1. STATISTIK PENDIDIKAN EDU5950 SEM1 2013-14
STATISTIK DESKRIPTIF: UKURAN KECENDERUNGAN MEMUSAT
Rohani Ahmad Tarmizi - EDU5950

2. 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 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.

3. 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).

4. TIGA JENIS UKURAN KECENDERUNGAN MEMUSAT
MOD
MEDIAN/PENENGAH
MIN/PURATA

5. 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)

6. Maklumat Demografi Pengetua
Latar 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
32.5
SPM 70
41.4
SRP 3
1.18

7. Umur Frekuensi Peratus 25-30 tahun 6 2.8 31-36 tahun 9 4.3 37-42 tahun
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

8. Kaum Frekuensi Peratus Melayu 154 73.0 Cina 41 19.4 India 14 6.6
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
Jumlah
211
100

9. MOD
Set A: mod adalah 75 (unimod)
Set B: mod adalah 75 dan 80 (dwimod)
Set C: taburan ini tidak mempunyai mod.
Kes 1:
Kes 2:

10. MEDIAN Median 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:
Kes 2:

11. Kes 1:
Skor ke (n+1)/2
Kes2:
Kes 2:
Skor ke 12/2- skor ke 6, skor ke-7
Purata kedua-dua skor – [ ] = 40.5
Purata bagi skor ke n/2 dan skor ke n/2 + 1

12. MIN
Min 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.

13. MIN
Kes 1:
345/9 =
38.33
Kes 2:
467/12 =
38.92

14. UKURAN KECENDERUNGAN MEMUSAT BAGI TABURAN BERKUMPUL
MOD – KATEGORI YANG PALING KERAP
MEDIAN – SKOR TENGAH
MIN – SKOR PURATA

15. An instructor recorded the average number of absences for his students in one semester. For a random sample the data are:
Calculate the mean, the median, and the mode
Mean:
n = 9
As students which measure is the most representative of the center of the data.
Median: Sort data in order
The middle value is 3, so the median is 3.
Mode: The mode is 2 since it occurs the most.

16. Mean: The average value is 4
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?
Mean: The average value is 4
As students which measure is the most representative of the center of the data.
Median:
The middle value is 3, so the median is 4.
Mode: The mode is 2 and 4 since it occurs the most.

17. Symmetric Uniform Skewed right Skewed left mean = median
Measures of central tendency and its location in a distribution
Shapes of Distributions
Symmetric
Uniform
mean = median
Both curves at the top are symmetric. Note that the fulcrum is placed at the mean of each distribution.
Skewed right
Skewed left
Mean > median
Mean < median

18. 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.

19. Jenis data:
X f 6 9 12 17 15 8
► Data mentah – skala ordinal /sela/nisbah
► Data berkumpul (secara individu) X f
► Data berkumpul (berselang)
Group f
Group f

20. Raw / Individual Data

21. Individual Grouped Data
X f fX 6 9 12 17 15 8

22. Grouped Data
Group f

23. 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
For a population:
For a sample:
There are other measures of central tendency (ex. midrange), but these three are the most commonly used. Be sure to discuss the advantages and disadvantages for each.

24. Activity I - Calculating MCT
Calculate mode, median, and mean for the three data sets
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
Data set:
3 7
4 7
5 7
5 8
6 8
6 9 7
X = ΣX n 96 15 =
6.4

25. Activity II - Calculating MCT
GROUPED Frequency distribution
♠ Mode – The value with the highest frequency (17) is 34
Mode = 34
♠ Median
Md = (71+1) / 2 = 36
The 36th value is corresponding to 34
Md = 34
♠ Mean
Data set:
X f cf
Total
X =
ΣfX n
2434 71 = =

26. Mean – Calculated based on class mid-point (m)
Activity III - GROUPED Frequency distribution
Mean – Calculated based on class mid-point (m)
→ n = =
Σfm
Σfm n
X =
2370.5 71 =
Data set:
Group f cf m
21 –
31 –
41 –
Total =

27. f …Cont. n 2 F = 30.5 + 10 (0.2656) Md = L + i = 30.5 + 2.656 = 33.156
♠ Median
Md = (71+1) / 2 = 36
The value 36th is located in the 31 – 40 class
→ L = i = F = = 32
Data set:
Group f cf m
21 –
31 –
41 – md f n 2 F md f
= (0.2656) = =
Md = L + i 71 2 27 32
Md =

28. 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
Data set: ΣX n
333 21
X = = =
15.857

29. 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
Data set:
X X f cf 68
Total
X =
ΣfX n
5377 68 = =

30. X f cf
f . X 65 10
650 74 13 23
962 78 21 44
1638 86 15 59
1290 93 9 68
837
Total
5377

31. WORKED EXAMPLE 3: Calculating Measures of Central Tendency
3. Grouped Frequency distribution
♠ Modal class – class 51-75
♠ Median
ML = (55+1) / 2 = 28
The value 28th is located in the 51 – 75 class
→ L = 51 i = F = = 23
Data set:
Group f cf m
26 –
51 –
76 – 55
Total md f n 2 F md f
= (0.5435) = =
Md = L + i 55 2 15 23
Md = 51+25

32. …Cont. ♠ Mean – Calculated based on class mid-point (m)
Σfm
Σfm n
X =
3515 55 =
Group
Midpoint
Frequency
F . Xmidpt
26-50 38 15
570
51-75 63 23
1449
76-100 88 17
1496 55
3515 =

33. WORKED EXAMPLE 4: Calculating Measures of Central Tendency
Minutes Spent on the Phone
This data will be used for several examples. You might want to duplicate it for your students.

34. f Calculate the mean, the median, and the mode of this grouped data
Class 3 5 8 9 f
Midpoints
72.5
84.5
96.5
108.5
120.5
f x Midpoint
217.5
422.5
772.0
976.5
602.5
This histogram is labeled at the class boundaries. Explain that midpoints could have been labeled instead. = 30 =

35. F f f 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 interval
n number of cases (sample size)
F cumulative frequency before the
median class
frequency of the class with median n 2 F md f f md

36. f Calculate the mean, the median, and the mode of this grouped data
Class 3 5 8 9 f
Midpoints
72.5
84.5
96.5
108.5
120.5
Cumulative f 3 8 16 25 30
This histogram is labeled at the class boundaries. Explain that midpoints could have been labeled instead.
L = 90.5
I = 12
n = 30
F = 8
fmd = 8

37. 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.

38. MIN BAGI DATA BERKUMPUL
L1: 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.

39. LATIHAN PENGIRAAN MIN (DATA BERKUMPUL)
KELAS
FREKUENSI
TITIK TENGAH
5-9 2 7
10-14 11 12
15-19 26 17
20-24 22
25-29 8 27
30-34 6 32

40. PENGIRAAN MIN DATA BERKUMPUL
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 22
17X22=374
25-29 8 27
8X27=216
30-34 6 32
6X32=192 70
1370

41. 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

42. LATIHAN PENGIRAAN MEDIAN (DATA BERKUMPUL)
KELAS
FREKUENSI
FREK.
KUMULATIF
5-9 2
10-14 11 13
15-19 26 39
20-24 17 56
25-29 8 64
30-34 6 70

43. 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.

44. 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.

45. 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
Frequency
Distribution
Grouped Freq.
distribution
X = ΣX n
ΣfX n
X =
Σfm n
X =

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 Stock B
Use this example to review the measures of central tendency. Both sets of data have the same mean, the same median and the same mode. Students clearly see that the data sets are vastly different though. A good lead in for measures of variation.

47. 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 Stock B
Use this example to review the measures of central tendency. Both sets of data have the same mean, the same median and the same mode. Students clearly see that the data sets are vastly different though. A good lead in for measures of variation.

48. 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.