1.7.4 Mode Mode s the value whch occurs most frequency. The mode may not exst, and even f t does, t may not be unque. For ungrouped data, we smply count the largest frequency of the gven value. If all are of the same frequency, no mode exts. If more than one values have the same largest frequency, then the mode s not unque. Example 7 The value for the mode of the data n Example 5 s 15 (unmodal) Example 8 {,,, 4, 5, 6, 7, 7, 7} Mode = or 7 (Bmodal) For grouped data, the mode can be found by frst dentfy the largest frequency of that class, called modal class, then apply the followng formula on the modal class: mode = d L + ( L L ) 1 1 d1+ d 1 where: L 1 s the lower class boundary of the modal class; d 1 s the dfference of the frequences of the modal class wth the prevous class and s always postve; d s the dfference of the frequences of the modal class wth the followng class and s always postve; L s the upper class boundary of the modal class. Geometrcally the mode can be represented by the followng graph and can be obtaned by usng smlar trangle propertes. The formula can be derved by nterpolaton usng second degree polynomal. 19
Note that the mode s ndependent of extreme values and t may be appled n qualtatve data. 1.7.5 Concluson For symmetrcally dstrbuted data, the mean, medan and mode can be used almost nterchangeably. For moderately skewed dstrbuton data, ther relatonshp can be gven by Mean - Mode 3 (Mean - Medan) Physcally, mean can be nterpreted as the center of gravty of the dstrbuton. Medan dvdes the area of the dstrbuton nto two equal parts and mode s the hghest pont of the dstrbuton. 1.8 Dsperson and Skewness Sometmes mean, medan and mode may not be able to reflect the true pcture of some data. The followng example explans the reason. 0
Example 9 There were two companes, Company A and Company B. Ther salares profles gven n mean, medan and mode were as follow: Company A Company B Mean $30,000 $30,000 Medan $30,000 $30,000 Mode (Nl) (Nl) However, ther detal salary ($) structures could be completely dfferent as that: Company A 5,000 15,000 5,000 35,000 45,000 55,000 Company B 5,000 5,000 5,000 55,000 55,000 55,000 Hence t s necessary to have some measures on how data are scattered. That s, we want to know what s the dsperson, or varablty n a set of data. 1.8.1 Range Range s the dfference between two extreme values. The range s easy to calculate but can not be obtaned f open ended grouped data are gven. 1.8. Decles, Percentle, and Fractle Decle dvdes the dstrbuton nto ten equal parts whle percentle dvdes the dstrbuton nto one hundred equal parts. There are nne decles such that 10% of the data are D 1 ; 0% of the data are D ; and so on. There are 99 percentles such that 1% of the data are P 1 ; % of the data are P ; and so on. Fractle, even more flexble, dvdes the dstrbuton nto a convenence number of parts. 1.8.3 Quartles Quartles are the most commonly used values of poston whch dvdes dstrbuton nto four equal parts such that 5% of the data are Q 1 ; 50% of the data are Q ; 75% of the data are Q 3. The frst quarter s conventonally denoted as Q 1, whle the second and thrd quarters grouped together s Q and the last quarter s Q 3. Note that Q ncludes the medan, contans half of the frequency and excludes extreme values. It s also denoted the value (Q 3 - Q 1 ) / as the Quartle Devaton, Q D, or the semnterquartle range. 1
1.8.4 Mean Absolute Devaton Mean absolute devaton s the mean of the absolute values of all devatons from the mean. Therefore t takes every tem nto account. Mathematcally t s gven as: f x µ f where: f s the frequency of the th tem; x s the value of the th tem or class mark; µ s the arthmetc mean. 1.8.5 Varance and Standard Devaton The varance and standard devaton are two very popular measures of varaton. Ther formulatons are categorzed nto whether to evaluate from a populaton or from a sample. The populaton varance, σ, s the mean of the square of all devatons from the mean. Mathematcally t s gven as: ( x - ) f µ f where: f s the frequency of the th tem; x s the value of the th tem or class mark; µ s the populaton arthmetc mean. The populaton standard devaton σ s defned as σ = σ. The sample varance, denoted as s gves: f ( x x) ( f ) 1 where: f s the frequency of the th tem; x s the value of the th tem or class mark; x s the sample arthmetc mean. The sample standard devaton, s, s defned as s = For ungrouped data, s. ( x x) x ( x) / Σ Σ Σ s = = n 1 n 1 n
For grouped data, ( Σfx) Σf Σf( x x) Σfx s = = Σf 1 Σf 1 where f = n Note that when calculatng the sample varance, we have to subtract 1 from the total frequency whch appears n the denomnator. Although when the total frequency s large, s σ, the subtracton of 1 s very mportant. Example 10 Measures of Grouped Data (Refers to the followngs Data Set) Gas Frequency Class Class fx fx Consumpton ( f ) boundary mark ( x ) 10 19 1 9.5 19.5 14.5 14.5 10.5 0 9 0 19.5 9.5 4.5 0 0 30 39 1 9.5 39.5 34.5 34.5 1190.5 40 49 4 39.5 49.5 44.5 178 791 50 59 7 49.5 59.5 54.5 381.5 0791.75 60 69 16 59.5 69.5 64.5 103 66564 70 79 19 69.5 79.5 74.5 1415.5 105454.8 80 89 0 79.5 89.5 84.5 1690 14805 90 99 17 89.5 99.5 94.5 1606.5 151814.3 100 109 11 99.5 109.5 104.5 1149.5 101.8 110 119 3 109.5 119.5 114.5 343.5 39330.75 10 19 1 119.5 19.5 14.5 14.5 15500.5 1. x x f =, n = f n 1 14.5 + 0 4.5 + + 1 14.5 = 100 = 79.7 100 7970 671705 3
. 3. 50 48 medan = 79.5 + 10 = 80.5 0 5 13 Q1 = 59.5 + 10 16 67 Q 3 75 68 = 89.5 + 10 17 93.6 0 19 mode = 79.5 + 10 (0 19) + (0 17) = 8 4. sample s.d., n( x f ) ( x f s = n( n 1) ) = = 19. 100(671705) (7970) 100(100 1) 1.8.6 Coeffcent of Varaton The coeffcent of varaton s a measure of relatve mportance. It does not depend on unt and can be used to make comparson even two samples dffer n means or relate to dfferent types of measurements. The coeffcent of varaton gves: Standard Devaton Mean 100% Example 11 x S Salesman salary $916.76/month $86.70 Clercal salary $98.50/week $0.55 4
86.70 CVs = 100% = 31% 916.76 0.55 CVc = 100% = 1% 98.50 1.8.7 Skewness The skewness s an abstract quantty whch shows how data pled-up. A number of measures have been suggested to determne the skewness of a gven dstrbuton. One of the smplest one s known as Pearson s measure of skewness: Skewness = Mean Mode Standard Devaton 3 (Mean Medan) Standard Devaton If the longer tal s on the rght, we say that t s skewed to the rght, and the coeffcent of skewness s postve. Skewed to the rght (postvely skewed) 5
If the longer tal s on the left, we say that s skewed to the left and the coeffcent of skewness s negatve. Skewed to the left (negatvely skewed) Example 1 We are gong to use Example 9 to evaluate the dfferent measurements of varaton. As stated above, the salary ($) scales of the two companes are: Company A: 5,000 15,000 5,000 35,000 45,000 55,000 Company B: 5,000 5,000 5,000 55,000 55,000 55,000 Range Company A: $55,000 - $5,000 = $50,000 Company B: $55,000 - $5,000 = $50,000 6
Mean absolute devaton Company A: $ ( 5,000-30,000 + 15,000-30,000 + 5,000-30,000 + 35,000-30,000 + 45,000-30,000 + 55,000-30,000 ) / 6 = $15,000 Company B: $ ( 5,000-30,000 + 5,000-30,000 + 5,000-30,000 + 55,000-30,000 + 55,000-30,000 + 55,000-30,000 ) / 6 = $5,000 Varance Company A: {(5,000-30,000) + (15,000-30,000) + (5,000-30,000) + (35,000-30,000) + (45,000-30,000) + (55,000-30,000) } / 6 = 91,666,667 (dollar square) Company B: {(5,000-30,000) + (5,000-30,000) + (5,000-30,000) + (55,000 - Standard devaton 30,000) + (55,000-30,000) + (55,000-30,000) } / 6 = 65,000,000 (dollar square) Company A: $ 91,666,667 = $17,078 Company B: $ 65,000,000 = $5,000 Coeffcent of varaton Company A: $17,078 / $30,000 100% = 56.93% Company B: $5,000 / $30,000 100% = 83.33% 7
Coeffcent of Skewness Pearson s 1 st coeffcent of skewness, SK 1 Mean Mode = Standard devaton Pearson s nd coeffcent of skewness SK 3(Mean Medan) = Standard devaton Chebyshev s Theorem For any set of data, the proporton of data that les between the mean plus and mnus k 1 standard devatons s at least 1 k 1.e. Pr( µ kσ x µ + kσ ) 1 k Symbols Populaton Sample Sze N n Mean µ x Standard devaton σ s Varance σ s 8