Ukuran Sebaran (Keragaman) Data

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1 Ukuran Sebaran (Keragaman) Data Dr. Akhmad Rizali Ukuran keragaman Dari tiga ukuran pemusatan, belum dapat memberikan deskripsi yang lengkap bagi suatu data Perlu juga diketahui seberapa jauh pengamatanpengamatan 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 1

2 Measures of Dispersion and Variability These are measurements of how spread the data is around the center of the distribution f f x x RANGE (Kisaran = Rentang) Difference between lowest and highest numbers Place numbers in order of magnitude, then range = X n -X = X 1 = X = X 3 = X 4 = X 5 Range = 5 - = 3 Problem - no information about how clustered the data is

3 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. (X i -X) = 0 So, take an absolute value to avoid this Problem the more numbers in the data set, the higher the SS Contoh Misal, jumlah buku tulis yang dibawa 5 mahasiswa adalah 3, 5, 7, 7, 8. Rerata (mean) data tersebut adalah 30/5 = 6. Simpangan dihitung dengan mengurangi setiap nilai pengamatan dengan reratanya No. Nilai observasi Simpangan (x x) Simpangan kuadrat (x x) = = = = = 4 Jumlah 0 16 Agar nilainya tidak negatif, dapat di kuadratkan yang kemudian disebut simpangan kuadrat. Simpangan kuadrat bermanfaat untuk penghitungan varian. 3

4 Simpangan Rerata (Mean Deviation) Sample mean deviation = X i -X Essentially the average deviation from the mean n No. Nilai observasi Simpangan (x - x) = = = = = Jumlah 0 Simpangan rerata (x - x)/n (3-6)/5 = -3/5 (5-6)/5 = -1/5 (7-6)/5 = 1/5 (7-6)/5 = 1/5 (8-6)/5 = /5 0 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 Sample SS = (X i -X) = Xi ( Xi) /n SS is much more common than mean deviation 4

5 Example = X 1 = X = X 3 = X 4 = X 5 Sample SS = (X i -X) X = 3. SS = ( - 3.) + ( - 3.) + (3-3.) + (4-3.) + (5-3.) = = 6.8 Problem the more numbers in the data set, the higher the SS VARIAN (Ragam) Dalam prakteknya, simpangan jarang digunakan karena sulit dimanipulasi secara matematis Sebagai gantinya diperlukan kuadrat semua simpangan tersebut kemudian dibagi derajad bebas n 1, dan disebut dengan varian (ragam) Digunakan pembagi n 1 agar menjadi penduga tak bias Ragam populasi dilambangkan dengan σ², sedang ragam contoh dilambangkan dengan s 5

6 The mean SS is known as the variance Population Variance ( ): = (X i - ) N This is just SS N Our best estimate of is sample variance (s ): S = (X i -X) = n -1 Note : divide by n-1 Xi ( Xi) /n known as degrees of freedom n 1 Problem - units end up squared Mengapa (n 1) disebut derajad bebas (kebebasan)? Perhatikan ilustrasi berikut: Apabila seseorang hendak mengangkat 100 kg beras dari lantai 1 ke lantai 3 dan ia harus mengangkat maksimal sebanyak 5 kali, maka orang tersebut dapat memilih menyelesaikannya dalam kali angkat, 3 kali atau sampai (n 1) kali Sampai dengan 4 (n 1) kali, orang tersebut bebas memilih berapa kg yang diangkat ke lantai 3 Namun pada angkatan terakhir (1 kali), mau tidak mau, orang tersebut harus mengangkat semua beras yang tersisa. Artinya kebebasan memilih jumlah yang diangkat hanya (n 1) kali 6

7 Standar Deviasi 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 diperoleh satuan yang sama dengan satuan asalnya, maka varian tersebut diakarkan. Akar dari ragam disebut simpangan baku (s) atau dikenal dengan standar deviasi Standard Deviation (Standar Deviasi) => square root of variance For a population: = For a sample: s = s = (X i - ) N s = (X i -X ) n -1 7

8 Hitung standar deviasi Contoh = X 1 = X = X 3 = X 4 = X 5 s = (X i -X ) n -1 s = ( - 3.) + ( - 3.) + (3-3.) + (4-3.) + (5-3.) 5-1 X = 3. = = Varian adalah kuadrat dari standar deviasi. Berapa nilainya?? Coefficient of Variation (Koefisien Keragaman) KK (V or sometimes CV) Variance (s ) 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) CV = s X X 100% Sometimes expressed as a % Note that there are no units emphasizes that it is a relative measure 8

9 Example: = X 1 = X = X 3 = X 4 = X 5 X = 3. g s = g CV = s X (X 100%) CV = g 3. g CV = or CV = 40.75% Attention there is not any UNIT, or % The Normal Distribution (Distribusi Normal) 68.7% 95.44% f 99.73% 3 X 3 There is an equation which describes the height of the normal curve in relation to its standard dev ( ) 9

10 Normal distribution with σ = 1, with varying means μ = 0 μ = 1 μ = If you get difficulties to keep this term, read statistics books Normal distribution with μ = 0, with varying standard deviations σ = 1 σ = 1.5 σ =

11 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 Mean, median and mode 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 11

12 Symmetry mode median mean 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 1

13 Latihan Banyaknya buah pisang yang terserang hama dari 16 tanaman adalah 4, 9, 0, 1, 3, 4, 1, 3, 30, 1, 7, 13, 18, 4, 5, dan 15. Dengan menganggap data tersebut sebagai contoh, hitunglah varian, simpangan baku dan koefisien keragamannya. Statistik mana yang paling tepat untuk menggambarkan keragaman data tersebut? To study how first grade students utilize their time when assigned to a math task, researcher observes 4 students and records their time off task out of 0 minutes. Times off task (minutes) : 4, 0,,, 4, 1, 4, 6, 9, 7,, 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 andleaf diagram Determine the intervals x ±s, x ±s, x ±3s Find the proportion of the measurements that lie in each of this intervals. Compare your finding with empirical guideline of bell shaped distribution 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) : Plot a histogram of the data! Find the relative frequency of the usage time that did not exceed 80 Calculate the mean, variance and the standard deviation Calculate the median and quartiles The mean of corn weight is 78 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. 100, 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 13

14 The employment s salary at seed company, abbreviated, as follows : 18, 15, 1, 19, 13, 15, 14, 3, 18 and 16 rupiah. If these abbreviation is real salary divide Rp ,, find the mean, variance and deviation standard of them. 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 Excel, Minitab and SPSS program. Use these software to find x, s, s, and coefisien of variation (CV) for data set in exercise b. Histogram and another illustration can also be created Some properties of the standard deviation if a fixed number c is added to all measurements in a data set, will the deviations (x i 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 (x i 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 14

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