tag:blogger.com,1999:blog-73438956724755605742024-02-19T22:53:54.386-08:00MatematikaRoby PoenyAhttp://www.blogger.com/profile/10615253628895024102noreply@blogger.comBlogger1125tag:blogger.com,1999:blog-7343895672475560574.post-11401076029494240532010-05-13T01:48:00.000-07:002010-05-13T01:49:20.784-07:00Kuartil<p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><b><i>JANGKAUAN (RANGE)</i></b> </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Notasi: <b>J</b></span></p> <p> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Untuk data yang </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> tidak dikelompokkan, </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> jangkauan adalah </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> selisih antara nilai terbesar dan nilai terkecil. </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Untuk data yang </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> dikelompokkan, </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> jangkauan adalah </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> selisih antara titik tengah kelas tertinggi dengan titik </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> tengah </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> kelas terendah.</span> </p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"><b><br /> <i>KUARTIL</i></b> Notasi: <b>q</b></span></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> </span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;">Kuartil membagi data (n) yang berurutan atas 4 bagian yang sama banyak.</span></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;">------|------|-------|-------<br /> Q1 Q2 Q3</span></p> <p> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Q1 = kuartil bawah (1/4n )<br /> </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Q2 </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> = kuartil tengah/median (1/2n)<br /> </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Q3 </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> = kuartil atas (1/4n )</span></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;">Untuk data yang tidak dikelompokkan terlebih dahulu dicari mediannya, kemudian kuartil bawah dan kuartil atas.</span></p> <p> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Untuk data yang dikelompokkan rumusan kuartil identik dengan rumusan mencari median.</span></p> <p align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><b>Q<sub>1</sub> = L<sub>1</sub> + [(1/4n - (</b></span><b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"><span style="font-family:Symbol;">å</span></span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> f)<sub>1</sub>)/f<sub>Q1</sub>] . c</span></b></p> <p align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><b>Q<sub>3</sub> = L<sub>3</sub> + [(3/4n - (</b></span><b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"><span style="font-family:Symbol;">å</span></span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> f)<sub>3</sub>)/f<sub>Q3</sub>] . c</span></b></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><br /> <b><i>DESIL</i></b> Notasi: <b>D</b></span></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;">Desil membagi data (n) yang berurutan atas 10 </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> <i>bagian yang sama besar. </i> </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> (D,, D2, D3, . . . . . . , D9)</span></p> <p align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><b>Di = Li + ((i/10)n - (</b></span><b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"><span style="font-family:Symbol;">å</span></span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> f)i)/fi . c </span></b></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><b><br /> <i>PERSENTIL</i></b> </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Notasi: <b>P</b></span></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;">Persentil membagi data (n) yang berurutan atas 100 </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> <i><b>bagian yang sama besar. </b></i> </span> <b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> (P<sub>1</sub>, P<sub>2</sub>, </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> P<sub>3</sub>, . . . . . . ,P<sub>99</sub></span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;">)</span></b></p> <p align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><b>Pi = Li +( i/100 n - (</b></span><b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"><span style="font-family:Symbol;">å</span></span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;">f)i)/fi . c</span></b></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Cara mencari Desil dan Persentil identik dengan cara mencari kuartil.</span></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><br /> <b>SIMPANGAN</b></span></p> <p><b><i> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> SIMPANGAN KUARTIL</span></i></b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Notasi: </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> <b>Qd<br /> </b></span><i><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;">(JANGKAUAN SEMI INTERKUARTIL)</span></i></p> <p align="center"><b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Qd = (Q3 - Q1) / 2</span></b></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><b><i>SIMPANGAN BAKU</i></b> Notasi: <b>S</b><br /> </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> <i>(STANDAR DEVIASI)</i></span></p> <p align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><b> S = </b></span><b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"><span style="font-family:Symbol;">Ö</span></span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;">((</span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"><span style="font-family:Symbol;">å</span></span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;">fi(xi-x bar)²)/n)<br /> </span></b></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><br /> </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> <i>atau CARA <b>CODING</b></i></span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"><br /> ___________________<br /> <b>S = <span style="font-family:symbol;">Ö</span> <span style="font-family:symbol;">(å</span> f<sub>i</sub>d<sub>i</sub>² / n) - (f<sub>i</sub>d<sub>i</sub>/n)²<br /> __________________<br /> = c <span style="font-family:symbol;">Ö</span> (<span style="font-family:symbol;">å</span> f<sub>i</sub>u<sub>i</sub>² / n) - (fiu<sub>i</sub>/n)²</b></span></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><b><br /> RAGAM (VARIANSI) </b></span> <b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> </span></b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;">Notasi:</span><b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> S²</span></b></p> <p><b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><br /> KOEFISIEN KERAGAMAN V = S / x bar . 100%</span></b></p> <p><br /> <i><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">Contoh:</span></i></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">1. Data tidak dikelompokkan<br /> Diketahui data</span></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> 95, 84, 86, 90, 93, 88, 97, 98, 89, 94</span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> </span></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Data diurutkan terlebih dahulu, menjadi:<br /> </span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> 84 86 818 89 90 93 94 915 97 98</span></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Q1 = 88 ; Q2 = 90 93 ; Q3 = 95</span></p> <p> <span style="font-family:Times Roman;font-size:85%;color:#000000;"> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;"> a. Jangkauan J = 98 - 84 = 14</span></span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"><br /> <br /> b. Kuartil Q1=88 ; Q2 = (90+93)/2 = 91,5 ; Q3 = 95<br /> <br /> Simpangan kuartil = Qd = (95 - 88) / 2 = 3,5<br /> <br /> <br /> c. Rata-Rata<br /> <br /> = (88+86+88+89+90+93+95+97+98)/10 = 91,4<br /> <br /> Simpangan baku = <span style="font-family:Symbol;">Ö</span>(((84-91,4)² + ...... + (98-91,4)²)/10) = 4,72</span></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">2. Data dikelompokkan</span><br /> </p> <table align="center" border="1" width="60%"> <tbody><tr bg style="color:#ffffff;"> <td width="22%" height="30"> <div align="center"><b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">Skor</span></b></div> </td> <td width="53%" height="30"> <div align="center"><b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">Titik Tengah</span></b></div> </td> <td width="25%" height="30"> <div align="center"><b><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">Frekuensi</span></b></div> </td> </tr> <tr> <td width="22%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">50-54</span></div> </td> <td width="53%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">52</span></div> </td> <td width="25%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">4</span></div> </td> </tr> <tr> <td width="22%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">55-59</span></div> </td> <td width="53%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">57</span></div> </td> <td width="25%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">6</span></div> </td> </tr> <tr> <td width="22%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">60-64</span></div> </td> <td width="53%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">62</span></div> </td> <td width="25%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">8</span></div> </td> </tr> <tr> <td width="22%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">65-69</span></div> </td> <td width="53%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">67</span></div> </td> <td width="25%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">16</span></div> </td> </tr> <tr> <td width="22%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">70-74</span></div> </td> <td width="53%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">72</span></div> </td> <td width="25%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">10</span></div> </td> </tr> <tr> <td width="22%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">75-79</span></div> </td> <td width="53%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">77</span></div> </td> <td width="25%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">3</span></div> </td> </tr> <tr> <td width="22%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">80-84</span></div> </td> <td width="53%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">82</span></div> </td> <td width="25%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">2</span></div> </td> </tr> <tr> <td width="22%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">85-89</span></div> </td> <td width="53%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">87</span></div> </td> <td width="25%"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">1</span></div> </td> </tr> <tr> <td width="22%"> <br /></td> <td width="53%"> <br /></td> <td bg width="25%" style="color:#ffffff;"> <div align="center"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"><b>n = 50</b></span></div> </td> </tr> </tbody></table> <p> <span style="font-family:Garamond;font-size:85%;color:#000000;"> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;">a. Jangkauan = Titik tengah kelas tertinggi - Titik tengah kelas terendah = 87-52 =35 </span></span></p> <p><span style="font-family:Garamond;font-size:85%;color:#000000;"> </span> <span style="font-family:Garamond;font-size:85%;color:#000000;"> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;">b. Kuartil bawah (¼n )</span><br /> <br /> Q1 = </span><span style="font-family:Garamond;font-size:85%;"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;">59,5 + ((12,5 - 10)/8 . (5)</span></span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">)</span><span style="font-family:Garamond;font-size:85%;color:#000000;"> = </span><span style="font-family:Garamond;font-size:85%;"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;">61,06</span></span><span style="font-family:Garamond;font-size:85%;color:#000000;"><br /> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;"><br /> Kuartil bawah (¾n )</span></span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"><br /> <br /> Q3 = 69,5 + (37,5 - 34)/10 . 5 = 71,25<br /> <br /> </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;color:#000000;"> Simpangan Kuartil </span> <span style="font-family:Garamond;font-size:85%;color:#000000;"><br /> <br /> </span><span style="font-family:Garamond;font-size:85%;"> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;"> Qd </span></span><span style="font-family:Garamond;font-size:85%;color:#000000;">= </span><span style="font-family:Garamond;font-size:85%;"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;">(Q3 - Q1) </span></span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">/ 2 </span><span style="font-family:Garamond;font-size:85%;color:#000000;">= </span><span style="font-family:Garamond;font-size:85%;"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;">(71,25 - 61,06) / 2 = 5,09</span></span></p> <p> <span style="font-family:Garamond;font-size:85%;color:#000000;"> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;">c. Rata-rata</span><br /> </span><span style="font-family:Garamond;font-size:85%;"> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;"> _<br /> x = ((4)(52) + (6)(57) + ... + (1)(870)</span></span><span style="font-family:Garamond;font-size:85%;color:#000000;"> </span><span style="font-family:Garamond;font-size:85%;"> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;">/ 50 = 66,4</span></span></p> <p><span style="font-family:Garamond;font-size:85%;color:#000000;"> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;">d. Simpangan Baku</span><br /> <br /> ___________________________________<br /> </span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> <span style="font-family:Symbol;"> Ö</span>((52-66,4)² + ...... + (87-66,4)²)/50 = 7,58</span></p> <p><i><span style="font-family:Garamond;font-size:85%;"><span style="font-family:Verdana, Arial, Helvetica, sans-serif;">CATATAN:</span></span></i><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> </span></p> <p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> </span></p><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> <li>Bila pada suatu kumpulan data, <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">setiap </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> data </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> ditambah / dikurangi </span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">dengan suatu bilangan, maka:<br /> </span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">- </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> nilai statistik yang </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> berubah: </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> Rata-rata, Median, Modus, Kuartil.<br /> </span><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">- nilai statistik yang </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> tetap : J </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> angkauan, Simpangan Kuartil, Simpangan baku.<br /> </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> <br /> </span></li> <li><span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> Bila pada suatu kumpulan data, </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> setiapp </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> data </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> dikali ldibagi </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;"> dengan suatu bilangan, maka: </span> <span style="font-family:Verdana, Arial, Helvetica, sans-serif;font-size:85%;">semua nilai statistiknya berubah.</span></li></span>Roby PoenyAhttp://www.blogger.com/profile/10615253628895024102noreply@blogger.com0