Matakuliah
Tahun
Versi
: A0064 / Statistik Ekonomi
: 2005
: 1/1
Pertemuan 3
Statistik Deskriptif-1
1
Learning Outcomes
Pada akhir pertemuan ini, diharapkan mahasiswa
akan mampu :
• Menghitung beberapa persoalan dalam
ukuran pemusatan (mean, modus, dan
median) dan ukuran keragaman (renrang,
jarak antar kuartil , ragam dan simpangan
baku)
2
Outline Materi
• Ukuran-ukuran Pemusatan
• Ukuran-ukuran Keragaman
• Pengelompokkan Data dan Histogram
3
COMPLETE
5th edi tion
1-4
BUSINESS STATISTICS
Summary Measures: Population Parameters
Sample Statistics

Measures of Central
Tendency
Median
Mode
Mean

McGraw-Hill/Irwin

Measures of Variability
 Range
 Interquartile range
 Variance
 Standard Deviation
Other summary
measures:
 Skewness
 Kurtosis
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
5th edi tion
1-5
1-3 Measures of Central Tendency
or Location
Median
 Middle value when
sorted in order of
magnitude
 50th percentile
Mode
 Most frequentlyoccurring value
Mean
 Average
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-6
BUSINESS STATISTICS
Example – Median (Data is used from
Example 1-2)
Sales
9
6
12
10
13
15
16
14
14
16
17
16
24
21
22
18
19
18
20
17
McGraw-Hill/Irwin
Sorted Sales
6
9
10
12
13
14
14
15
16
16
16
17
17
18
18
19
20
21
22
24
See slide # 19 for the template output
Median
50th Percentile
(20+1)50/100=10.5
16 + (.5)(0) = 16
Median
The median is the middle
value of data sorted in
order of magnitude. It is
the 50th percentile.
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
5th edi tion
1-7
Example - Mode (Data is used from
Example 1-2)
See slide # 19 for the template output
.
.
. . . . : . : : : . . . .
.
--------------------------------------------------------------6
9 10 12 13 14 15 16 17 18 19 20 21 22 24
Mode = 16
The mode is the most frequently occurring value. It
is the value with the highest frequency.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-8
BUSINESS STATISTICS
Arithmetic Mean or Average
The mean of a set of observations is their average the sum of the observed values divided by the
number of observations.
Population Mean
Sample Mean
N
m=
McGraw-Hill/Irwin
n
x
x=
i =1
N
Aczel/Sounderpandian
x
i =1
n
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
5th edi tion
1-9
Example – Mean (Data is used from
Example 1-2)
Sale
s
9
6
12
10
13
15
16
14
14
16
17
16
24
21
22
18
19
18
20
17
n
x=
x
i =1
n
=
317
= 1585
.
20
See slide # 19 for the template output
317
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
5th edi tion
1-10
Example - Mode (Data is used from
Example 1-2)
.
.
. . . . : . : : : . . . .
.
--------------------------------------------------------------6
9 10 12 13 14 15 16 17 18 19 20 21 22 24
Mean = 15.85
Median and Mode = 16
See slide # 19 for the template output
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
1-11
BUSINESS STATISTICS
5th edi tion
1-4 Measures of Variability or
Dispersion

Range
 Difference between maximum and minimum values

Interquartile Range
 Difference between third and first quartile (Q3 - Q1)

Variance
 Average*of the squared deviations from the mean

Standard Deviation
 Square root of the variance

Definitions of population variance and sample variance differ slightly.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
1-12
5th edi tion
Example - Range and Interquartile Range (Data
is used from Example 1-2)
Sales
9
6
12
10
13
15
16
14
14
16
17
16
24
21
22
18
19
18
20
17
Sorted
Sales
6
9
10
12
13
14
14
15
16
16
16
17
17
18
18
19
20
21
22
24
McGraw-Hill/Irwin
Maximum - Minimum =
Range
Rank
24 - 6 = 18
1
Minimum
2
3
4
5
Q1 = 13 + (.25)(1) = 13.25
6 First Quartile
7
8
9
10
See slide # 19 for the template output
11
12
13
14
Q3 = 18+ (.75)(1) = 18.75
15
16 Third Quartile
17
Q3 - Q1 =
Interquartile
18
18.75 - 13.25 = 5.5
19
Range
Maximum
20
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-13
BUSINESS STATISTICS
Variance and Standard Deviation
Population Variance
Sample Variance
2
m
(x )
s 2 = i=1
x
2
s=
McGraw-Hill/Irwin
( x)
-
i=1
s
s =
2
i =1
N
N
=
(x - x)
n
N
N

i =1
2
N
2
(n - 1)
(
)
x n
=
N
2
2
n
x
i =1
2
n
i =1
(n - 1)
s= s
2
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-14
BUSINESS STATISTICS
Calculation of Sample Variance
x
6
9
10
12
13
14
14
15
16
16
16
17
17
18
18
19
20
21
22
24
317
x-x
-9.85
-6.85
-5.85
-3.85
-2.85
-1.85
-1.85
-0.85
0.15
0.15
0.15
1.15
1.15
2.15
2.15
3.15
4.15
5.15
6.15
8.15
0
McGraw-Hill/Irwin
(x - x) 2
97.0225
46.9225
34.2225
14.8225
8.1225
3.4225
3.4225
0.7225
0.0225
0.0225
0.0225
1.3225
1.3225
4.6225
4.6225
9.9225
17.2225
26.5225
37.8225
66.4225
378.5500
x
n
2
36
81
100
144
169
196
196
225
256
256
256
289
289
324
324
361
400
441
484
576
5403
Aczel/Sounderpandian
s =
2
=
(x - x)
i =1
(n - 1)
378.55
=
(20 - 1)
378.55
= 19.923684
19
n
=
2
x 2
i =1
 n x
 i =1 
2
n
(n - 1)
2
100489
317
5403 5403 20 =
20
=
19
(20 - 1)
5403 - 5024.45 378.55
=
= 19.923684
19
19
s = s = 19.923684 = 4.46
=
2
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
1-15
5th edi tion
Example: Sample Variance Using the
Template
(n+1)P/100
Quartiles
Note: This is
just a
replication
of slide #19.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-16
BUSINESS STATISTICS
1-5 Group Data and the Histogram


Dividing data into groups or classes or
intervals
Groups should be:
Mutually exclusive
• Not overlapping - every observation is assigned to
only one group
Exhaustive
• Every observation is assigned to a group
Equal-width (if possible)
• First or last group may be open-ended
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
5th edi tion
1-17
BUSINESS STATISTICS
Frequency Distribution

Table with two columns listing:
 Each and every group or class or interval of values
 Associated frequency of each group
• Number of observations assigned to each group
• Sum of frequencies is number of observations
– N for population
– n for sample


Class midpoint is the middle value of a group or
class or interval
Relative frequency is the percentage of total
observations in each class
 Sum of relative frequencies = 1
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
1-18
BUSINESS STATISTICS
5th edi tion
Example 1-7: Frequency Distribution
x
Spending Class ($)
0 to less than 100
100 to less than 200
200 to less than 300
300 to less than 400
400 to less than 500
500 to less than 600
f(x)
Frequency (number of customers)
f(x)/n
Relative Frequency
30
38
50
31
22
13
0.163
0.207
0.272
0.168
0.120
0.070
184
1.000
• Example of relative frequency: 30/184 = 0.163
• Sum of relative frequencies = 1
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
1-19
BUSINESS STATISTICS
5th edi tion
Cumulative Frequency Distribution
x
Spending Class ($)
0 to less than 100
100 to less than 200
200 to less than 300
300 to less than 400
400 to less than 500
500 to less than 600
F(x)
Cumulative Frequency
30
68
118
149
171
184
F(x)/n
Cumulative Relative Frequency
0.163
0.370
0.641
0.810
0.929
1.000
The cumulative frequency of each group is the sum of the
frequencies of that and all preceding groups.
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
1-20
5th edi tion
Histogram

A histogram is a chart made of bars of
different heights.
Widths and locations of bars correspond to
widths and locations of data groupings
Heights of bars correspond to frequencies or
relative frequencies of data groupings
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
5th edi tion
1-21
Histogram Example
Frequency Histogram
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
COMPLETE
BUSINESS STATISTICS
5th edi tion
1-22
Histogram Example
Relative Frequency Histogram
McGraw-Hill/Irwin
Aczel/Sounderpandian
© The McGraw-Hill Companies, Inc., 2002
Penutup
• Pembahasan materi dilanjutkan dengan
Materi Pokok 4 (Statistik Deskriptif-2)
23