the 1 the meda s at the posto the ordered lst. 2 Example 2 Legth of sta a short term hosptal Fd the meda of the legth of sta a short term hosptal

5 Example 3 Serum Cholesterol Sx me wth hgh serum cholesterol partcpated a stud to determe the effects of det o cholesterol level. At the begg of the stud the cholesterol levels are show below: What s the meda cholesterol level for these me? What s the mea cholesterol level for these me? Robust or resstat statstc: f the value of a statstc s uaffected b chages a small porto of the data, the t s sad to be ths tpe of statstc. What statstc, mea or meda could be descrbed as robust? Of the two data sets that we have studed legth of sta a short term hosptal ad cholesterol levels at the begg of a stud have smlar values for the mea ad meda? Wh do ou thk that there was a dfferece for the data set wth a dssmlar mea ad meda? If we have aother data set ad the hstogram for ths data set s left skewed, where do ou thk that the mea ad meda would be located?

6 Procedure 2 Fdg the populato mea for group data Whe there s a larger data set or a frequec dstrbuto for the data set, there s a equato to help fd the mea. Smbolcall, Where s the populato mea, N s the populato sze ad s equal to the ( f value occurs. f N s a product of how ma tmes the value of each - Example 4 Mles ru per week over a ear perod (b EKJ) Mles ru Number of per week weeks What s the mea for ths data set? What s the meda for ths data set? Are the mea ad meda the same? Wh or wh ot? f )

7 Costruct a hstogram of the data to help Mode of a data set: s the value(s) of the data set wth the largest frequec Procedure to fd the mode of a data set Fd the frequec of each value the data set. If o value occurs more tha oce, the the data set has o mode. Otherwse, a value that occurs wth the greatest frequec s a mode of the data set. Example 5 Rug data Fd the mode(s) of ths data set.