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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

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

Summary Measures

Variasi

Ukuran Pemusatan

Quartile

Mean

Mode

Koefisien Variasi

Median

Range

Varians

Standard Deviasi

Geometric Mean


Ukuran pemusatan
Ukuran Pemusatan

Ukuran Pemusatan

Mean

Median

Mode

Geometric Mean


Mean arithmetic mean
Mean (Arithmetic Mean)

  • Rata-rata contoh

  • Rata-rata Populasi

Sample Size

Population Size


Mean arithmetic mean1
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
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
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
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


Example1

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


Modus
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
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).


Example2
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
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
Extreme Values

Symmetric: Mean = Median

Skewed right: Mean > Median

Skewed left: Mean < Median


Key concepts
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 concepts1
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
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
Ukuran Variasi

Variasi

Varians

Standard Deviasi

Koefisien Variasi

Range

Varians Populasi

Standart deviasi untuk populasi

Varians Contoh

Standart deviasi untuk contoh

Interquartile Range


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
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

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

  • Varians / Ragam Contoh :

  • Varians / Ragam Populasi :


Standard deviasi
Standard Deviasi

  • Sample Standard Deviation:

  • Population Standard Deviation:


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
Two Ways to Calculate the Sample Variance

Use the Definition Formula:


Two ways to calculate the sample variance1
Two Ways to Calculate the Sample Variance

Use the Calculational Formula:


Some notes
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
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
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
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
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
Distribution Shape & Box-and-Whisker

Left-Skewed

Symmetric

Right-Skewed



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