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Pertemuan 02 Ukuran Numerik Deskriptif

Pertemuan 02 Ukuran Numerik Deskriptif. Materi: Ukuran Pemusatan dan Posisi (Letak) Ukuran Variasi. Ukuran Pemusatan Mean (rata-rata), Median, Modus, Geometrik mean, Kuartil, Desil, Persentil Measure of Variation

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Pertemuan 02 Ukuran Numerik Deskriptif

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  1. Pertemuan 02 Ukuran Numerik Deskriptif .

  2. Materi: • Ukuran Pemusatan dan Posisi (Letak) • Ukuran Variasi

  3. . • Ukuran Pemusatan • Mean (rata-rata), Median, Modus, Geometrik mean, Kuartil, Desil, Persentil • Measure of Variation • Range, Interquartile Range, Varians/ragam dan Standard Deviasi, Koefisien variasi • Bentuk • Simetris, Skewenes, Using Box-and-Whisker Plots

  4. Summary Measures Summary Measures Variasi Ukuran Pemusatan Quartile Mean Mode Koefisien Variasi Median Range Varians Standard Deviasi Geometric Mean

  5. Ukuran Pemusatan Ukuran Pemusatan Mean Median Mode Geometric Mean

  6. Mean (Arithmetic Mean) • Rata-rata contoh • Rata-rata Populasi Sample Size Population Size

  7. Mean (Arithmetic Mean) (continued) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 . Mean = 5 Mean = 6 • Rata-rata hitung (arithmetic mean) disebut rata-rata • - Rata-rata data tidak berkelompok • - Rata-rata data berkelompok • Dimana : fi = frekuensi kelas ke i • k = jumlah kelas • xi= nilai tengah kelas ke i

  8. Example • The set: 2, 9, 1, 5, 6 If we were able to enumerate the whole population, the population mean would be called m(the Greek letter “mu”).

  9. Median (nilai tengah) 0 1 2 3 4 5 6 7 8 9 10 12 14 0 1 2 3 4 5 6 7 8 9 10 • Data paling tengah setelah data disusun menurut nilainya. • Median dat tidak berkelompok • Jika N ganjil maka median adalah data paling tengah. • Jika N genap maka median adalah dua data tengah dibagi 2. Median = 5 Median = 5

  10. Median data berkelompok dimana : Me = Median Bd = Tepi bawah kelas median Id = Interval kelas median n = jumlah frekuensi F(d-1) = frekuensi kumulatif sebelum kelas median fd = frkuensi kelas median

  11. Median = 4th largest measurement Median = (5 + 6)/2 = 5.5 — average of the 3rd and 4th measurements Example • The set: 2, 4, 9, 8, 6, 5, 3 n = 7 • Sort: 2, 3, 4, 5, 6, 8, 9 • Position:.5(n + 1) = .5(7 + 1) = 4th • The set: 2, 4, 9, 8, 6, 5 n = 6 • Sort: 2, 4, 5, 6, 8, 9 • Position: .5(n + 1) = .5(6 + 1) = 3.5th

  12. Modus • Nilai/harga/data terbanyak • Modus data tidak berkelompok 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Tak ada modus Modus = 9

  13. Modus data berkelompok Dimana : Mo = modus Bo = Tepi kelas bawah modus Io = Panjang kelas modus fo = frekuensi kelas modus f1 = frekuensi sebelum kelas modus f2 = frekuensi sesudah kelas modus

  14. Mode • The mode is the measurement which occurs most frequently. • The set: 2, 4, 9, 8, 8, 5, 3 • The mode is 8, which occurs twice • The set: 2, 2, 9, 8, 8, 5, 3 • There are two modes—8 and 2 (bimodal) • The set: 2, 4, 9, 8, 5, 3 • There is no mode (each value is unique).

  15. Example The number of quarts of milk purchased by 25 households: 0 0 1 1 1 1 1 2 2 2 2 2 2 2 2 2 3 3 3 3 3 4 4 4 5 • Mean? • Median? • Mode? (Highest peak)

  16. Rata-rata ukur (Geometric Mean) • Jika perbandingan tiap dua data berurutan tetap atau hampir tetap, banyak dipakai rata-rata ukur. • Geometric Mean Rate of Return • Measures the status of an investment over time

  17. Extreme Values Symmetric: Mean = Median Skewed right: Mean > Median Skewed left: Mean < Median

  18. Key Concepts I. Measures of Center 1. Arithmetic mean (mean) or average a. Population: m b. Sample of size n: 2. Median: position of the median =.5(n +1) 3. Mode 4. The median may preferred to the mean if the data are highly skewed. II. Measures of Variability 1. Range: R = largest - smallest

  19. Key Concepts 2. Variance a. Population of N measurements: b. Sample of n measurements: 3. Standard deviation 4. A rough approximation for s can be calculated as s»R/4. The divisor can be adjusted depending on the sample size.

  20. Quartiles 25% 25% 25% 25% • Split Ordered Data into 4 Quarters • Position of i-th Quartile • and are Measures of Noncentral Location • = Median, a Measure of Central Tendency Data in Ordered Array: 11 12 13 16 16 17 18 21 22

  21. Ukuran Variasi Variasi Varians Standard Deviasi Koefisien Variasi Range Varians Populasi Standart deviasi untuk populasi Varians Contoh Standart deviasi untuk contoh Interquartile Range

  22. Range • Measure of Variation • Difference between the Largest and the Smallest Observations: • Ignores How Data are Distributed Range = 12 - 7 = 5 Range = 12 - 7 = 5 7 8 9 10 11 12 7 8 9 10 11 12

  23. Interquartile Range • Measure of Variation • Also Known as Midspread • Spread in the middle 50% • Difference between the First and Third Quartiles • Not Affected by Extreme Values Data in Ordered Array: 11 12 13 16 16 17 17 18 21

  24. 4 6 8 10 12 14 The Variance • The variance is measure of variability that uses all the measurements. It measures the average deviation of the measurements about their mean. • Flower petals: 5, 12, 6, 8, 14

  25. Varians/Ragam • Varians / Ragam Contoh : • Varians / Ragam Populasi :

  26. Standard Deviasi • Sample Standard Deviation: • Population Standard Deviation:

  27. Standard Deviation • Approximating the Standard Deviation • Used when the raw data are not available and the only source of data is a frequency distribution

  28. Two Ways to Calculate the Sample Variance Use the Definition Formula:

  29. Two Ways to Calculate the Sample Variance Use the Calculational Formula:

  30. Some Notes • The value of s is ALWAYS positive. • The larger the value of s2 or s, the larger the variability of the data set. • Why divide by n –1? • The sample standard deviation s is often used to estimate the population standard deviation s. Dividing by n–1 gives us a better estimate of s. Applet

  31. Comparing Standard Deviations Data A Mean = 15.5 s = 3.338 11 12 13 14 15 16 17 18 19 20 21 Data B Mean = 15.5 s = .9258 11 12 13 14 15 16 17 18 19 20 21 Data C Mean = 15.5 s = 4.57 11 12 13 14 15 16 17 18 19 20 21

  32. Koefisien Variasi • Measure of Relative Variation • Always in Percentage (%) • Shows Variation Relative to the Mean • Used to Compare Two or More Sets of Data Measured in Different Units • Sensitive to Outliers

  33. Bentuk Sebaran • Describe How Data are Distributed • Measures of Shape • Symmetric or skewed Right-Skewed Left-Skewed Symmetric Mean< Median < Mode Mean= Median =Mode Mode < Median < Mean

  34. Exploratory Data Analysis • Box-and-Whisker • Graphical display of data using 5-number summary Median( ) X X largest smallest 12 4 6 8 10

  35. Distribution Shape & Box-and-Whisker Left-Skewed Symmetric Right-Skewed

  36. Selamat Belajar Semoga Sukses.

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