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1 Pertemuan 02 Ukuran Numerik Deskriptif.. 2 Materi: Ukuran Pemusatan dan Posisi (Letak) Ukuran Pemusatan dan Posisi (Letak) Ukuran Variasi Ukuran Variasi.

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Presentasi berjudul: "1 Pertemuan 02 Ukuran Numerik Deskriptif.. 2 Materi: Ukuran Pemusatan dan Posisi (Letak) Ukuran Pemusatan dan Posisi (Letak) Ukuran Variasi Ukuran Variasi."— Transcript presentasi:

1 1 Pertemuan 02 Ukuran Numerik Deskriptif.

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

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

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

5 5 Ukuran Pemusatan MeanMedianMode Geometric Mean

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

7 7 Mean (Arithmetic Mean). (continued) Mean = 5 Mean = 6 -Rata-rata hitung (arithmetic mean) disebut rata-rata - Rata-rata data tidak berkelompok - Rata-rata data tidak berkelompok - Rata-rata data berkelompok Dimana : fi = frekuensi kelas ke i Dimana : fi = frekuensi kelas ke i k = jumlah kelas k = jumlah kelas xi= nilai tengah kelas ke i xi= nilai tengah kelas ke i

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

9 9 Median (nilai tengah) Data paling tengah setelah data disusun menurut nilainya. Data paling tengah setelah data disusun menurut nilainya. Median dat tidak berkelompok Median dat tidak berkelompok –Jika N ganjil maka median adalah data paling tengah. –Jika N genap maka median adalah dua data tengah dibagi Median = 5

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

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

12 12Modus Nilai/harga/data terbanyak Nilai/harga/data terbanyak Modus data tidak berkelompok Modus data tidak berkelompok Modus = Tak ada modus

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

14 14Mode The mode is the measurement which occurs most frequently. The mode is the measurement which occurs most frequently. The set: 2, 4, 9, 8, 8, 5, 3 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 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 The set: 2, 4, 9, 8, 5, 3 –There is no mode (each value is unique).

15 15Example Mean? Mean? Median? Median? Mode? (Highest peak) Mode? (Highest peak) The number of quarts of milk purchased by 25 households:

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

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

18 18 Key Concepts I. Measures of Center 1. Arithmetic mean (mean) or average a. Population:  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 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. The divisor can be adjusted depending on the sample size.

20 20Quartiles Split Ordered Data into 4 Quarters Split Ordered Data into 4 Quarters Position of i-th Quartile Position of i-th Quartile and are Measures of Noncentral and are Measures of NoncentralLocation = Median, a Measure of Central Tendency = Median, a Measure of Central Tendency 25% Data in Ordered Array:

21 21 Ukuran Variasi Variasi VariansStandard DeviasiKoefisien Variasi Varians Populasi Varians Contoh Standart deviasi untuk populasi Standart deviasi untuk contoh Range Interquartile Range

22 22 Range Measure of Variation Measure of Variation Difference between the Largest and the Smallest Observations: Difference between the Largest and the Smallest Observations: Ignores How Data are Distributed Ignores How Data are Distributed Range = = Range = = 5

23 23 Measure of Variation Measure of Variation Also Known as Midspread Also Known as Midspread –Spread in the middle 50% Difference between the First and Third Quartiles Difference between the First and Third Quartiles Not Affected by Extreme Values Not Affected by Extreme Values Interquartile Range Data in Ordered Array:

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

25 25 –Varians / Ragam Contoh : –Varians / Ragam Populasi : Varians/Ragam

26 26 Standard Deviasi - Sample Standard Deviation: - Population Standard Deviation:

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

28 28 Two Ways to Calculate the Sample Variance Sum45060 Use the Definition Formula:

29 29 Two Ways to Calculate the Sample Variance Sum45465 Use the Calculational Formula:

30 30 The value of s is ALWAYS positive. The value of s is ALWAYS positive. The larger the value of s 2 or s, the larger the variability of the data set. The larger the value of s 2 or s, the larger the variability of the data set. Why divide by n – 1? Why divide by n – 1? –The sample standard deviation s is often used to estimate the population standard deviation  Dividing by n – 1 gives us a better estimate of  Some Notes Applet

31 31 Comparing Standard Deviations Mean = 15.5 s = Data B Data A Mean = 15.5 s = Mean = 15.5 s = 4.57 Data C

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

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

34 34 Exploratory Data Analysis Box-and-Whisker Box-and-Whisker –Graphical display of data using 5-number summary Median( ) X largest X smallest

35 35 Distribution Shape & Box-and-Whisker Right-SkewedLeft-SkewedSymmetric

36 36 Selamat Belajar Semoga Sukses. Selamat Belajar Semoga Sukses.


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