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Statistika

Statistika. Konsep dasar dan metoda penggunaannya dalam penelitian. Tujuan :.

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Statistika

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  1. Statistika Konsep dasar dan metoda penggunaannya dalam penelitian

  2. Tujuan : • Untuk memajukan pemikiran yang tertib, runut dan jelas, terutama yang berhubungan dengan pengumpulan dan interpretasi data numerik, serta menyediakan sejumlah teknik statistika yang mempunyai kegunaan yang luas dalam penelitian. • Melakukan penyajian, peringkasan dan pencirian data Statistika adalah cara berpikir perihal ketidakpastian.

  3. Penelitian : • Penyelidikan terencana untuk mendapatkan fakta baru, untuk memperkuat atau menolak hasil hasil percobaan terdahulu. Penyelidikan demikian ini akan membantu pengambilan keputusan

  4. Pertanyaan yang harus dijawab : • Untuk setiap perhitungan statistik, selalu muncul pertanyaan mengenai ketelitiannya, berapa angka yang masih dapat dipercaya sebagai akhir dari serangkaian perhitungan yang kita lakukan

  5. Aplikasi Statistik dibagi menjadi dua bagian: • Statistik Deskriptif Menjelaskan / menggambarkan berbagai karakteristik data seperti mean, std dev, variansi dan sebagainya • Statistik Induktif (Inferensi) Membuat berbagai inferensi terhadap sekumpulan data yang berasal dari suatu sampel. Tindakan inferensi tersebut seperti melakukan perkiraan, peramalan, pengambilan keputusan dan sebagainya.

  6. Dalam prakteknya kedua bagian statistik tersebut digunakan bersama-sama, umumnya dimulai dengan statistik deskriptif lalu dilanjutkan dengan berbagai analisis statistik untuk inferensi.

  7. Elemen Statistik. 1. Populasi Sekumpulan data yang mengidentifikasikan suatu fenomena yang tergantung dari kegunaan dan relevansi data yang dikumpulkan. 2. Sampel Sekumpulan data yang diambil / diseleksi dari suatu populasi. (sampel adalah bagian dari populasi).

  8. 3. Statistik Inferensi Suatu keputusan, perkiraan atau generalisasi tentang suatu populasi berdasarkan informasi yang terkandung dari suatu sampel 4. Pengukuran Reabilitasdari StatistikInferensi. Tujuan dari statistik pada dasarnya adalah melakukan deskripsi terhadap data sampel, kemudian melakukan inferensi terhadap populasi data berdasar pada informasi (hasil statistik deskriptif) yang terkandung dalam sampel. Catatan : Karena sampel yang diambil hanya sebagian dari populasi, dapat terjadi bias dalam kesimpulannya. Sebagai konsekuensi dari kemungkinan timbulnya berbagai bias dalam inferensi, perlu diukur reabilitas dari setiap inferensi yang telah dibuat.

  9. Tipe Data Statistik. • DataKualitatif a.  Nominal Mis gender, tgl lahir dsb yang untuk mudahnya dapat dikategorikan dengan angka. (level sama) b. Ordinal Misal selera, dsb (level tidak sama)

  10. Tipe Data Statistik. • Data Kuantitatif a.   Data Interval Data yang memiliki jangkauan Mis pengukuran suhu, Cukup panas antara 50 – 80 derajat C, Panas antara 80 – 110 C, dan Sangat panas antara 110 – 140 C b. Data Rasio. Data dengan tingkat pengukuran ter “tinggi” diantara jenis lainnya. Sehingga dapat dilakukan operasi matematika. Mis jumlah barang, berat badan dsb.

  11. Statistik Deskriptif Bagian ini lebih berhubungan dengan pengumpulan dan peringkasan data, serta penyajian hasil peringkasan tersebut. Penyajian tabel dan grafik misalnya     1. Distribusi Frekuensi     2. Histogram, Pie chart dsb Dua ukuran penting yang sering digunakan dalam pengambilan keputusan adalah :     1. Mencari Central Tendency (mean, median, modus)     2. Mencari Ukuran Dispersi (std deviasi, variansi) Ukuran lain yang sering digunakan adalah Skewness dan Kurtosis untuk mengetahui kemiringan data.

  12. Statistical Notation • Variabel biasanya ditulis sbg “x” dan “y” • Untuk populasi dinotasikan dg huruf besar “N” (“N” for populations and “n” for samples) • Sigma ( ) mewakili operasi penjumlahan

  13. Statistical Notation X y xy x+1 x2 2 3 6 3 4 3 5 15 4 9 6 8 48 7 36 4 2 8 5 16

  14. Statistical Notation Sxindicates that scores on variable “x” are to be added; Sy indicates that scores on variable “y” are to be added In the previous table, S x = 2 + 3 + 6 + 4 = 15 S y = 3 + 5 + 8 + 2 = 18 S x S y = 15*18 = 270

  15. Statistical Notation S xyindicates that the 2 variables (x and y) are to be multiplied together, then summed. S xy = (2*3) + (3*5) + (6*8) + (4*2) = 77 ( 6 + 15 + 48 + 8 = 77) • Note that S xy (does not equal) S x S y (77 vs. 270)

  16. Statistical Notation S(x+1) indicates that a constant value of 1 is added to each score, then each score is added up • Remember that operations in parenthesis are always done first S(x+1) = (2+1) + (3+1) + (6+1) + (4+1) = 19 = ( 3 + 4 + 7 + 5 = 19) • Notice that S(x+1) Sx+1 (19 vs. 16)

  17. Statistical Notation X2 indicates to square each of the x values, then add them up • X2 = 2² + 3² + 6² + 4² = 65 (= 4 + 9 + 36 + 16=65) • Notice that x2 (x) 2 (65 vs. 225)

  18. Dalam statistika hal yang paling penting adalah PENGAMATAN

  19. Perhatikan data berikut : 8, 8, 9, 10, 11, 12, 12 5, 6, 8, 10, 12, 14, 15 1, 2, 5, 10, 15, 18, 19 Dan 8, 9, 10, 10, 10, 11, 12 5, 7, 9, 10, 11, 13, 15 1, 5, 8, 10, 12, 15, 19

  20. Perhatikan data berikut : 8, 8, 9, 10, 11, 12, 12 s: 1.6 5, 6, 8, 10, 12, 14, 15 s: 3.58 1, 2, 5, 10, 15, 18, 19 s: 6.96 Dan 8, 9, 10, 10, 10, 11, 12 s: 1.19 5, 7, 9, 10, 11, 13, 15 s: 3.16 1, 5, 8, 10, 12, 15, 19 s: 5.60

  21. Central Tendency CENTRAL TENDENCY: • A statistical measure that identifies a single score that is most typical or representative of the entire group; a single score or measurement used to describe an entire distribution • Usually, a value that reflects the middle of the distribution is used, because this is where most of the scores pile up • No single measure of central tendency works best in all circumstances, so there are 3 different measures -- mean, median, and mode. Each works best in a specific situation

  22. Central Tendency (Mode) MODE: • The score or category that has the greatest frequency; the most common score • To find the mode, simply locate the score that appears most often • In a frequency distribution table, it will be the score with the largest frequency value • In a frequency graph, it will be the tallest bar or point

  23. Central Tendency (Mode) Example: A sample of class ages is given. . . Ages f * The age with the highest 23 1 frequency is 19, with a 22 0 frequency of 3; therefore, the 21 1 mode is 19. 20 0 19 3 18 2

  24. Central Tendency (Mode) • A distribution may have more than one mode, or peak: A distribution with 2 modes is said to be bimodal; A distribution with more than 2 modes is said to be multimodal Example: A sample of class ages. . . Age f * age 22 and age 19 both 23 1 have a frequency of 3; if 22 3 this distribution were 21 1 graphed, there would be 20 1 2 peaks; therefore this 19 3 distribution is bimodal -- 18 2 both 22 and 19 are modes

  25. Central Tendency (Mode) Advantages: • Easiest to determine • The only measure of central tendency that can be used with nominal (categorical) data Disadvantages • Sometimes is not a unique point in the distribution (bimodal or multimodal) • Not sensitive to the location of scores in a distribution • Not often used beyond the descriptive level

  26. Central Tendency (Median) MEDIAN: • The score that divides the distribution exactly in half; 50% of the individuals in a distribution have scores at or below the median

  27. Central Tendency (Median) Method 1: (Use when N (or n) is an odd number) List the scores from lowest to highest; the middle score on the list is the median Example: The ages of a sample of class members are 24, 18, 19, 22, and 20. What is the median value? • List the scores from lowest to highest: 18, 19, 20, 22, 24 • The middle score is 20 - therefore, that is the median

  28. Central Tendency (Median) Method 2: (Use when N (or n) is an even number) List scores in order from lowest to highest and locate the point halfway between the middle two scores Example: The ages of a sample of class members are 18, 19, 20, 22, 24 and 30. What is the median age? The scores are already listed from lowest to highest; select the middle two scores (20, 22) and find the middle point:

  29. Central Tendency (Mean) MEAN (µ, x): • The mathematical average of the scores • The amount that each individual would receive if the total (x) were divided up equally between everyone in the distribution • Computed by adding all of the scores in the distribution and dividing that sum by the total number of scores • Population mean: • Sample mean:

  30. Central Tendency (Mean) • Note that, while the computations would yield the same answer, the symbols differ for a population (, N) and a sample (x,n) Example:

  31. Central Tendency (Mean) • If a constant is added or subtracted from each score used to compute the mean, the mean will change by the value of that constant Example: The class scores on a 15-point quiz are 8, 4, 12, 14, 4, and 6. The mean of these scores is. . .

  32. Central Tendency (Mean) Suppose the instructor made an error on the quiz and decided to add 1 point to everyone’s score. How would that change the mean? X+1 * The new scores are 9, 5, 13, 15, 5, &7 8+1= 9 * The new mean is: 4+1= 5 12+1=13 14+1=15 * so, adding a constant to each score 4+1= 5 (x+a) and calculating mean has the 6+1= 7 same result as adding that constant to X = 54 the mean (x+ a)

  33. Central Tendency (Mean) • If all of the scores are multiplied or divided by a constant value, and the mean is then computed, the result will be the same as if the mean were multiplied or divided by that constant Example: A 10-point quiz is given to a class. Their scores are 6, 5, 7, 8, 10 and 6. The mean is computed. . . X=42

  34. Central Tendency (Mean) The instructor then decides to change the value of the quiz from 10 points to 20 points. What is the new mean for the class? X(2) * The new scores are 12, 10, 14, 6(2)=12 16, 20, 12 5(2)=10 * The new mean is: 7(2)=14 8(2)=16 * thus, multiplying each score by 10(2)=20 a constant is the same as 6(2)=12 multiplying the mean by the X=84 constant

  35. Central Tendency (Mean) • Sensitive to extreme scores, and therefore may not be desirable when working with highly skewed distributions Example: compare 2 samples of class ages 1) 18, 19, 20, 22, 24 2) 18, 19, 20, 22, 47 Vs.

  36. Mode Median Mean Mean – Mode = 3(Mean – Median)

  37. Persentase luas daerah antara Mean dan Std Dev X 1 2 3 | x +/- 1s | = 68.27% | x +/- 2s | = 95.45% | x +/- 3s | = 99.73%

  38. Measures of Variability • range- (highest score - lowest score) +1 • deviation- score - mean • mean deviation- the average absolute deviation score

  39. Measures of Variability • sum of squares- the sum of the squared deviation scores • definitional

  40. Measures of Variability • variance- the average sum of the squared deviation scores • sample- • population-

  41. Statistik Inferensi Setelah dilakukan uji terhadap suatu distribusi data, dan terbukti bahwa data yang diuji berdistribusi normal atau mendekati normal, maka pada data tersebut dapat dilakukan berbagai Inferensi dengan metode statistik parametrik. Tetapi jika dalam pengujian terbukti distribusi data tidak berdistribusi normal atau jauh dari normal maka Inferensi yang dilakukan harus dengan metode statistik non parametrik.

  42. Uji Hipotesa Dalam melakukan uji hipotesis, ada banyak faktor yang menentukan seperti apakah sampel yang diambil berjumlah banyak atau hanya sedikit, apakah std deviasi populasi diketahui, apakah variansi dari populasi diketahui, apa metode parametrik yang digunakan, dst. 1.   Prosedur Uji Hipotesis a.   Menentukan H0 dan H1 b.   Menentukan nilai statistiknya 1. Tingkat kepercayaan        2. Derajat kebebasan        3. Jumlah sampel yang didapat

  43. c.  Menentukan Statistik hitung, nilai ini • tergantung pada metode parametrik • yang digunakan. b.       d. Mengambil keputusan, hal ini ditentukan dengan membandingkan nilai statistik hitung dengan nilai statistik tabel atau nilai kritisnya.

  44. Kesalahan : • Jika kita menolak hipotesa, sedangkan hipotesa benar dikatakan kita melakukan kesalahan type I • Sebaliknya jika kita menerima hipotesa sedangkan hipotesa salah, dikatakan kita melakukan kesalahan type II

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