STATISTIK PENDIDIKAN EDU5950 SEM1 2013-14

1. STATISTIK PENDIDIKANEDU5950SEM1 2013-14 STATISTIK DESKRIPTIF:UKURAN KECENDERUNGAN MEMUSAT Rohani Ahmad Tarmizi - EDU5950

2. UKURAN KECENDERUNGAN MEMUSAT • Teknikpenggambaran data telahmemberikitasatucaramemperihal data dalambentukjadualfrekuensi, cartapalangataupai, histogram, poligonfrekuensi, danjadualsilang. • Analisisinimenjelaskanpolataburanskor-skorataupunfrekuensibagikategori-kategoritertentu. • Iamemberigambaran yang menyeluruhtetapitidakmenunjukkansesuatutumpuanataukecenderungan. • Iajugatidakmerupakanbentuk 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 –ukuranskor/nilai/kategori yang paling kerapdalamsesuatutaburan, yang jugamenunjukkanskor/nilai/kategori yang lazim (“typical”). • Mod bagi data kategorikal – adalahkategori yang terkerap (sekolahmenengahbiasa)

6. Maklumat Demografi Pengetua

7. Jadual 1: Taburan Responden Guru Kanan Berdasarkan Umur

8. Jadual 30: Taburan Responden Guru Kanan Berdasarkan Kaum

9. 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

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: 30 35 28 42 45 36 40 41 48 • Kes 2: 30 30 34 35 28 45 45 45 40 41 46 48

11. 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

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: 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

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: 2 4 2 0 40 2 4 3 6 Calculate the mean, the median, and the mode Mean: n = 9 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 2since it occurs the most.

16. 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 4since it occurs the most.

17. Measures of central tendency and its location in a distribution Shapes of Distributions Symmetric Uniform mean = median 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 • 45 4 ► Data mentah – skala ordinal /sela/nisbah 5 8 9 7 6 8 7 6 5 3 7 8 ► Data berkumpul (secara individu) X 25 28 30 34 38 43 45 f 6 9 12 17 15 8 4 ► Data berkumpul (berselang) Group 21-30 31-40 41-50 f 27 32 12 Group f 21-30 27 31-40 32 41-50 12

20. Raw / Individual Data 5 8 9 7 6 8 7 6 5 3 7 8

21. Individual Grouped Data • X f fX • 6 • 9 • 12 • 17 • 15 • 8 • 45 4

22. Grouped Data Group f 21-30 27 31-40 32 41-50 12

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:

24. ΣX n X = 96 15 = = 6.4 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 8 6 9 7

25. Data set: Xfcf 25 25 6 28 9 15 30 12 27 34 17 44 38 15 59 43 8 67 45 4 71 ΣfX n X = 2434 71 Total = = 34.282 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

26. Data set: Group f cf m 21 – 30 27 27 25.5 31 – 40 32 59 35.5 41 – 50 12 71 45.5 71 71 Total Σfm Activity III - GROUPED Frequency distribution Mean – Calculated based on class mid-point (m) → n = 71 = 2370.5 Σfm n X = 2370.5 71 = = 33.387

27. n 2 71 2 F 27 32 md f md f Data set: Group f cf m 21 – 30 27 27 25.5 31 – 40 32 59 35.5 41 – 50 12 71 45.5 71 71 …Cont. ♠ Median Md = (71+1) / 2 = 36 The value 36th is located in the 31 – 40 class → L = 30.5 i = 10 F = 27 = 32 = 30.5 + 10 (0.2656) = 30.5 + 2.656 = 33.156 Md = L + i Md = 30.5 + 10

28. Data set: 10 12 14 17 20 21 10 14 15 18 20 11 14 15 19 20 12 14 17 19 21 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 ΣX n 333 21 = X = = 15.857

29. Data set: XXfcf 65 65 10 10 74 74 13 23 78 78 21 44 86 86 15 59 93 93 9 68 68 ΣfX n X = 5377 68 = = 79.074 Total 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

31. n 2 55 2 F 15 23 md f md f Data set: Group f cf m 26 – 50 15 15 38 51 – 75 23 38 63 76 – 100 17 55 88 55 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 = 25 F = 15 = 23 Total = 51 + 25 (0.5435) = 51 + 13.587 = 64.587 Md = L + i Md = 51+25

32. Σfm …Cont. ♠ Mean – Calculated based on class mid-point (m) → n = 55 = 3515 Σfm n X = 3515 55 = = 63.909

33. WORKED EXAMPLE 4: Calculating Measures of Central Tendency Minutes Spent on the Phone 102 124 108 86 103 82 71 104 112 118 87 95 103 116 85 122 87 100 105 97 107 67 78 125 109 99 105 99 101 92

34. 3 5 8 9 5 Class 67 - 78 79 - 90 91 - 102 103 -114 115 -126 f Calculate the mean, the median, and the mode of this grouped data Midpoints 72.5 84.5 96.5 108.5 120.5 f x Midpoint 217.5 422.5 772.0 976.5 602.5 = 2991 30 = 99.7

35. n 2 F md • 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 f f md

36. 3 5 8 9 5 Class 67 - 78 79 - 90 91 - 102 103 -114 115 -126 f Calculate the mean, the median, and the mode of this grouped data Midpoints 72.5 84.5 96.5 108.5 120.5 Cumulative f 3 8 16 25 30 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. LATIHANPENGIRAAN MIN (DATA BERKUMPUL)

40. PENGIRAAN MIN DATA BERKUMPUL

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. LATIHANPENGIRAAN MEDIAN (DATA BERKUMPUL)

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. ΣX n Σfm n X = X = 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 ΣfX n X =

46. The closing prices for two stocks were recorded on ten successive Fridays. Calculate the mean, the median and the mode for each. Descriptive Statistics Stock AStock B 46 33 56 42 57 48 58 52 61 57 63 67 63 67 67 77 77 82 77 90

47. The closing prices for two stocks were recorded on ten successive Fridays. Calculate the mean, the median and the mode for each. Descriptive Statistics Stock AStock B 56 33 56 42 57 48 58 52 61 57 63 67 63 67 67 77 67 82 67 90

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.