5. Ukuran Sebaran ( keragaman )

1. 5. UkuranSebaran (keragaman) Kuswanto2013

2. Ukuran keragaman • Dari tiga ukuran pemusatan, belum dapat memberikan deskripsi yang lengkap bagi suatu data. • Perlu juga diketahui seberapa jauh pengamatan-pengamatan tersebut menyebar dari rata-ratanya. • Ada kemungkinan diperoleh rata-rata dan median yang sama, namun berbeda keragamannya. • Beberapa ukuran keragaman yang sering kita temui adalah range (rentang=kisaran=wilayah), simpangan (deviasi), varian (ragam), simpangan baku (standar deviasi) dan koefisien keragaman.

3. f f X X Measures of Dispersion and Variability These are measurements of how spread the data is around the center of the distribution

4. 2 2 3 4 5 = X1 = X2 = X3 = X4 = X5 • Range Kisaran = Rentang • difference between lowest and highest numbers Place numbers in order of magnitude, then range = Xn - X1. Range = 5 - 2 = 3 Problem - no information about how clustered the data is

5. 2. SIMPANGAN (deviation) You could express dispersion in terms of deviation from the mean, however, a sum of deviations from the mean will always = 0. i.e. (Xi- X) = 0 So, take an absolute value to avoid this Problem –the more numbers in the data set, the higher the SS

6. Contoh • Misal, jumlahbukutulis yang dibawa5mahasiswaadalah3, 5, 7, 7, 8. Rerata (mean) data tersebutadalah30/5 = 6. Simpangandihitungdenganmengurangisetiapnilaipengamatandenganreratanya Agar nilainya tidak negatip, dapat di kuadratkan yang kemudian disebut simpangan kuadrat. Simpangan kuadrat bermanfaat untuk penghitungan varian.

7. Sample mean deviation = | Xi- X | n 3. SimpanganRerata (Mean Deviation) Essentially the average deviation from the mean

8. Sample SS = (Xi - X)2 = 4. Sum of square Another way to get around the problem of zero sums is to square the deviations. Known as sum of squares or SS Xi2 - (Xi)2/n SS is much more common than mean deviation

9. Example Sample SS = (Xi - X)2 2 2 3 4 5 = X1 = X2 = X3 = X4 = X5 X = 3.2 = 1.44 + 1.44 + 0.04 + 0.64 + 3.24 = 6.8 SS = (2 - 3.2)2 + (2 - 3.2)2 + (3 - 3.2)2 + (4 - 3.2)2 + (5 -3.2)2 Problem –the more numbers in the data set, the higher the SS

10. 5. VARIAN (RAGAM) • Dalamprakteknya, simpanganjarangdigunakankarenasulitdimanipulasisecaramatematis. • Sebagaigantinyadiperlukankuadratsemuasimpangantersebutkemudiandibagiderajadbebasn-1, dandisebutdenganvarian (ragam) • Digunakanpembagi n-1 agar menjadipendugatak bias. • Ragampopulasidilambangkandenganσ², sedangragamcontohdilambangkandengans2,

11. Population Variance (2 ): This is just SS 2 = (Xi -  )2 N N Our best estimate of 2is sample variance (s2): S2 =  (Xi - X)2 n - 1 Note : divide by n-1 known as degrees of freedom The mean SS is known as the variance  Xi2 - (Xi)2/n = n - 1 Problem - units end up squared

12. Mengapa (n-1) disebutderajadbebas(kebebasan)? • Perhatikanilustrasiberikut. • Apabilaseseoranghendakmengangkat 100 kg berasdarilantai 1 kelantai 3 daniaharusmengangkatmaksimalsebanyak 5 kali, makaorangtersebutdapatmemilihmenyelesaikannyadalam 2 kali angkat, 3 kali atausampai (n-1) kali. • Sampaidengan 4 (n-1) kali, orangtersebutbebasmemilihberapa kg yang diangkatkelantai 3. • Namunpadaangkatanterakhir (1 kali), mautidakmau, orangtersebutharusmengangkatsemuaberas yang tersisa. Artinyakebebasanmemilihjumlah yang diangkathanya (n-1) kali.

13. 6. StandarDeviasi • Penggunaan ragam untuk mengukur keragaman, diperoleh satuan kuadrat dari satuan semula. • Apabila yang dihitung keragamannya adalah bobot buah melon dengan satuan kg, maka ragamnya akan mempunyai satuan kg². • Apabila yang diukur keragamannya adalah jumlah petani dengan satuan orang, maka ragamnya akan mempunyai satuan orang²??. Tentu saja hal ini sangat tidak logis. • Agar diperolehsatuan yang samadengansatuanasalnya, makavariantersebutdiakarkan. Akardariragamdisebutsimpanganbaku(s)ataudikenaldenganstandardeviasi

14.  = (Xi -  )2 N  = 2 s = s2 s = (Xi - X )2 n - 1 Standard Deviation (StandarDeviasi) => square root of variance For a population: For a sample:

15. Contoh 2 2 3 4 5 = X1 = X2 = X3 = X4 = X5 X = 3.2 s = (2 - 3.2)2 + (2 - 3.2)2 + (3 - 3.2)2 + (4 - 3.2)2 + (5 -3.2)2 5 - 1 = 1.44 + 1.44 + 0.04 + 0.64 + 3.24 4 = 1.304 Hitungstandardeviasi s = (Xi - X )2 n - 1 Varian adalahkuadratdaristandardeviasi. Berapanilainya??

16. s CV = X 7. Coefficient of Variation = KoefisienKeragaman = KK (V or sometimes CV): Variance (s2) and standard deviation (s) have magnitudes that are dependent on the magnitudes of the data. The coefficient of variation is a relative measure, so variability of different sets of data may be compared (stdev relative to the mean) Note that there are no units – emphasizes that it is a relative measure X 100% Sometimes expressed as a %

17. Example: 2 2 3 4 5 = X1 = X2 = X3 = X4 = X5 X = 3.2 g s CV = X 1.304 g CV = 3.2 g CV = 0.4075 or CV = 40.75% (X 100%) s = 1.304 g Attention  there is not any UNIT, or %

18. 68.27% 95.44% f 99.73% 3 2   2 3 X 8. The Normal Distribution (Distribusi Normal) : There is an equation which describes the height of the normal curve in relation to its standard dev ()

19. Normal distribution with σ=1, with varying means μ= 1 μ= 2 μ= 0 ƒ 4 5 -3 -2 -1 0 1 2 3 If you get difficulties to keep this term, read statistics books

20. σ = 1.5 σ = 2 Normal distribution with μ= 0, with varying standard deviations σ = 1 ƒ -5 -4 -3 -2 -1 0 1 2 3 4 5

21. Mean, median and mode 9. Symmetry and Kurtosis Symmetry means that the population is equally distributed around the mean i.e. the curve to the right side of the mean is a mirror image of the curve to the left side ƒ

22. ƒ ƒ Symmetry Data may be positively skewed (skewed to the right) Or negatively skewed (skewed to the left) So direction of skew refers to the direction of longer tail

23. mode ƒ median mean Symmetry

24. ƒ ƒ Kurtosis refers to how flat or peaked a curve is (sometimes referred to as peakedness or tailedness) The normal curve is known as mesokurtic A more peaked curve is known as leptokurtic A flatter curve is known as platykurtic

25. Latihandandiskusi • Banyaknyabuahpisang yang tersengathamadari 16 tanamanadalah 4, 9, 0, 1, 3, 24, 12, 3, 30, 12, 7, 13, 18, 4, 5, dan 15. Denganmenganggap data tersebutsebagaicontoh, hitunglahvarian, simpanganbakudankoefisienkeragamannya. Statistikmana yang paling tepatuntukmenggambarkankeragaman data tersebut? • To study how first-grade students utilize their time when assigned to a math task, researcher observes 24 students and records their time off task out of 20 minutes. Times off task (minutes) : 4, 0, 2, 2, 4, 1, 4, 6, 9, 7, 2, 7, 5, 4,13, 7, 7, 10, 10, 0, 5, 3, 9 and 8. For this data set, find : • Mean and standard deviation, median and range • Display the data in the histogram plot, dot diagram and also stem-and-leaf diagram • Determine the intervals x ± s, x ± 2s, x ± 3s • Find the proportion of the meausurements that lie in each of this intervals. • Compare your finding with empirical guideline of bell-shaped distribution

26. 3. The data below were obtained from the detailed record of purchases over several month. The usage vegetables (in weeks) for a household taken from consumer panel were (gram) : • 84 58 62 65 75 76 56 87 68 77 87 55 65 66 76 78 74 81 83 78 75 74 60 50 86 80 81 78 74 87 a. Plot a histogram of the data! b. Find the relative frequency of the usage time that did not exceed 80. c. Calculate the mean, variance and the standard deviation d. Calculate the median and quartiles. 4. The mean of corn weight is 278 g by ear and deviation standard is 9,64 g, and than we have 10 ears. If they are gotten from ten different fields, mean of plant height is Rp. 1200,- and its deviation standard is Rp 90,-, which one have more homogenous, the weight of corn ear or the plant height? Explain your answer! Verify your results by direct calculation with the other data.

27. 5. The employment’s salary at seed company, abbreviated, as follows : 18, 15, 21, 19, 13, 15, 14, 23, 18 and 16 rupiah. If these abbreviation is real salary divide Rp. 100.000,-, find the mean, variance and deviation standard of them. 6. Computer-aided statistical calculations. Calculation of the descriptive statistic such as x and s are increasingly tedious with large data sets. Modern computers have come a long way in alleviating the drudgery of hand calculation. Microsoft Exel, Minitab or SPSS are three of computing packages those are easy accessible to student because its commands are in simple English. Find these programs and install its at your computers. Bellow main and sub menu of Microsoft Exel, Minitab and SPSS program. Use these software to find x, s, s2, and coefisien of variation (CV) for data set in exercise b. Histogram and another illustration can also be created.

28. 7. Some properties of the standard deviation • if a fixed number c is added to all measurements in a data set, will the deviations (xi -x) remain changed? And consequently, will s² and s remain changed, too? Take data sample. • If all measurements in a data set are multiplied by a fixed number d, the deviation (xi -x) get multiplied by d. Is it right? What about the s² and s? Take data sample. • Apply your computer software to explain your data sample. Verify your results by other data.

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