statistical techniques for ordinal data
TRANSCRIPT
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
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TOOLS WILL NEED Ordinal scales Probability (the unit normal table) Introduction to hypothesis testing Correlation
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Preview People have a passion for rankings thingsNile is the longest river in the worldThe Labrador retriever is the number one
registered dog in the USEnglish is the fourth common native
language in the world (after Chinese, Hindi, and Spanish)
Universitas Indonesia is number 201 top university in the world (THES, 2009)
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
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Preview Part of fascination with rank is that they are
easy to obtain and they are easy to understand
What is your favorite ice-cream flavor? Ordinal scales are less demanding and less
sophisticated than the interval or ratio scales easier to use ordinal scales can cause some problems for statistical analysis
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Preview Because ordinal data (ranks) provide
limited information they must be used and interpreted carefully
Standard statistical methods such as means, t test, or analysis F variance should not be used when data are measured on an ordinal scale
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DATA FROM AN ORDINAL SCALE Ordinal values (ranks) only tell the
direction from one score to another, but provide no information about the distance between scores
In a horse race, for example, we know that the second-place horse is somewhere between the first- and the third-place horses a rank of second is not necessarily halfway between first and third
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OBTAINING ORDINAL MEASUREMENT1. Ranks are simpler.
“He is little taller than I am”2. The original score may violate some of the
basic assumption that underlie certain statistical procedures. the homogeneity of variance assumption
3. The original score may have unusually high variance
4. Occasionally, an experiment produce undetermined, or infinite, score
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Rank the following scores3 4 4 7 9 9 9 12
14 3 4 0 3 5 14 3
Boy’s score 8, 17, 14, 21Girl’s score 18, 25, 23, 21, 34, 28, 32, 30, 13
LEARNING CHECK
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THE MANN-WHITNEY U TEST
An Alternative toThe Independent-Measures t Test
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THE MANN-WHITNEY U TEST… is designed to use the data from two
separate samples to evaluate the difference between two treatment (or two population)
The calculations for this test require that the individual scores in the two samples
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THE MANN-WHITNEY U TEST A real difference between the two
treatments should cause the scores in one sample to be generally larger than the score in the other sample
If the two sample are combined and all the scores placed in rank order on a line, the scores from one sample should be concentrated at one end of the line, and the scores from the other sample should be concentrated at the other end
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The scores from the two samples are clustered at opposite ends of the rank ordering
1 2 173 4 5 6 14 167 8 9 10 11 12 1513 18
In this case, the data suggest a systematic difference between the two treatment (or
two samples)
Sample from treatment A
Sample from treatment B
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THE MANN-WHITNEY U TEST On the other hand, if there is no treatment
difference, the large and small scores will be mixed evenly in the two samples because there is no reason for one set of scores to be systematically larger or smaller than the other
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The scores from the two samples are intermixed evenly along the scale
1 2 173 4 5 6 14 167 8 9 10 11 12 1513 18
In this case, the data indicating no consistent difference between the two
treatment (or two samples)
Sample from treatment A
Sample from treatment B
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THE NULL HYPOTHESIS FORTHE MANN-WHITNEY U TEST
Because the Mann-Whitney test compares two distributions (rather than two means), the hypotheses tend to be somewhat vague
H0 : There is no difference between treatments therefore, there is no tendency for the ranks
in one treatment condition to be systematically higher (or lower) than the ranks in the other treatment condition
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CALCULATION OF THE MANN-WHITNEY U
Sample A : 27, 2, 9, 48, 6, 15Sample B : 71, 63, 18, 68, 94, 8 Combine the two samples and all 12 scores
are placed in rank order
2, 6, 8, 9, 15, 18, 27, 48, 63, 68, 71, 94 Each individual in sample A is assigned 1
point every score in sample B that has a higher rank
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CALCULATION OF THE MANN-WHITNEY U
RANKORDERED SCORES
POINTS FOR SAMPLE A
POINTS FOR SAMPLE BSCORE SAMPLE
1 2 A 6 points
2 6 A 6 points
3 8 B 4 points
4 9 A 5 points
5 15 A 5 points
6 18 B 2 points
7 27 A 4 points
8 48 A 4 points
9 63 B 0 points
10 68 B 0 points
11 71 B 0 points
12 94 B 0 points
UA + UB = nAnB
30 + 6 = 6(6)
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RANKING WITHOUT SCORES We assumed we had obtained a score for
each individual. However, it is not necessary to have a set previously obtained scores
For example, a researcher could observe a group of 12 preschool children (6 boys and 6 girls) and rank them in terms aggressive behavior
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COMPUTING U FOR LARGE SAMPLES
RANKORDERED SCORES
SCORE SAMPLE
1 2 A
2 6 A
3 8 B
4 9 A
5 15 A
6 18 B
7 27 A
8 48 A
9 63 B
10 68 B
11 71 B
12 94 B
UA = nAnB +nA(nA+1)
2- Σ RA
UB = nAnB +nB(nB+1)
2- Σ RB
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THE MANN-WHITNEY U
… is the smaller USee Table B.9ATo be significant for any given nA and nB,
the obtained U must be equal to or less than the critical value in the table.
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Reporting in Literature
The original scores measured in questionaire score, were rank-ordered and a Mann-Whitney U-test was used to compared the ranks for the n=6 group A versus n=6 group B. The results indicate there is significant difference between group A and group B, U=6, p<.05 one-tailed with the sum of the ranks equal to 27 for group A and 61 for group B.
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HYPOTHESIS TESTS WITHTHE MANN-WHITNEY U
A large difference between the two treatments (or samples) causes all the ranks from sample A to cluster at one end of the scale all the ranks from sample B to cluster at the other.
At the extreme, there is no overlap between two sample the Mann-Whitney U will be zero because one of the sample gets no point at all
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Psychosis such as schizophrenia often expressed in the artistic work produced by patients. To test the reliability of this phenomenon, a psychologist collected 10 painting done by schizophrenic patient and another 10 painting by normal college student. A professor in art department was asked to rank order all 20 paintings in term bizarreness.
Schizophrenic patents: 1, 3, 4, 5, 6, 8, 9, 11, 12, 14 Student: 2, 7, 10, 13, 15, 16, 17, 18, 19, 20Test at the .01 level of significance
LEARNING CHECK
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The KRUSKAL-WALLIS Test
An Alternative toThe Independent-Measures ANOVA
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Kruskal-Wallis Test
Berbeda dengan analisis Mann-Whitney U Test yang terbatas untuk membandingkan
2 kelompok (treatment) yang terpisah, analisis Kruskal-Wallis T Test digunakan untuk mengevaluasi perbedaan urutan
individu dari 3 kelompok atau lebih yang independen (between subjects)
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Ilustrasi PenelitianSeorang peneliti ingin mengetahui pengaruh
kelembapan (kadar air) udara terhadap performa kerja karyawan dalam mengetik.3(tiga) kelompok karyawan dipilih secara
acak untuk ditempatkan pada 3(tiga) ruangan secara terpisah. Ketiga ruangan tersebut diatur agar memiliki kelembapan udara
rendah (60%), sedang (75%), dan tinggi (90%).Kecepatan mengetik diukur dengan urutan
menyelesaikan mengetik ulang suatu tulisan yang diberikan peneliti.
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Kruskal-Wallis Test Apabila hasil pengukuran (original data) DV
memiliki varians yang besar dan terdapat undetermined/infinite score, maka akan lebih tepat menggunakan analisis Kruskal-Wallis dibandingkan dengan One-Way ANOVA
Skala pengukuran DV merupakan skala ordinal (rank-order) atau data numerik (skala interval/rasio) diubah dalam bentuk rank-order (ordinal)
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Kruskal-Wallis TestTujuan penelitiannya adalah untuk
mengetahui apakah kelompok treatment yang satu akan memiliki ranking yang secara
konsisten lebih tinggi (atau lebih rendah) dibandingkan kelompok lainnya.
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Hipotesis Kruskal-Wallis Test H0: tidak ada kecenderungan bahwa ranking
pada kelompok treatment tertentu akan secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada kelompok treatment yang lain.Dengan demikian, tidak ada perbedaan yang signifikan di antara kelompok treatment
HA: sedikitnya ranking pada satu kelompok treatment secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada kelompok treatment yang lain.Dengan demikian, terdapat perbedaan yang signifikan di antara kelompok treatment.
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Hipotesis Kruskal-Wallis Test H0: tidak ada kecenderungan bahwa ranking pada kelompok
treatment tertentu akan secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada kelompok treatment yang lain.Dengan demikian, tidak ada perbedaan yang signifikan di antara kelompok treatment
HA: sedikitnya ranking pada satu kelompok treatment secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada kelompok treatment yang lain.Dengan demikian, terdapat perbedaan yang signifikan di antara kelompok treatment. 30
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
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Kelembapan Udara
Rendah Sedang Tinggi
Rank?
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
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Kelembapan Udara
Rendah Sedang Tinggi
1
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
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Kelembapan Udara
Rendah Sedang Tinggi
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14
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Kelembapan Udara
Rendah Sedang Tinggi
12
3,53,5
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14
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Kelembapan Udara
Rendah Sedang Tinggi
12 5
3,53,5
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14
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Kelembapan Udara
Rendah Sedang Tinggi
12 5
63,5
3,5
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14
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Kelembapan Udara
Rendah Sedang Tinggi
12 5
63,5 7
3,5
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14
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Kelembapan Udara
Rendah Sedang Tinggi
9 12 9 5
6 93,5 7
3,5
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14
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Kelembapan Udara
Rendah Sedang Tinggi
9 12 9 5
6 93,5 711 3,5
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14
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Kelembapan Udara
Rendah Sedang Tinggi
9 12 9 5
6 93,5 7 1211 3,5
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14
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Kelembapan Udara
Rendah Sedang Tinggi
9 12 9 5
6 93,5 7 1211 3,5 13
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14
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Kelembapan Udara
Rendah Sedang Tinggi
9 12 9 5
14 6 93,5 7 1211 3,5 13
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Pengaruh Kelembapan terhadap Kecepatan Mengetik (menit:detik)
Kelembapan Udara
Rendah Sedang Tinggi
5:07 4:12 8:254:28 5:07 4:406:39 4:49 5:074:35 4:58 5:435:16 4:35 6:14
43
Kelembapan Udara
Rendah Sedang Tinggi
9 1 152 9 5
14 6 93,5 7 1211 3,5 13
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Analisis Kruskal-Wallis T Test
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Kelembapan Udara
Rendah Sedang Tinggi
9 1 152 9 5
14 6 93,5 7 1211 3,5 13
T1 = 39,5 T2 = 26,5 T3 = 54
n = 5 n = 5 n = 5
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Analisis Kruskal-Wallis T Test
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Kelembapan Udara
Rendah Sedang Tinggi
T1 = 39,5 T2 = 26,5 T3 = 54
n = 5 n = 5 n = 5
T =12
N(N+1)
ΣT2
n-
3(N+1)
T =12
15(16)+
39,52
5- 3(16)+
26,52
5542
5
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Analisis Kruskal-Wallis T TestT = 3,785 Signifikan?
Table B.8 Chi Square; df = k-1
df = k-1 = 2; critical value = 5,993,785 < 5,99 TIDAK Signifikan
Tidak Ada pengaruh kelembapan terhadap kecepatan mengetik
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
47
The FRIEDMAN Test
An Alternative toThe Repeated-Measures ANOVA
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Kruskal-Wallis Test
Berbeda dengan analisis Wilcoxon T Test yang terbatas untuk membandingkan 2
kelompok (treatment) yang terpisah, analisis Friedman Test digunakan untuk
mengevaluasi perbedaan urutan individu dari 3 treatment atau lebih dari satu
kelompok (within subjects)
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
49
Ilustrasi PenelitianTiara ingin melihat pengaruh pemberian pelatihan Empati terhadap keterampilan
berkomunikasi karyawan.Seorang atasan diminta untuk meranking
keterampilan berkomunikasi setiap karyawan pada ketiga pengukuran.
Pengukuran keterampilan berkomunikasi dilakukan sebelum pemberian traning, 3
bulan setelah training, dan 6 bulan setelah training.
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Friedman Test Apabila hasil pengukuran (original data) DV
memiliki varians yang besar dan terdapat undetermined/infinite score, maka akan lebih tepat menggunakan analisis Friedman Test dibandingkan dengan Repeated-Measures ANOVA
Skala pengukuran DV merupakan skala ordinal (rank-order) atau data numerik (skala interval/rasio) diubah dalam bentuk rank-order (ordinal)
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Friedman TestTujuan penelitiannya adalah untuk
mengetahui apakah treatment yang satu akan memiliki ranking yang secara konsisten lebih
tinggi (atau lebih rendah) dibandingkan treatment lainnya.
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Hipotesis Kruskal-Wallis Test H0: tidak ada kecenderungan bahwa ranking
pada treatment tertentu akan secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada treatment yang lain.Dengan demikian, tidak ada perbedaan yang signifikan di antara treatment
HA: sedikitnya ranking pada satu treatment secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada treatment yang lain.Dengan demikian, terdapat perbedaan yang signifikan di antara treatment. 52
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Hipotesis Kruskal-Wallis Test H0: tidak ada kecenderungan bahwa ranking pada treatment
tertentu akan secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada treatment yang lain.Dengan demikian, tidak ada perbedaan yang signifikan di antara treatment
HA: sedikitnya ranking pada satu treatment secara sistematis lebih tinggi (atau lebih rendah) dibandingkan ranking pada treatment yang lain.Dengan demikian, terdapat perbedaan yang signifikan di antara treatment. 53
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Skor Keterampilan Berkomunikasi Karyawan Sebelum 3-bulan 6-bulan
A 24 22 30B 19 25 28C 22 34 30D 25 28 34E 20 29 29F 26 24 33
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Rank Keterampilan Berkomunikasi Karyawan Sebelum 3-bulan 6-bulan
A 2 1 3B 1 2 3C 1 3 2D 1 2 3E 1 2,5 2,5F 2 1 3
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Rank Keterampilan Berkomunikasi Karyawan Sebelum 3-bulan 6-bulan
A 2 1 3B 1 2 3C 1 3 2D 1 2 3E 1 2,5 2,5F 2 1 3
Total 8 11,5 16,5
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Analisis Friedman Test
57
Keterampilan Berkomunikasi Rendah Sedang Tinggi
R1 = 8 R2 = 11,5 R3 = 16,5
n = 6 n = 6 n = 6
χ2r=
12nk(k+1)
ΣT2 - 3n(k+1)
H =12
(6)(3)(4)(8)2 + (11,5)2 + (16,5)2 - (3)(6)(4)
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STATISTICAL TECHNIQUES FOR ORDINAL DATA
Analisis Friedman Testχ2
r= 6,08 Signifikan?
Table B.8 Chi Square; df = k-1
df = 2; critical value = 5,996,08 > 5,99 SIGNIFIKAN
Ada pengaruh pelatihan terhadap keterampilan berkomunikasi