Neryškioji dichotominių testo klausimų ir socialinių rodiklių diferencijavimo savybių klasifikacija

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1 Neryškoj dchotomnų testo klausmų r socalnų rodklų dferencjavmo savybų klasfkacja Aleksandras KRYLOVAS, Natalja KOSAREVA, Julja KARALIŪNAITĖ Technologcal and Economc Development of Economy Receved 9 May 07; accepted 07 Aprl 08

2 Man dea: To construct an estmate of a test tem solvablty We propose to separate all test takers nto three fuzzy subsets: W weak students; A average students; S strong students. For each subset we assess the solvablty by a trangular fuzzy number: T r W, r A T T, rs These numbers are compared to each other, and on that bass classfcaton of test tems s performed.

3 . Order relatons of the fuzzy trangular numbers If S s a crsp set, defne the solvablty of the test tem k(s) of the set S of all testees as the percent of correctly answered students number t(s): t( S) k ( S) 00(%), S () where S s a number of elements of the set S. If S s a fuzzy set, we wll defne the solvablty of the test tem by the fuzzy trangular number Tr(L, T, R) (0 L T R 00), whch n the specfc case concdes wth (): k(s) = L = T = R. In general Tr(L, T, R) s a fuzzy number of the set x[0, 00] whch has the trangular membershp functon (Zadeh 965) x L, for x [ L, T ], T L R x ( x), for x ( T, R], R T 0, otherwse. It s clear that 0 ( x).

4 Suppose that Tr (L, T, R ) and Tr (L, T, R ) are two trangular fuzzy numbers. We denote that Tr Tr, when L L & T T & R R. () Tr Tr, when R < L. (3) We suggest such classfcatons of the test tem dfferentaton property: The test tem well dfferentates all students, when: The test tem well dfferentates strong students, when: The test tem well dfferentates weak students, when: Tr T W r A T r S ; Tr W T r T A r S ; Tr T W r A T r S ; The test tem badly dfferentates students, when: Tr W T T r S ; r A The test tem napproprate s n all other cases.

5 . Algorthm of makng the trangular fuzzy numbers Suppose, that A = {a S: 0 µ A (a) } s a fuzzy set (the fuzzy subset of the testees (unversal) set S), µ A (a) ts membershp functon. The α cuts ( level sets) of the set A are called (crsp) sets A α = {a S: µ A (a) }. We wll take the nonempty cuts calculate the values A, p A,..., A n of the set A. For each cut A we t A 00(%), where t number of the testees from the set A who correctly answered the test tem, A number of elements of the set A (we apply () formula). We get number T of the trangular fuzzy number Tr(L, T, R) when = : T = p.0.

6 Fg.. The values L are obtaned by the least squares method mnmzng by L the sum n d, where ) ( p p d and L L T p ) ( ) (. Smlarly are obtaned the values R.

7 The values L and R are obtaned by the least squares method. Denote n,...,, that <.0 values for whch T p, and correspondngly denote n,...,,, when T p ; n - + n + = n. Defne the objectve functons for the parameters L and R:, ) ( ) ( n L L T p L f. ) ( ) ( n T R R p R g The meanng of the functons f(l), g(r) s sum of the squares of the values d shown n the Fg. Further we solve the equatons 0 ) ( L f and 0 ) ( R g :, ) ( n n T p L. ) ( n n T p R

8 3. Fndng fuzzy subsets of the testees set The fuzzy subsets W, A, S of the testees set we defne by the trapezodal membershp functons t a, b a ) ( d t ( a, b, c, d t), d c, 0, when t [ a, b], when t [ c, d], when t ( b, c), otherwse, where a b c d. Let us suppose that knowledge (or other consdered property) of all tested students s valued by some scores b, b,, b n. Denote the least and the greatest b values as mn and max.

9 Let us take four numbers mn < α < β < γ < δ < max and let us defne the membershp functons of the subsets W, A, S as: Parameters t get such values t A S )( t), (,,, )( t),,,max, max) ( t). t t 3 t W (mn, mn,, t 4 t, t3 t4, and the meanng of the parameters α, β, γ, δ s the followng: when b α then the student for sure s weak, when β b γ average, when b δ strong. Fg.. Membershp functons of the weak, average and strong students W A S ( t), ( t), ( ). ). (mn, mn,, t )( ( t,,, t3 ) t4,,max, max) t

10 For each subset W, A, S we construct the trangles Tr(L, T, R) and analyze all parameters t, whch follow restrctons. From the obtaned fuzzy trangular numbers we wll make optmstc trangles (we take averages of L, T, R, when all t values are reselected) and pessmstc trangles (wder ntervals, when we take mnl, maxr and average of T values, when all t values are reselected). Thus whle classfyng test tems of the test we pay attenton not only to the average (optmstc), but also to the pessmstc fuzzy assessments.

11 4. Analyss of the test tems of one partcular test In ths secton we wll show some test tems from the mathematcs test. The test was gven to 06 students from Vlnus Gedmnas Techncal Unversty faculty of Cvl Engneerng. Ther knowledge was evaluated by the results of the test made of 0 test tems, each valued by one score. We present as an example of obtaned results by the Table. Table. A part of the obtaned results In the frst column there are students numbers, n the frst row test tems numbers. The consdered test tems are shaded. In the last column there are total numbers of correctly answered test tems of each student scores b. Here we present only a part of one table. There were 4 groups wth 9, 6, 8 and 3 students correspondngly n each group.

12 Parameters were: mn =, max = 0, α = 9, β =, γ = 4, δ = weak students ( b 9); 30 - average ( b 4); 9 - strong (7 b 0). For the rest of the students we cannot unambguously state that they are strong, average or weak. We have 33 = 06 (4+30+9) such students. We vared wth assgnng them to one of the three subsets, changng parameters t, constructng trapezods. Next, we present the test tems from the consdered test: 6 lm x 4 ln x x0 ) 4; 3) ; 5) lmt does not exst; ) 0; 4) ln 4; 6). The solvablty of the test tem s 74(%) (that s 74% of all students gave correct answer to that test tem). The fuzzy trangular numbers of the groups and for all students are shown n the Fgs. 4 8.

13 Notce that only for one ( nd ) group test tem 6 badly dfferentates students n the pessmstc case (relatons T ). For all other groups separately and for all T r W r A T r S groups together the test tem well dfferentates weak students both n optmstc and pessmstc cases (relatons Tr T W r T r ). A S Queston No 6, group No. Blue trangles - pesmstc case, red - optmstc Queston No 6, group No. Blue trangles - pesmstc case, red - optmstc Fg. 4. Fuzzy trangles test tem 6, group (blue trangles pessmstc, red optmstc cases) Fg. 5. Fuzzy trangles test tem 6, group (blue trangles pessmstc, red optmstc cases).

14 Queston No 6, group No 3. Blue trangles - pesmstc case, red - optmstc Fg. 6. Fuzzy trangles test tem 6, group 3 (blue trangles pessmstc, red optmstc cases). Queston No 6, all groups. Blue trangles - pesmstc case, red - optmstc Queston No 6, group No s. Blue trangles - pesmstc case, red - optmstc Fg. 7. Fuzzy trangles test tem 6, group 4 (blue trangles pessmstc, red optmstc cases) Fg. 8. Fuzzy trangles test tem 6, all groups (blue trangles pessmstc, red optmstc cases).

15 5. Concluson and future research Method of establshng the test s tems dfferentaton property was proposed. Consdered method does not requre the strct evaluaton of the testees knowledge. To apply t s enough to have only relatve achevement scores, for example raw test scores. The same method can be appled both for the relatvely small groups of testees and for combned groups. An experment allows to expect to get a stable test tems classfcaton havng not bg (0 30 students) groups of testees, however to get statstcally relable conclusons t s approprate to carry out Monte Carlo type experments. Ths s the object of our future research. Also pay attenton that n ths survey all test tems are consdered ndependently from another, though the test tems blocks essentally make one task and t would be rght to study them together (Krylovas, Kosareva 0).

16 AČIŪ UŽ DĖMESĮ!

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22 Example. Suppose that the fuzzy set of testees s: S = {(a,.0), (b, 0.5), (c,.0), (d, 0.3), (e, 0.), (f, 0.3)}. Suppose that only a and b answered correctly to the test tem. Then the set S has four dfferent cuts and correspondng p values: S { a, c}, p 00 50, S S S { a, b, c},.0 p 0.5 { a, b, c, d, f }, p { a, b, c, d, e, f }, p 00 67, , Thus n =, = 0., p = 33, = 0.3, p = 40; n + =, = 0.5, p parameters of the fuzzy trangular number Tr(L, T, R): =67. We fnd the T=50, ( ) (.0 0.) ( ) (.0 0.3) L (.0 0.) (.0 0.3) ( ) (.0 0.5) R 84. (0.5) 3.85,