SAMPLE VERSUS POPULATION. Population - consists of all possible measurements that can be made on a particular item or procedure.

Transcription

1 SAMPLE VERSUS POPULATION Populatio - cosists of all possible measuremets that ca be made o a particular item or procedure. Ofte a populatio has a ifiite umber of data elemets Geerally expese to determie good estimates for populatio mea ad variace Sample - a subset of data selected from the populatio Ca be gathered i a ecoomical fashio May or may ot be represetative of populatio PLATE -1

2 METHODS OF ANALYZING DATA I. Numerical methods a. Rage - highest value mius the lowest value i the data set b. Dispersio - differece betwee two elemets i a data set c. Measures of cetral tedecy d. Measures of variatio II. Graphical Methods Frequecy Histogram - a bar graph based o a selected class width. Also kow as a histogram. Class - a subregio of the data. Class width - rage of a sigle class PLATE -

3 METHODS OF ANALYZING DATA Measures of cetral tedecy: 1. Mea - arithmetic mea of data. Symbolized by: A. µ for populatio, true value B. y estimate of µ determied from a sample, where y i 1. Media - physical midpoit of a data set whe data arraged i umerical order. A. If data set has odd umber of elemets, the the media is the physical midpoit of the arraged data. B. If the data set has a eve umber of elemets, the the media is the average of the two elemets that are earest the physical midpoit. 3. Mode - the most frequetly occurig elemet i the data set. Data sets ca have more oe mode. PLATE -3

4 NUMERICALLY ORDERED DATA SET Mea = 3.5 Rage = = 6.0 To create 7 uiform width classes divide rage by 7. Class width = 6.0/7 = 0.86 Add to mimum data elemet to get bouds for first class. First iterval: 0.1 x < 0.96 = Secod iterval : 0.96 x < 1.8 = etc. PLATE -4

5 A CLASS FREQUENCY TABLE Class Class Class relative iterval frequecy frequecy /50 = /50 = /50 = /50 = /50 = /50 = /50 = 0.14 = 50/50 = 1 Note that sum of class relative frequecs always oe. Use class frequecy table to costruct histogram. Histogram PLATE -5

6 COMMON HISTOGRAM SHAPES (a) (b) (c) (d) (e) Thigs that histograms idicate about data: 1. a is less precise tha b. c is bimodal 3. d is skewed to the right 4. e is skewed to the left PLATE -6

7 NUMERICALLY ORDERED DATA SET Measures of Cetral Tedecy Mea = 3.5 Media = Mode = 3.8 (occurs 3 times) PLATE -7

8 DEFINITIONS True value - A quatities theoretically correct or exact value. This value is ever kow. Error, - The differece betwee a measured quatity ad its true value. = y - µ i Residual, v - The differece betwee ay measured value ad the most probable value for a data set (the mea). Degrees of freedom, redudacies - The umber of measuremets that are i excess of the umber ecessary to solve for the ukows. PLATE -8

9 MEASURES OF DATA VARIATION Variace - a value by which the precisio of a populatio is expressed. Populatio variace, i Sample variace, v S i 1 STANDARD ERROR, - Rage withi mea that 68.3% of all observatios lie. Stadard deviatio, S - is a estimate for the stadard error of a populatio. Stadard deviatio of the mea, S y S PLATE -9

10 PRECISION VERSUS ACCURACY a b c d Results a is accurate but ot precise. b is either precise or accurate. c is precise but ot accurate. errors) d is accurate ad precise. Observer s viewpoit (Never kow its accurate) (Assumed poor) (Caused by systematic (The goal i data collectio) PLATE -10

11 ALTERNATE FORMULA FOR SAMPLE VARIANCE Expadig S i 1 ( y ) 1 S 1 1 [(y y 1 ) (y y ) (y y ) ] Substitutig i 1 for y S 1 1 y 1 y y Expadig S 1 1 y 1 y 1 y y y 1 y y PLATE -11

12 ALTERNATE FORMULA FOR SAMPLE VARIANCE Recogizig that occurs times S 1 1 ( y 1 y y ) y 1 y y Or S 1 1 ( ) ( ) Factorig ad regroupig, S 1 1 ( ) 1 ( ) 1 1 ( ) 1 ( ) Multiplyig the last term by, S 1 1 ( ) PLATE -1

13 ALTERNATE FORMULA FOR SAMPLE VARIANCE Fially, recogizig that y S ( ) y 1 NOTE: This formula ca overwhelm a hadheld calculator whe the measuremets ivolve large umbers. PLATE -13

14 EXAMPLE y = 3.5 Determiatio of stadard deviatio usig residuals. # y v v # y v v # y v v # y v v = S 50 i 1 v i ±1.37 PLATE -14

15 USING ALTERNATE FORMULA ( ) = 7, SO S 7, ( 3.5 ) sec OR S = 1.88 = ±1.37" PLATE -15