Download UKURAN KECENDERUNGAN MEMUSAT

STATISTIK DESKRIPTIF:
UKURAN KECENDERUNGAN MEMUSAT
Rohani Ahmad Tarmizi - EDU5950
1
UKURAN KECENDERUNGAN MEMUSAT
 Teknik penggambaran data telah memberi kita
satu cara memperihal data dalam bentuk
jadual frekuensi, carta palang atau pai,
histogram, poligon frekuensi, dan jadual
silang.
 Analisis ini menjelaskan pola taburan skorskor ataupun frekuensi bagi kategori-kategori
tertentu.
 Ia memberi gambaran yang menyeluruh tetapi
tidak menunjukkan sesuatu tumpuan atau
kecenderungan.
 Ia juga tidak merupakan bentuk yang ringkas.
 Oleh itu bagi mendapatkan gambaran yang
ringkas serta kecenderungan kepada sesuatu
nilai/kategori, maka UKURAN
KECENDERUNGAN MEMUSAT boleh
digunakan.
 Ukuran ini merupakan ukuran tumpuan bagi
sesuatu taburan.
 Ia boleh mengambil ukuran tumpuan sebagai
skor/nilai (data kuantitatif) ataupun kategori
(data kualitatif).
TIGA JENIS UKURAN KECENDERUNGAN
MEMUSAT
 MOD
 MEDIAN/PENENGAH
 MIN/PURATA
MOD
 MOD –ukuran skor/nilai/kategori yang
paling kerap dalam sesuatu taburan,
yang juga menunjukkan
skor/nilai/kategori yang lazim
(“typical”).
 Mod bagi data kategorikal – adalah
kategori yang terkerap (sekolah
menengah biasa)
Maklumat Demografi Pengetua
Latar Belakang
Jantina
Kumpulan Etnik
Frekuensi
%Frekuensi
Lelaki
119
68.4
Perempuan
55
31.6
Melayu
121
69.5
Cina
42
24.1
India
4
2.3
7
4.0
Bacelor
12
7.1
Diploma
29
17.2
STPM
55
32.5
SPM
70
41.4
SRP
3
1.18
Bumiputra Sabah/Sarawak
Pencapaian
Akademik
Jadual 1: Taburan Responden Guru Kanan Berdasarkan Umur
Umur
Frekuensi
Peratus
25-30 tahun
6
2.8
31-36 tahun
9
4.3
37-42 tahun
68
32.2
43-48 tahun
91
43.1
49-54 tahun
33
15.6
Lebih 55 tahun
4
2.0
Jumlah
211
100
Jadual 30: Taburan Responden Guru Kanan Berdasarkan
Kaum
Kaum
Frekuensi
Peratus
Melayu
154
73.0
Cina
41
19.4
India
14
6.6
Lain-lain
2
1.0
211
100
Jumlah
MOD
 Set A:91 68 85 75 75 77 90 80 95 mod adalah 75
(unimod)
 Set B:60 80 80 75 75 67 90 80 75 mod adalah 75 dan
80 (dwimod)
 Set C: 70 70 84 84 80 80 20 20 56 56 taburan ini
tidak mempunyai mod.
 Kes 1: 30 35 28 42 45 36 40 41 48
 Kes 2: 30 30 34 35 28 45 45 45 40 41 46 48
MEDIAN
 Median adalah skor yang di tengah-tengah sesuatu
taburan.
 Ia merupakan skor di mana terletaknya 50% skorskor di bawahnya dan 50% skor-skor di atasnya.
 Median dapat ditentukan dengan menyusun skorskor mengikut aturan menurun atau menaik dan
skor di tengah di kenal pasti.
 Kes 1: 30 35 28 42 45 36 40 41 48
 Kes 2: 30 30 34 35 28 45 45 45 40 41 46 48
 Kes 1:
 30 35 28 42 45 36 40 41 48
28 30 35 36 40 41 42 45 48
28 30 35 36 40 41 42 45 48
 Skor ke (n+1)/2
 Kes2:
 Kes 2: 30 30 34 35 28 45 45 45 40 41 46 48
Skor ke 12/2- skor ke 6, skor ke-7
 28 30 30 34 35 40 41 45 45 45 46 48
 Purata kedua-dua skor – [ 40 + 41 ] = 40.5
 Purata bagi skor ke n/2 dan skor ke n/2 + 1
MIN
 Min adalah ukuran pukul rata dengan itu mulamula lagi dipanggil purata.
 Ia ditentukan dengan mengambil jumlah kesemua
skor-skor dalam taburan dan dibahagikan dengan
bilangan skor-skor.
 Ia sangat kerap digunakan untuk data kuantitatif
seperti IQ, kecergasan fizikal, tahap kebimbangan,
tahap pengetahuan..
 Min juga boleh digunakan untuk membuat
perbandingan antara dua atau lebih set data yang
diperoleh.
MIN
 Kes 1: 30 35 28 42 45 36 40 41 48
 345/9 = 38.3333
 38.33
 Kes 2: 30 30 34 35 28 45 45 45 40 41 46 48
 467/12 = 38.9166
 38.92
UKURAN KECENDERUNGAN MEMUSAT BAGI
TABURAN BERKUMPUL
 MOD – KATEGORI YANG PALING KERAP
 MEDIAN – SKOR TENGAH
 MIN – SKOR PURATA
An instructor recorded the average number of
absences for his students in one semester. For a
random sample the data are:
2 4 2 0 40 2 4 3
6
Calculate the mean, the median, and the mode
Mean:
63
x
x

7
n
=
9
x  63
x
9
n
Median:
Sort data in order
0 2 2 2 3
4 4 6
40
The middle value is 3, so the median is 3.
Mode:
The mode is 2 since it occurs the most.
15
An instructor recorded the average number of
absences for his students in one semester. For a
random sample the data are:
2 4 3 0 10 2 5 4
6
Which is the most appropriate measure of central tendency?
Mean: The average value is 4
Median: The middle value is 3, so the median is 4.
Mode:
The mode is 2 and 4 since it occurs the most.
16
Measures of central tendency and its location in a distribution
Shapes of Distributions
Symmetric
1
2
3
4
5
6
Uniform
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
mean = median
Skewed right
1
2
3
4
5
6
7
8
Mean > median
Skewed left
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
Mean < median
17
KEPENCONGAN
 Data yang digambarkan boleh dianggarkan bentuk
taburannya dengan mengguna skor-skor min,
median dan mod.
 Bagi taburan yang mana min=median=mod maka
taburan ini dipanggil normal.
 Bagi taburan yang mana min>median>mod maka
taburannya dipanggil pencong ke kanan atau
positif.
 Bagi taburan yang mana min<median<mod maka
taburannya dipanggil pencong kiri atau negatif.
Jenis data:
► Data mentah – skala ordinal /sela/nisbah
5
7
8
6
9
5
7
3
6
7
8
8
► Data berkumpul (secara individu)
X
f
25
6
28
9
30 34 38 43 45
12 17 15 8 4
► Data berkumpul (berselang)
Group
f
21-30 31-40 41-50
27
32
12
X
25
28
30
34
38
43
45
Group
21-30
31-40
41-50
f
6
9
12
17
15
8
4
f
27
32
12
19
Raw / Individual Data
5
7
8
6
9
5
7
3
6
7
8
8
20
Individual Grouped Data
X
25
28
30
34
38
43
45
f
6
9
12
17
15
8
4
fX
21
Grouped Data
Group
21-30
31-40
41-50
f
27
32
12
22
Measures of Central Tendency
Mode: The value with the highest frequency
Median: The point at which an equal number of values fall
above and fall below it.
Mean: The sum of all data values divided by the number of
values
x
For a population:

For a sample:
N
x
x
n
fx
x
n
23
Activity I - Calculating MCT
Calculate mode, median, and mean for the three data sets
1.
RAW SCORES
♠ Mode -The value with the highest frequency (4) is 7
Mode = 7
♠ Median - Data must be arranged in an array
ML = (15+1) / 2 = 8
i.e. Median is the average of the 8th values
Median = 7
♠ Mean
X=
=
ΣX
n
96
15
= 6.4
Data set:
3
7
4
7
5
7
5
8
6
8
6
8
6
9
7
24
Activity II - Calculating MCT
2.
GROUPED Frequency distribution
♠ Mode – The value with the highest frequency (17) is 34
Mode = 34
Data set:
♠ Median
Md = (71+1) / 2 = 36
The 36th value is corresponding to 34
Md = 34 ΣfX
X=
♠ Mean
=
n
2434
71
= 34.282
X
25
28
30
34
38
43
45
f
25
9
12
17
15
8
4
cf
6
15
27
44
59
67
71
Total
25
Activity III - GROUPED Frequency distribution
Mean – Calculated based on class mid-point (m)
→ n = 71
Σfm = 2370.5
X=
=
Σfm
n
2370.5
71
= 33.387
Data set:
Group
21 – 30
31 – 40
41 – 50
Total
f
27
32
12
71
cf
27
59
71
71
m
25.5
35.5
45.5
26
Data set:
…Cont.
Group
21 – 30
31 – 40
41 – 50
f
♠ Median
Md = (71+1) / 2 = 36
The value 36th is located in the 31 – 40 class
→ L = 30.5 i = 10 F = 27 f = 32
27
32
12
71
cf
27
59
71
71
m
25.5
35.5
45.5
md
n
Md = L + i
Md = 30.5 + 10
F
2
= 30.5 + 10 (0.2656)
= 30.5 + 2.656
= 33.156
f md
71
2
27
32
27
WORKED EXAMPLE 1: Calculating Measures of Central Tendency
Calculate mode, median and mean for the data sets
1. Raw data
♠ Mode – The value with the high
frequency (4) is 14
Mode = 14
Data set:
10
10
11
12
♠ Median – Data must be arranged in array
ML = (21+1) / 2 = 11
i.e. median is the average of the 11th value
Md = 15
♠ Mean
X =
ΣX
n
=
333
21
12
14
14
14
14
15
15
17
17
18
19
19
20
20
20
21
21
= 15.857
28
WORKED EXAMPLE 2: Calculating Measures of Central Tendency
2.
Frequency distribution
♠ Mode – The value with the highest frequency (21) is 78
Mode = 78
♠ Median
ML = (68+1) / 2 = 34.5
The 36th value is corresponding to 78
Md = 78
♠ Mean
ΣfX
X=
=
n
5377
68
= 79.074
Data set:
X
X
65 65
74 74
78 78
86 86
93 93
Total
f
10
13
21
15
9
68
cf
10
23
44
59
68
29
X
65
74
f
10
13
cf
10
23
f.X
650
962
78
86
93
Total
21
15
9
68
44
59
68
1638
1290
837
5377
30
WORKED EXAMPLE 3: Calculating Measures of Central Tendency
Data set:
3. Grouped Frequency distribution
♠ Modal class – class 51-75
Group
f
26 – 50
15
51 – 75
23
♠ Median
76 – 100
17
ML = (55+1) / 2 = 28
Total
55
The value 28th is located in the 51 – 75 class
→ L = 51 i = 25 F = 15 fmd = 23
n
Md = L + i
Md = 51+25
F
2
f md
55
2
cf
15
38
55
m
38
63
88
= 51 + 25 (0.5435)
= 51 + 13.587
= 64.587
15
23
31
…Cont.
♠ Mean – Calculated based on class mid-point (m)
Σfm = 3515
→ n = 55
X=
=
Σfm
n
3515
55
= 63.909
Group
Midpoint
Frequency
F . Xmidpt
26-50
38
15
570
51-75
63
23
1449
76-100
88
17
1496
55
3515
32
WORKED EXAMPLE 4: Calculating Measures of Central Tendency
Minutes Spent on the Phone
102
71
103
105
109
124
104
116
97
99
108
112
85
107
105
86
118
122
67
99
103
82
87
95
87 100
78 125
101
92
33
Calculate the
mean, the median, and the mode of this grouped data
Class
f
Midpoints
f x Midpoint
67 - 78
3
72.5
217.5
79 - 90
5
84.5
422.5
91 - 102
8
96.5
772.0
103 -114
9
108.5
976.5
115 -126
5
120.5
602.5
fx
x
n
= 2991
= 99.7
30
34
 Grouped frequency distribution
♠ Locate the median class that contains the ML
♠ Then calculate median using the formula
Md = L + i
where:
L
i
n
F
f md
n
2
F
fmd
lower boundary of the class with median
class interval
number of cases (sample size)
cumulative frequency before the
median class
frequency of the class with median
35
Calculate the mean, the
data
median, and the mode of this grouped
Class
f
Midpoints
Cumulative f
67 - 78
3
72.5
3
79 - 90
5
84.5
8
91 - 102
8
96.5
16
103 -114
9
108.5
25
115 -126
5
120.5
30
L=
90.5
I =
12
n=
30
F=8
fmd = 8
36
MIN BAGI DATA BERKUMPUL
 Min masih lagi jumlah semua skor dan dibahagikan
dengan bilangan skor-skor.
 Oleh itu, bagi setiap skor/kelas yang berkumpul maka
perlu ditentukan jumlah pada skor/kelas tersebut,
kemudian jumlahkan kesemua skor-skor tersebut dan
dibahagikan dengan jumlah bilangan bagi taburan
tersebut.
MIN BAGI DATA BERKUMPUL
 L1: Tentukan nilai-nilai titik-tengah bagi
sela/kelas - X titik-tengah
 L2: Kirakan jumlah skor bagi setiap
– f x X titik-tengah
 L3: Jumlahkan semua nilai f x X titik-tengah
 L4: Bahagikan jumlah tersebut dengan
skor dalam taburan.
setiap
sela/kelas
bilangan
LATIHAN
PENGIRAAN MIN (DATA BERKUMPUL)
KELAS
FREKUENSI TITIK
TENGAH
5-9
2
7
10-14
11
12
15-19
26
17
20-24
17
22
25-29
8
27
30-34
6
32
PENGIRAAN MIN DATA BERKUMPUL
KELAS
FREKUEN TITIK
SI
TGH
FREK X TITIK
TENGAH
5-9
2
7
2X7=14
10-14
11
12
11X12=132
15-19
26
17
26X17=442
20-24
17
22
17X22=374
25-29
8
27
8X27=216
30-34
6
32
6X32=192
70
1370
MEDIAN BAGI DATA
BERKUMPUL ATAU SEKUNDER
 L1: Tentukan bilangan skor dan bahagi dengan 2 –
 L2: Tentukan kelas yang mengandungi median –
 L3: Tentukan had bawah sebenar (sempadan kelas) bagi





kelas tersebut:
L4: Tentukan F –nilai frekuensi bagi kelas sebelum
terdapat median
L5: Tentukan fm – bilangan skor dalam kelas yang
terdapat median
L6: Tentukan n bilangan skor dalam taburan
L7: Tentukan saiz atau sela kelas
L8: Masukkan nilai-nilai yang didapati dalam formula
LATIHAN
PENGIRAAN MEDIAN (DATA BERKUMPUL)
KELAS
FREKUENSI
5-9
2
FREK.
KUMULATIF
2
10-14
11
13
15-19
26
39
20-24
17
56
25-29
8
64
30-34
6
70
Use of Mode
 Relevant for raw and frequency distribution data.
 Mode corresponds to value with the highest frequency.
 For raw data, count frequency for each value – where mode
is the value with the highest frequency.
 For frequency distribution data, locate the value the
highest frequency.
 Mode is not susceptible to extreme values.
 A data can have one (unimodal), two (bimodal) or multiple
modes.
43
Use of Median
 Relevant for raw and frequency distribution data.
 Median corresponds to the middle value in the
distribution.
 Median is not susceptible to extreme values.
 Median is useful for skewed distribution or distribution
with extreme scores.
 Median does change in value when there exist extreme
scores, unlikely mean, which will be affected by extreme
scores.
44
Use of Mean
 The most frequently used MCT
 However it is very much susceptible to the presence of
extreme values
 Mean is used when the distribution is normal.
 Mean is also used in calculation of the statistic.
ex. t-test
 Formula:
Raw data
X=
ΣX
n
Frequency
Distribution
X=
ΣfX
n
Grouped Freq.
distribution
X=
Σfm
n
45
Descriptive Statistics
The closing prices for two stocks were recorded on ten
successive Fridays. Calculate the mean, the median and
the mode for each.
Stock A
46
56
57
58
61
63
63
67
77
77
Stock B
33
42
48
52
57
67
67
77
82
90
46
Descriptive Statistics
The closing prices for two stocks were recorded on ten
successive Fridays. Calculate the mean, the median and
the mode for each.
Stock A
56
56
57
58
61
63
63
67
67
67
Stock B
33
42
48
52
57
67
67
77
82
90
47
Measures of Central Tendency
and Variability
 Both these measures allow description of a distribution as a
whole in a quantitative (numerical) manner.
 MEASURES OF CENTRAL TENDENCY indicate central
measurement representing the distribution of data - MEAN,
MEDIAN ,MODE.
 MEASURES OF VARIABILITY indicate the extent to which
scores are different from each other, are dispersed, or
spread out - RANGE, MEAN DEVIATION, VARIANCE,
STANDARD DEVIATION.
48