# Pertemuan 02 Ukuran Numerik Deskriptif - PowerPoint PPT Presentation

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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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## 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

• Range, Interquartile Range, Varians/ragam dan Standard Deviasi, Koefisien variasi

• Bentuk

• Simetris, Skewenes, Using Box-and-Whisker Plots

### Summary Measures

Summary Measures

Variasi

Ukuran Pemusatan

Quartile

Mean

Mode

Koefisien Variasi

Median

Range

Varians

Standard Deviasi

Geometric Mean

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 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

### 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

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, 3n = 7

• Sort:2, 3, 4, 5, 6, 8, 9

• Position:.5(n + 1) = .5(7 + 1) = 4th

• The set: 2, 4, 9, 8, 6, 5n = 6

• Sort:2, 4, 5, 6, 8, 9

• Position: .5(n + 1) = .5(6 + 1) = 3.5th

### 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

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).

### 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

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

### 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

• 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

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

### 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:

### Two Ways to Calculate the Sample Variance

Use the Calculational Formula:

### 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

### 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

### Exploratory Data Analysis

• Box-and-Whisker

• Graphical display of data using 5-number summary

Median( )

X

X

largest

smallest

12

4

6

8

10

### Distribution Shape & Box-and-Whisker

Left-Skewed

Symmetric

Right-Skewed

• Selamat Belajar Semoga Sukses.