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

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.

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 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

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

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

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% 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 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 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.

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: n = 9 As students which measure is the most representative of the center of the data. 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.

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: 2 4 3 0 10 2 5 4 6 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.

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

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: 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

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

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

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

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.

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

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

Mean – Calculated based on class mid-point (m) Activity III - GROUPED Frequency distribution Mean – Calculated based on class mid-point (m) → n = 71 = 2370.5 Σfm Σfm n X = 2370.5 71 = 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 = 33.387

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 = 30.5 i = 10 F = 27 = 32 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 md f n 2 F md f = 30.5 + 10 (0.2656) = 30.5 + 2.656 = 33.156 Md = L + i 71 2 27 32 Md = 30.5 + 10

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

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 65 65 10 10 74 74 13 23 78 78 21 44 86 86 15 59 93 93 9 68 68 Total X = ΣfX n 5377 68 = = 79.074

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

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

…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 = 63.909

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 This data will be used for several examples. You might want to duplicate it for your students.

f Calculate the mean, the median, and the mode of this grouped data Class 67 - 78 79 - 90 91 - 102 103 -114 115 -126 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. = 2991 30 = 99.7

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

f Calculate the mean, the median, and the mode of this grouped data Class 67 - 78 79 - 90 91 - 102 103 -114 115 -126 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

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 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.

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

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

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 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

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.

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.

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 =

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 46 33 56 42 57 48 58 52 61 57 63 67 67 77 77 82 77 90 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.

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 56 33 56 42 57 48 58 52 61 57 63 67 67 77 67 82 67 90 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.

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.