DESCRIBING DATA

DUA UKURAN UTAMA

STATISTIKA

• Gejala pemusatan (central tendencies)

• Ukuran penyebaran (measures of dispersion)

GEJALA PEMUSATAN

• Rata-rata (mean)

• Nilai tengah (median)

• Modus (mode)

UKURAN PENYEBARAN

• Jangkauan (range)

• Simpangan kuartil

• Variansi (variance)

• Simpangan baku (standard deviation)

RATA-RATA HITUNG

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CONTOH

MENGHITUNG ARITHMETIC

MEAN

Hitunglah arithmetic mean dari data berikut:

• 37, 45, 73, 10, 17, 90, 77

• 340 cm, 890 cm, 178 cm, 322 cm, 100 cm

MENGHITUNG

ARITHMETIC MEAN

86,497

77901710734537

X

WEIGHTED

ARITHMETIC MEAN

CONTOH KASUS (1)

WEIGHTED ARITHMETIC MEAN

• UTS : 30%

• TUGAS : 20%

• UAS : 50%

Seorang mahasiswa mendapat nilai UTS = 50,

nilai TUGAS = 80, nilai UAS = 30. Berapa

nilai rata-rata mahasiswa tersebut?

CONTOH KASUS (2)

WEIGHTED ARITHMETIC MEAN

The Loris Healthcare System employs 200

persons on the nursing staff. Fifty are nurse’s

aides, 50 are practical nurses, and 100 are

registered nurses. Nurse’s aides receive $8 an

hour, practical nurses $15 an hour, and

registered nurses $24 an hour. What is the

mean hourly wage?

CONTOH KASUS (3)

WEIGHTED ARITHMETIC MEAN

Nilai Ujian Banyaknya

Mahasiswa

13 28

27 20

39 15

51 10

65 8

72 4

88 1

THE MEDIAN

• is the midpoint of the values after they have

been ordered from the smallest to the largest,

or the largest to the smallest.

• is not affected by extremely large or small

values.

• can be computed for ordinal-level data or

higher.

CONTOH PENGHITUNGAN

MEDIAN (1)

Diketahui data berikut, tentukan nilai tengahnya

13, 8, 25, 19, 39, 51, 4, 19, 45

• Urutkan data dari yang terkecil:

4 8 13 19 19 25 39 45 51

• Penggal data menjadi 2 bagian sama banyak:

4 8 13 19 19 25 39 45 51

• Data yang di tengah = nilai median = 19

CONTOH PENGHITUNGAN

MEDIAN (2)

Diketahui data berikut:

13, 8, 25, 19, 39, 51, 4, 19, 45, 55

• Urutkan data dari yang terkecil:

4 8 13 19 19 25 39 45 51 55

• Penggal data mjd 2 bagian sama banyak:

4 8 13 19 19 25 39 45 51 55

• Median = (19 + 25)/2 = 22

CONTOH PENGHITUNGAN

MEDIAN (3)A sample of single persons in Towson, Texas,

receiving Social Security payments revealed

these monthly benefits: $852, $598, $580,

$1,374, $960, $878, and $1,130. What is the

median monthly benefit?

CONTOH PENGHITUNGAN

MEDIAN (4)A sample of single persons in Towson, Texas,

receiving Social Security payments revealed

these monthly benefits: $852, $700, $598,

$580, $1,374, $960, $878, and $1,130. What is

the median monthly benefit?

THE MODE

• is the value of the observation that appears

most frequently.

• is useful in summarizing nominal-level data

• is not affected by extremely high or low

values.

• In some data sets there can be more than one

mode.

CONTOH MENGHITUNG

MODUS (1)• Data:

2 8 3 9 8 8 1 6 7 7 5

• Modus = Mo = 8

• Data ini bersifat unimodal

CONTOH MENGHITUNG MODUS (2)

• Data:

2 8 3 9 8 1 6 7 5 3

• (Mo)1 = 3 (Mo)2 = 8

• Data ini bersifat bimodal

JANGKAUAN (Range)• Merupakan selisih antara data terbesar dengan

data terkecil.

• R = Xmaks – Xmin

• Misalkan diketahui data:

32 21 24 68 65 71 78 41

• Xmaks = 78 Xmin = 21

• R = 78 – 21 = 57

VARIANCE AND STANDARD

DEVIATION• Variance: the arithmetic mean of the squared

deviations from the mean.

• Standard deviation: the square root of the

variance.

VARIANSI POPULASI

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VARIANSI SAMPEL

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CONTOH KASUS

PERHITUNGAN VARIANSI (1)

The number of traffic citations issued last year by month in Beaufort County, South Carolina, is reported below. Determine the population variance and standard deviation.

Jan Feb Mar Apr May Jun Jul Aug Sept Oct Nov Des

19 17 22 18 28 34 45 39 38 44 34 10

CONTOH KASUS

PERHITUNGAN VARIANSI (2)

The hourly wages for a sample of part-time

employees at Home Depot are: $12, $20, $16,

$18, and $19. What is the sample variance and

standard deviation?

RATA-RATA DATA

TERKELOMPOK

CONTOH MENGHITUNG

RATA-RATA DATA

TERKELOMPOK

NO. Tinggi Badan (cm) f M f.M

1 155 hingga 165 17 160 2720

2 165 hingga 175 30 170 5100

3 175 hingga 185 13 180 2340

4 185 hingga 195 10 190 1900

JUMLAH 70 12060

Rata-rata = 172cm

VARIANSI DAN SIMPANGAN

BAKU SAMPEL

DATA TERKELOMPOK

CONTOH MENGHITUNG VARIANSI

DAN SIMPANGAN BAKU DATA

TERKELOMPOK

NO. Tinggi Badan (cm) f M M-Mean (M-Mean)^2 f.(M-Mean)^2

1 155 hingga 165 17 160 -12 144 2448

2 165 hingga 175 30 170 -2 4 120

3 175 hingga 185 13 180 8 64 832

4 185 hingga 195 10 190 18 324 3240

JUMLAH 70 6640

Rata-rata (mean) = 172cm

Variansi = 96cm.cm

Simpangan baku = 9,8cm

SKEWNESS

• is a measure of the degree of asymmetry of a

distribution

• can be positive, negative, or zero

• Rumus koefisien kemencengan Pearson:

Karl Pearson

(1857 – 1936)

KEMENCENGAN (SKEWNESS)

(sumber: http://www.southalabama.edu/coe/bset/johnson/lectures/lec15.htm

diakses tanggal 18 September 2012 pukul 9:35 WIB)

CONTOH MENGHITUNG

KOEFISIEN KEMENCENGAN

PEARSONFollowing are the earnings per share for a sample of 15 software companies for the year 2010.

$0.09 $0.13 $0.41 $0.51 $1.12 $1.20 $1.49

3.18 3.50 6.36 7.83 8.92 10.13 12.99

16.40

Find the coefficient of skewness using Pearson’s estimate. What is your conclusion regarding the shape of the distribution?