Bagian ini membahas dua macam ukuran penting di dalam statistika, yaitu tendensi sentral dan ukuran penyebaran.

1. DESCRIBING DATA

2. DUA UKURAN UTAMA
STATISTIKA
• Gejala pemusatan (central tendencies)
• Ukuran penyebaran (measures of dispersion)

3. GEJALA PEMUSATAN
• Rata-rata (mean)
• Nilai tengah (median)
• Modus (mode)

4. UKURAN PENYEBARAN
• Jangkauan (range)
• Simpangan kuartil
• Variansi (variance)
• Simpangan baku (standard deviation)

5. RATA-RATA HITUNG
n
X
X
n
i
i
 1

6. 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

7. MENGHITUNG
ARITHMETIC MEAN
86,49
7
77901710734537


X

8. WEIGHTED
ARITHMETIC MEAN

9. 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?

10. 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?

11. CONTOH KASUS (3)
WEIGHTED ARITHMETIC MEAN
Nilai Ujian Banyaknya
Mahasiswa
13 28
27 20
39 15
51 10
65 8
72 4
88 1

12. 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.

13. 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

14. 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

15. 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?

16. 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?

17. 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.

18. CONTOH MENGHITUNG
MODUS (1)
• Data:
2 8 3 9 8 8 1 6 7 7 5
• Modus = Mo = 8
• Data ini bersifat unimodal

19. 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

20. 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

21. VARIANCE AND STANDARD
DEVIATION
• Variance: the arithmetic mean of the squared
deviations from the mean.
• Standard deviation: the square root of the
variance.

22. VARIANSI POPULASI
 
n
XX
n
i
i

 1
2
2


23. VARIANSI SAMPEL
 
1
1
2
2




n
XX
s
n
i
i

24. 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

25. 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?

26. RATA-RATA DATA
TERKELOMPOK

27. 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

28. VARIANSI DAN SIMPANGAN
BAKU SAMPEL
DATA TERKELOMPOK

29. 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

30. SKEWNESS
• is a measure of the degree of asymmetry of a
distribution
• can be positive, negative, or zero
• Rumus koefisien kemencengan Pearson:

31. Karl Pearson
(1857 – 1936)

32. KEMENCENGAN (SKEWNESS)
(sumber: http://www.southalabama.edu/coe/bset/johnson/lectures/lec15.htm
diakses tanggal 18 September 2012 pukul 9:35 WIB)

33. CONTOH MENGHITUNG
KOEFISIEN KEMENCENGAN
PEARSON
Following 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?