statistik deskriptif: ukuran kecenderungan memusat rohani ahmad tarmizi - edu5950 1
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STATISTIK DESKRIPTIF:UKURAN
KECENDERUNGAN MEMUSAT
Rohani Ahmad Tarmizi - EDU5950 1
UKURAN KECENDERUNGAN MEMUSATTeknik 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 skor-skor 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
MODMOD –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 PengetuaLatar Belakang Frekuensi %Frekuensi
Jantina Lelaki 119 68.4
Perempuan 55 31.6
Kumpulan Etnik Melayu 121 69.5
Cina 42 24.1
India 4 2.3
Bumiputra Sabah/Sarawak 7 4.0
Pencapaian Akademik
Bacelor 12 7.1
Diploma 29 17.2
STPM 55 32.5
SPM 70 41.4
SRP 3 1.18
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 1: Taburan Responden Guru Kanan Berdasarkan Umur
Kaum Frekuensi Peratus
Melayu 154 73.0
Cina 41 19.4
India 14 6.6
Lain-lain 2 1.0
Jumlah 211 100
Jadual 30: Taburan Responden Guru Kanan Berdasarkan Kaum
MODSet 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
MEDIANMedian adalah skor yang di tengah-tengah
sesuatu taburan.Ia merupakan skor di mana terletaknya 50%
skor-skor di bawahnya dan 50% skor-skor di atasnya.
Median dapat ditentukan dengan menyusun skor-skor 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 48Skor 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-728 30 30 34 35 40 41 45 45 45 46 48Purata kedua-dua skor – [ 40 + 41 ] = 40.5Purata bagi skor ke n/2 dan skor ke n/2 +
1
MINMin adalah ukuran pukul rata dengan itu
mula-mula 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.
MINKes 1: 30 35 28 42 45 36 40 41 48345/9 = 38.333338.33
Kes 2: 30 30 34 35 28 45 45 45 40 41 46 48
467/12 = 38.916638.92
UKURAN KECENDERUNGAN MEMUSAT BAGI TABURAN BERKUMPUL
MOD – KATEGORI YANG PALING KERAP
MEDIAN – SKOR TENGAH MIN – SKOR PURATA
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2 4 2 0 40 2 4 3 6
Calculate the mean, the median, and the mode
n
xx
63x n = 9 7
9
63x
Mean:
Median: Sort data in order
0 2 2 2 3 4 4 6 40The middle value is 3, so the median is 3.
Mode: The mode is 2 since it occurs the most.
An instructor recorded the average number of absences for his students in one semester. For a random sample the data are:
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2 4 3 0 10 2 5 4 6
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.
An instructor recorded the average number of absences for his students in one semester. For a random sample the data are:
Which is the most appropriate measure of central tendency?
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Measures of central tendency and its location in a distributionShapes of Distributions
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12
UniformSymmetric
Skewed right Skewed left
Mean > median Mean < median
mean = median
KEPENCONGANData 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.
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Jenis data:► Data mentah – skala ordinal /sela/nisbah
5 8 9 7 6 87 6 5 3 7 8
► Data berkumpul (secara individu)
X 25 28 30 34 38 43 45f 6 9 12 17 15 8 4
► Data berkumpul (berselang)
Group 21-3031-4041-50f 27 32 12
X f25 628 930 1234 1738 1543 845
4
Group f 21-30 2731-40 3241-50 12
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5 8 9 7 6 87 6 5 3 7 8
21
X f fX
25 628 930 1234 1738 1543 845 4
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Group f21-30 2731-40 3241-50 12
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For a sample:
N
x
n
xx
Measures of Central Tendency
n
fxx
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 For a population:
Mode: The value with the highest frequency
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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 arrayML = (15+1) / 2 = 8i.e. Median is the average of the 8th valuesMedian = 7
♠ Mean X =ΣX n
9615= = 6.4
Data set:3 74 75 75 86 86 86 97
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Activity II - Calculating MCT
2. GROUPED Frequency distribution♠ Mode – The value with the highest frequency (17) is 34 Mode = 34
♠ MedianMd = (71+1) / 2 = 36The 36th value is corresponding to 34Md = 34
♠ Mean
X =ΣfX n2434 71
= = 34.282
Data set:
X f cf25 25 6 28 9 1530 12 2734 17 4438 15 5943 8 6745 4 71
Total
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Activity III - GROUPED Frequency distributionMean – Calculated based on class mid-point (m)
→ n = 71 = 2370.5
X =Σfm n
Σfm
=2370.5 71
= 33.387Data set:
Group f cf m21 – 30 27 27 25.531 – 40 32 59 35.541 – 50 12 71 45.5
71 71Total
27
…Cont.
♠ MedianMd = (71+1) / 2 = 36The value 36th is located in the 31 – 40 class→ L = 30.5 i = 10 F = 27 = 32
md f
n 2
F
mdfMd = L + i
Md = 30.5 + 1071 2
27
32
= 30.5 + 10 (0.2656)= 30.5 + 2.656= 33.156
Data set:
Group f cf m21 – 30 27 27 25.531 – 40 32 59 35.541 – 50 12 71 45.5
71 71
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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
♠ 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
33321= =15.857
Data set:
10 12 14 17 20 2110 14 15 18 2011 14 15 19 2012 14 17 19 21
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WORKED EXAMPLE 2: Calculating Measures of Central Tendency
2.Frequency distribution♠ Mode – The value with the highest frequency (21) is 78
Mode = 78
♠ MedianML = (68+1) / 2 = 34.5The 36th value is corresponding to 78Md = 78
♠ MeanX =
ΣfX n5377 68
= = 79.074
Data set:
X X f cf65 65 10 1074 74 13 2378 78 21 4486 86 15 5993 93 9 68 68 Total
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X f cf f . X
65 10 10 650
74 13 23 962
78 21 44 1638
86 15 59 1290
93 9 68 837
Total 68 5377
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WORKED EXAMPLE 3: Calculating Measures of Central Tendency
3. Grouped Frequency distribution ♠ Modal class – class 51-75
♠ MedianML = (55+1) / 2 = 28The value 28th is located in the 51 – 75 class→ L = 51 i = 25 F = 15 = 23md f
n 2
F
mdfMd = L + i
Md = 51+25
55 2
15
23
= 51 + 25 (0.5435)= 51 + 13.587= 64.587
Data set:
Group f cf m26 – 50 15 15 3851 – 75 23 38 6376 – 100 17 55 88
55Total
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…Cont.
♠ Mean – Calculated based on class mid-point (m)
→ n = 55 = 3515
X =Σfm n
=3515 55
= 63.909
Σfm
Group Midpoint
Frequency
F . Xmidpt
26-50 38 15 570
51-75 63 23 1449
76-100 88 17 1496
55 3515
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102 124 108 86 103 82 71 104 112 118 87 95103 116 85 122 87 100105 97 107 67 78 125109 99 105 99 101 92
Minutes Spent on the Phone
WORKED EXAMPLE 4: Calculating Measures of Central Tendency
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Class
67 - 78
79 - 90
91 - 102
103 -114
115 -126
3
5
8
9
5
f Midpoints
72.5
84.5
96.5
108.5
120.5
Calculate the mean, the median, and the mode of this grouped data
f x Midpoint
217.5
422.5
772.0
976.5
602.5
n
fxx
= 2991
30
= 99.7
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Grouped frequency distribution♠ Locate the median class that contains the ML♠ Then calculate median using the formula
Md = L + i
where: L lower boundary of the class with median
i class intervaln number of cases (sample size)F cumulative frequency before the
median classfrequency of the class with median
f
n 2
F
md
mdf
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Class
67 - 78
79 - 90
91 - 102
103 -114
115 -126
3
5
8
9
5
f Midpoints
72.5
84.5
96.5
108.5
120.5
Calculate the mean, the median, and the mode of this grouped data
Cumulative f
3
8
16
25
30
n = 30
I = 12
L = 90.5
F = 8 fmd = 8
MIN BAGI DATA BERKUMPULMin 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 BERKUMPULL1: Tentukan nilai-nilai titik-tengah bagi
setiap sela/kelas - X titik-tengah
L2: Kirakan jumlah skor bagi setiap sela/kelas – f x X titik-tengah
L3: Jumlahkan semua nilai f x X titik-tengah
L4: Bahagikan jumlah tersebut dengan bilangan skor dalam taburan.
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
KELAS FREKUENSI
TITIK 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 medianL5: Tentukan fm – bilangan skor dalam kelas yang
terdapat medianL6: Tentukan n bilangan skor dalam taburanL7: Tentukan saiz atau sela kelasL8: Masukkan nilai-nilai yang didapati dalam
formula
KELAS FREKUENSI FREK.KUMULATIF
5-9 2 2
10-14 11 13
15-19 26 39
20-24 17 56
25-29 8 64
30-34 6 70
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Use of ModeRelevant 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.
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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.
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Use of MeanThe most frequently used MCTHowever it is very much susceptible to the
presence of extreme valuesMean is used when the distribution is normal.Mean is also used in calculation of the statistic.
ex. t-testFormula:
Raw data FrequencyDistribution
Grouped Freq.distribution
X =X =ΣX n X =
Σfm n
ΣfX n
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Stock A Stock B 46 33 56 42 57 48 58 52 61 57 63 67 63 67 67 77 77 82 77 90
The closing prices for two stocks were recorded on ten successive Fridays. Calculate the mean, the median and the mode for each.
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Stock A Stock B 56 33 56 42 57 48 58 52 61 57 63 67 63 67 67 77 67 82 67 90
The closing prices for two stocks were recorded on ten successive Fridays. Calculate the mean, the median and the mode for each.
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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.