Pertemuan 02 Ukuran Numerik Deskriptif . Pertemuan 02 Ukuran Numerik Deskriptif

Materi: Ukuran Pemusatan dan Posisi (Letak) Ukuran Variasi

Ukuran Pemusatan Measure of Variation Bentuk . 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

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

Ukuran Pemusatan Ukuran Pemusatan Mean Median Mode Geometric Mean

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

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

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”).

Median (nilai tengah) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14 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

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

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

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

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

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).

The number of quarts of milk purchased by 25 households: 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)

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

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

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

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.

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

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

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

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

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 4 6 8 10 12 14

Varians/Ragam Varians / Ragam Contoh : Varians / Ragam Populasi :

Standard Deviasi Sample Standard Deviation: Population Standard Deviation:

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

Two Ways to Calculate the Sample Variance Use the Definition Formula: 5 -4 16 12 3 9 6 -3 8 -1 1 14 25 Sum 45 60

Two Ways to Calculate the Sample Variance Use the Calculational Formula: 5 25 12 144 6 36 8 64 14 196 Sum 45 465

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

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

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

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

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

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

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