Ukuran Berangka

Ukuran BerangkaUkuran BerangkaTerbahagi kepada dua:Ukuran Kecenderungan MemusatDilakukan dengan menggunakan 1 nilai utk mewakili satu set data.3 jenis pengukur :-MinDefinisi min.Digunakan ke atas jenis skala nominal/ordinal/selang/nisbah?Rumus purataBila min tidak sesuai digunakan?MinDefinisi min = nilai purata yg digunakan utk mewakili satu set nilai-nilai yg diperhatikan.Digunakan ke atas jenis skala selang/nisbah.Rumus, min = Jum semua skor/Bil skorBila min tidak sesuai digunakan?Apabila terdapat skor ekstrim (terlalu kecil/ besar) dalam set data.Skor ekstrim ini akan menyebabkan skor min bagi data kajian terpesong jauh drp skor-skor yg normal.MedianDefinisi medianDigunakan ke atas jenis skala nominal/ordinal/ selang/nisbah?Rumus median (bil.skor ganjil, bil.skor genap)Nilai median tidak dipengaruhi oleh skor-skor ekstrim.

MedianMedian = nilai tengah sst taburan skor yg disusun mengikut urutan menaik atau menurun.Digunakan ke atas jenis skala ordinal, selang dan nisbahRumus median (bil.skor ganjil, bil.skor genap)Nilai median tidak dipengaruhi oleh skor-skor ekstrim.ModDefinisi nilai yg wujud paling kerap dalam sst taburan.Digunakan ke atas jenis skala nominal/ordinal/ selang/nisbah?Biasanya digunakan utk menyatakan ciri-ciri demografi subjek kajian. Cth?Boleh terdiri sama ada 1/ >1/ tiada mod.

ModDefinisi nilai yg wujud paling kerap dalam sst taburan.Digunakan utk data skala nominal atau ordinal.Biasanya digunakan utk menyatakan ciri-ciri demografi subjek kajian. Cth?Boleh terdiri lebih dari satu mod atau tiada mod.

Example 1Find the mode and median for each collection of numbers.

a. 1, 2, 3, 3, 4, 6, 9 b. 1, 1, 2, 3, 4, 5, 10c. 0, 1, 2, 3, 4, 4, 5, 5 d. 1, 2, 3, 4Answera. The mode is 3, since it occurs more often than any other number. The median is also 3, since it is the middle score in this ordered list of numbers.b. The mode is 1 and the median is 3.c. There are two modes, 4 and 5. Here we have an even number of scores. Hence we average the two middle scores to compute the median. The median is 3.5. Note that the median is not one of the scores in this case.d. The median is 2.5. There is no mode, since each number occurs equally often.Example 2Find the mean, median, and mode for the following sets of data that represent the monthly salaries of two small companies. What do you observe about the mode, median, and mean for the two sets of data?

Company A: $3,300, $2,500, $4,200, $3,100, $6,200, $3,300, $3,500, $5,100Company B: $2,500, $9,200, $3,100, $5,100, $3,300, $3,500, $4,200, $3,300AnswerSolutionCompany AMode = $3300Median = $3400Mean = $3900

Company BMode = $3300Median = $3400Mean = $ 4275Data InterpretationIf the owner of company B wanted to promote how much her employees are paid, she would likely choose the mean as the measure of central tendency because one employees high salary makes the company mean higher.

If an employee representative wanted to make a different point, they would choose the mode or median salary because they are less affected by one large salary.Ukuran SerakanUntuk menghurai serakan variabel iaitu sama ada nilai-nilai dalam suatu kumpulan data berjauhan atau berdekatan antara satu sama lain.Menurut Stringer (2008), ukuran serakan terdiri daripada 4 jenis iaitu:-

Sela @JulatJulat menerangkan tentang luasnya skor sesuatu kumpulan data (Khalid Johari, 2003)Menerangkan jurang antara nilai yg paling rendah dan nilai yg paling tinggi.Julat dipengaruhi oleh nilai ekstrem kerana hanya melibatkan perbezaan antara dua nilai data sahaja (iaitu skor max dan skor min)Diperolehi dengan menolak nilai yg paling tinggi (max) dgn nilai yg paling rendah (min)

Sela@Julat = Skor maksimum Skor minimumKeputusan Peperiksaan Akhir SemesterMata PelajaranPelajar 1Pelajar 2Pelajar 3BM622571BI868874Sains747274Matematik659572Sejarah826578KH546470PJK849868JUMLAH507507507SKOR MIN72.4272.4272.42Perihalkan ketiga-tiga data pelajar di atas dan cadangkan ukuran yg sesuai bagi mewakili setiap taburan tersebut.Keputusan Peperiksaan Akhir SemesterMata PelajaranPelajar 1Pelajar 2Pelajar 3BM622571BI868874Sains747274Matematik659572Sejarah826578KH546470PJK849868JUMLAH507507507SKOR MIN72.4272.4272.42SELA86-54 = 3298-25=7378-68=10PerihalanJadual menunjukkan bahawa walaupun skor min bagi ketiga-tiga pelajar tersebut sama, sela skor bagi pelajar 2 (sela = 73) adalah jauh lebih besar daripada pelajar 1 (sela = 32) dan pelajar 3 (sela = 10).Ini bermakna skor pelajar 2 bertabur jauh antara satu sama lain, pelajar 2 mahir dalam m/pel BI (skor = 88), Math (skor = 95) dan PJK(skor=98) tetapi kurang mahir dala m/pel BM (skor = 25).Prestasi pelajar 3 adalah sederhana bagi semua mata pelajaran kerana nilai sela skornya kecil iaitu 10.Varians & Sisihan PiawaiVarians ialah kuasa dua bagi sisihan piawai.

Sisihan piawai ialah punca kuasa dua varians.

Varians didefinisikan sebagai keluasan serakan nilai-nilai daripada min kumpulan, manakala sisihan piawai menerangkan serakan atau perbezaan nilai-nilai daripada min kumpulan.

Sisihan piawai merupakan petunjuk pengukuran utama dalam py utk menyatakan keserakan skor-skor dalam sesuatu taburan. Digunakan pada skala selang dan nisbah.

Sisihan piawai ialah jumlah purata sesuatu nilai atau skor individu tersisih daripada skor min dalam suatu taburan.Apabila nilai sisihan piawai adalah BESAR, bererti skor-skor data berada jauh berbeza antara satu sama lain iaitu menjauhi min .Semakin KECIL nilai sisihan piawai, menunjukkan skor-skor dalam satu set data adalah terkumpul berhampiran satu sama lain iaitu menghampiri min (bermaksud data adalah hampir seragam/homogenus serakan skor terendah & tertinggi adalah kecil)Contoh: Sisihan piawai set data A ialah 2.7 manakala set data B ialah 8.3, maka skor-skor data bagi set data A adalah berhampiran satu sama lain berbanding set data B.Example 3Find the variance and standard deviation for the numbers

5, 7, 7, 8, 10, 11.Answer

The standard deviation is the square root of the variance, 4. Hence the standard deviation is 2.Example 4Table below gives the variance and standard deviation for Class 1 and Class 2 take a reading test, rounded to two decimal places.

Which class has more heterogenous ability of reading?

AnswerComparing the classes on the basis of the standard deviation shows that the scores in class 2 were more widely distributed than were the scores in class 1, since the greater the standard deviation, the larger the spread of scores. Hence class 2 is more heterogeneous in reading ability than is class 1. This finding may mean that more reading groups are needed in class 2 than in class 1 if students are grouped by ability. Although it is difficult to give a general rule of thumb about interpreting the standard deviation, it does allow us to compare several sets of data to see which set is more homogeneous.

Example 5Adrienne made the following scores on two achievement tests.

On which test did she perform better relative to the class?

AnswerComparing Adriennes scores only to the means seems to suggest that she performed equally well on both tests, since her score is 15 points higher than the mean in each case. However, using the standard deviation as a unit of distance, we see that she was 1.5 (15 divided by 10) standard deviations above the mean on test 1 and only 1 (15 divided by 15) standard deviation above the mean on test 2. Hence she performed better on test 1, relative to the whole class.

Contoh 6TarikhSuhu TertinggiPerbezaan dari MinKuasa dua Perbezaan dari Min02-Jan5903-Jan6004-Jan4305-Jan4206-Jan3507-Jan3208-Jan3209-Jan4610-Jan4111-Jan52JUMLAHNMinKirakan varians & sisihan piawai.TarikhSuhu TertinggiPerbezaan dari MinKuasa dua Perbezaan dari Min02-Jan5914.8219.0403-Jan6015.8249.6404-Jan43-1.21.4405-Jan42-2.24.8406-Jan35-9.284.6407-Jan32-12.2148.8408-Jan32-12.2148.8409-Jan461.83.2410-Jan41-3.210.2411-Jan527.860.84JUMLAH442931.6N10Min44.2= 931.60 = 103.51 10 - 1= sqrt (103.51) = 10.2JawapanFikirkanMengapakah nilai perbezaan antara nilai skor dan min perlu dikuasa duakan?Utk mendapatkan nilai yg positif bagi perbezaan yg terjadi antara nilai skor dan nilai min.

Mengapa formula varians sampel adalah n-1?N 1 adalah utk mengelakkan bias berlaku semasa menganggar varians populasi.Box and Whisker Plots A popular application of the median is a box and whisker plot or simply a box plot. To construct a box and whisker plot, we first find the lowest score, the median, the highest score, and two additional statistics, namely the lower and upper quartiles. We define the lower and upper quartiles using the median. To find the lower and upper quartiles, arrange the scores in increasing order.With an even number of scores, say 2n, the lower quartile is the median of the n smallest scores. The upper quartile is the median of the n largest scores. With an odd number of scores, say 2n+1, the lower quartile is the median of the n smallest scores, and the upper quartile is the median of the n largest scores.We will use the reading test scores from Class 1 as an illustration:4.3, 4.9, 4.9, 5.1, 5.2, 5.2, 5.3, 5.3, 5.3, 5.4,5.4, 5.6, 5.6, 5.7, 5.8, 5.8, 5.9, 6.1, 6.2, 6.9

Lowest score = 4.3Lower quartile median of 10 lowest scores = 5.2Median = 5.4Upper quartile median of 10 highest scores = 5.8Highest score = 6.9Next, we plot these five statistics on a number line, then make a box from the lower quartile to the upper quartile, indicating the median with a line crossing the box. Finally, we connect the lowest score to the lower quartile with a line segment, one whisker, and the upper quartile to the highest score with another line segment, the other whisker. The box represents about 50% of the scores, and each whisker represents about 25%.

The difference between the upper and lower quartiles is called the interquartile range (IQR). This statistic is useful for identifying extremely small or large values of the data, called outliers. An outlier is commonly defined as any value of the data that lies more than 1.5 IQR units below the lower quartile or more than 1.5 IQR units above the upper quartile. For the class scores, IQR 5.8-5.2 = 0.6, so that 1.5 IQR units (1.5)(0.6) = 0.9. Hence any score below 5.2-0.9 = 4.3 or above 5.8+ 0.9 = 6.7 is an outlier. Thus 6.9 is an outlier for these data; that is, it is an unusually large value given the relative closeness of the rest of the data. Later in this section, we will see an explanation of outliers using z-scores. Often outliers are indicated using an asterisk. In the case of the earlier reading test scores, 6.9 was identified to be an outlier. When there are outliers, the whiskers end at the value farthest away from the box that is still within 1.5 IQR units from the end.

We can visually compare the performances of class 1 and class 2 on the reading test by comparing their box and whisker plots. The reading scores from class 2 are 3.6, 3.7, 4.1, 4.4, 4.7, 4.7, 4.7, 4.9, 5.0, 5.0, 5.4, 5.5, 5.6, 5.6, 6.2, 6.3, 6.8, 7.5, 7.8, and 8.4. Thus we have Lowest score 3.6Lower quartile 4.7Median 5.2Upper quartile 6.25Highest score 8.41.5 IQR 2.325.The box and whisker plots for both classes appear with outliers.

From the two box and whisker plots, we see that the scores for class 2 are considerablymore widely spread; the box is wider, and the distances to the extreme scores are greater.