Chapter II Dstrbuton Analyss D... (Absolute and Relatve Frequences) Let X be a characterstc possessng the attrbutesa, =,,..., k. The absolute frequency of the attrbutea, =,,..., k s defned as follows: F( a ): = Number of cases n whch a occurs, =,,..., k. The relatve frequency of the attrbutea, =,,..., k, s defned as: R... f ( a): = F( a), =,,..., k. n n : number of observatons. F( a ) n, =,,..., k f ( a ), =,,..., k k F( a) = n, = k f ( a) =. = Ex... Denote by X : wages per hour n of 3 workers n a frm: 7.5, 7.8, 7.8, 4.7, 5.5, 8.3, 6., 6., 6.55, 7.5, 5.5, 5.6, 5.6, 5.6, 6., 4., 5., 5.6, 6.55, 8., 7.5, 7.5, 6.55, 5.6, 5.6, 5., 6., 8.3, 7.5, 5.6. n = 3, k =.
a F( a ) f ( a ) 4,,33 4,7,33 5,,67 5,5,67 5,6 7,33 6, 4,33 6,55 3, 7,5 3, 7,5,67 7,8,67 8,,33 8,3,67 Total 3. D... (Absolute and Relatve Frequency Dstrbutons) The sequence F( a ), F( a ),..., F( a k) s defned as the absolute frequency dstrbuton. The sequence f n ( a ), f ( a ),..., f ( a k) s defned as the relatve frequency dstrbuton. R... (Graphcal Representaton of Frequences) We dstngush. bar charts A bar chart depcts the frequences by a seres of bars.. polygons A frequency polygon s a lne graph of a frequency dstrbuton. 3. frequency curves A frequency curve s a smoothed frequency polygon. 4. hstograms A hstogram s a bar graph of a frequency dstrbuton 5. pe charts (crcle dagrams) A pe chart s a pe-shaped fgure n whch peces of the pe represent the frequences.
Ex... (cont.). Bar chart 3 Percent 4, 5, 5,5 6, 7,5 7,8 8,3 4,7 5,5 5,6 6,55 7,5 8, wage per hour. Polygon 3 Percent 4, 5, 5,5 6, 7,5 7,8 8,3 4,7 5,5 5,6 6,55 7,5 8, wage per hour 3
4. Hstogramm 8 6 4 Std. Dev =,4 Mean = 6,3 N = 3, 4, 5, 6, 7, 8, 4,5 5,5 6,5 7,5 8,5 wage per hour 5. Pe chart 8,3 8, 7,8 7,5 4, 4,7 5, 5,5 5,5 7,5 5,6 6,55 6, R.. 3. In terms of skewness, a frequency curve can be. negatvely skewed: nonsymmetrcal wth the «tal» to the left.. postvely skewed: nonsymmetrcal wth the «tal» to the rght. 3. symmetrcal. 4
In terms of kurtoss, a frequency curve can be. platykurtc: flat wth the observatons dstrbuted relatvely evenly.. leptokurtc: peaked wth the observatons concentrated wthn a narrow range of values. 3. mesokurtc: nether flat nor peaked, n terms of the dstrbuton of observed values. D.. 3. (Cumulatve Absolute and Relatve Frequences) The cumulatve absolute frequency of the attrbutea, =,,..., k, s defned as: F( a ) + F( a ) +... + F( a ) = F( a ), =,,..., k. = The cumulatve relatve frequency of the attrbutea, =,,..., k, s defned as: f ( a ) + f ( a ) +... + f ( a ) = f ( a ), =,,..., k = Ex... (cont.) a F( a ) f ( a ) = F( a ) = f ( a ) 4,,33,33 4,7,33,66 3 5,,67 4,33 4 5,5,67 6, 5 5,6 7,33 3,433 6 6, 4,33 7,566 7 6,55 3,,666 8 7,5 3, 3,766 9 7,5,67 5,833 7,8,67 7,9 8,,33 8,933 8,3,67 3, Total 3. R.. 4. (Ogve) An ogve shows data values on the horzontal axs and ether the cumulatve absolute frequences, the cumulatve relatve frequences, or the cumulatve percent frequences on the vertcal axs. 5
Ex... (cont.) 8 6 Cumulatve Percent 4 4, 5, 5,5 6, 7,5 7,8 8,3 4,7 5,5 5,6 6,55 7,5 8, wage per hour D.. 4. (Emprcal Dstrbuton Functon) The emprcal dstrbuton functon s defned as follows: for x a F( x): = f ( a ) for a < x a = for x> a + R.. 5. (Some Important Propertes of the Dstrbuton Functon) k.. 3. 4. 5. ( ) F( x) = P X < x ( P : proporton) F ( x) ( x ) F( ) x, x : x < x F x ( ) ( ) P( x X < x ) = F x F x F (x) s at least left-sded contnuous and has at most a fnte number of ump dscontnutes. 6
6. x F( x) x + F( x) R.. 6. A dstrbuton functon can be represented. n analytc form. n tabular form 3. as a graph. Ex... (cont.). Analytc form.33.66.33..433 F( x) =.566.666.766.833.9.933. < x 4. 4.< x 4.7 4.7< x 5. 5.< x 5.5 5.5< x 5.6 5.6< x 6. 6.< x 6.55 6.55< x 7.5 7.5< x 7.5 7.5< x 7.8 7.8< x 8. 8.< x 8.3 8.3< x<+ 7
. Tabular form Interval F (x) ], 4.]. ] 4., 4.7].33 ] 4.7, 5.].66 ] 5., 5.5].33 ] 5.5, 5.6]. ] 5.6, 6.].433 ] 6., 6.55].566 ] 6.55, 7.5].666 ] 7.5, 7.5].766 ] 7.5, 7.8].833 ] 7.8, 8.].9 ] 8., 8.3].933 ] 8.3, + [. Ex... (cont.) Calculate and nterpret. F (5.75). F(7.8) F(5.). Soluton:. F (5.75) =.433. 43.3% of the workers earn less than 5.75 an hour.. F(7.8) F(5.) =.766.33=.633. 63.3% of the workers earn at least 5. but less than 7.8 an hour. R.. 6. The process of dvdng the data nto dfferent groups (vz. classes) whch are homogeneous wthn but heterogeneous between themselves s called a classfcaton. D.. 5. (Absolute and Relatve Class Frequences) Let the data be dvded nto the classesc, =,,..., p. The absolute frequency of the class C s defned as follows: F := Number of observatons nc, =,,..., p The relatve frequency of the class C s defned as follows: 8
Ex... (cont.) f : =, =,,..., p. n H C F f [ 4., 5.[.67.67..67 4.5 [ 5., 6.[.367.434..367 5.5 3 [ 6. 7.[ 7.33.667..33 6.5 4 [ 7. 8.4[.333..4.38 7.7 Total 3. = f w h m R.. 6. (Graphcal Representaton of Class Frequences). Bar charts 4 3 Percent 3 4 Earnngs per hour 9
. Polygon 4 3 Percent 3 4 Earnngs per hour 3. Hstogramm (not normed) : 8 6 4 Std. Dev =,7 Mean =,8 N = 3,,, 3, 4, Earnngs per hour
4. Pe chart 4 3 R.. 7. (Emprcal Dstrbuton of Grouped Data) for x m F( x): = f ( a ) for m < x m = for x> mk + Ex... (cont.). for < x 4.5.67 for 4.5< x 5.5 F( x) =.434 for 5.5< x 6.5.667 for 6.5< x 7.7. for 7.7< x<+ Ex... (cont.) Calculate and nterpret. F (6.5). F(7.4) F(5.6).
Soluton:. F (6.5) =.434. About 43.4% of the workers earn less than 6.5.. F(7.4) F(5.6) =.667.434=.33. About 3.3% of the workers earn at least 5.6 but less than 7.4 an hour.
(Last revsed: 3.3.9) 3