chi square goodness of fit test the goodness of fit of a supposed freaquencies to sample data. 1©...
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Chi square goodness of fit test
• The goodness of fit of a supposed freaquencies to sample data.
1© V.Čekanavičius, G.Murauskas
goodness of fit
• Data. One categorical (nominal) sample.
• All data is divided into k categories.
• At least 5 respondents in each category.
• We make a conjecture about ratios between categories.
© V. Čekanavičius, G. Murauskas
goodness of fit
• Statistical hypothesis
H0 : Conjecture is correct.
H1 : Conjecture is incorrect.
© V. Čekanavičius, G. Murauskas
H0 is rejected (data contradicts conjecture), if
H0 is accepted (data does not contradict conjecture), if
Here is the level of significance.
Conclusion based on p - value
05.0 p
05.0 p
05.0
© V. Čekanavičius, G. Murauskas
SPSS goodness of fit test Is ratio between national majority and
national minority 7:2 ?
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data
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Here
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Supposed ratio
variable
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Supposed ratio
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Frequencies
0 No 276 282,3 -6,3
1 Yes 87 80,7 6,3
363
1
2
Total
Category Observed N Expected N Residual
minority Minority Classification
ObservedExpected
difference
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Test Statistics
,639
1
,424
Chi-Squarea
df
Asymp. Sig.
minority Minority
Classification
0 cells (,0%) have expected frequencies less than5. The minimum expected cell frequency is 80,7.
a.
test statistic
p-value
Data does not contradict the ratio 7:2.
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ConcIusion
• Application of the goodness of fit test showed that there is no statistically significant difference between the supposed ratio of national majority/minority and sample data.
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SPSS Special caseA marketing analyst claims that 25% of the
customers will by certain type of sweets packed in large boxes, 25% in medium boxes, 30% in small boxes and 20% in very small boxes.
Data: 50 bought large boxes, 40 medium, 72 small and 19 very small.
Does data contradict analyst‘s claim statistically
significantly?
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datais numeric
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Here
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Weight by
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Supposedratio
Weight isleft alone
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RUSIS
50 45.3 4.8
40 45.3 -5.3
72 54.3 17.7
19 36.2 -17.2
181
1.00
2.00
3.00
4.00
Total
Observed N Expected N Residual
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Test Statistics
15.050
3
.002
Chi-Squarea
df
Asymp. Sig.
RUSIS
0 cells (.0%) have expected frequencies less than5. The minimum expected cell frequency is 36.2.
a.
Data statistically significantly
contradicts the supposed ratio.19© V.Čekanavičius, G.Murauskas
CHI SQUARE TEST FORINDEPENDENCE
Test of association for categorical data
test
• Two categorical (nominal) variables.
• We test if those categorical variables are dependent.
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Examples
• Does smoking depend on respondents religion;
• Do men and women vote similarly;• Is percent of male students the same in
all courses.
2
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Data
All data is organized in cells according to two categorical variables.
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Statistical hypothesis
H0 : variables are independent.
H1 : variables are dependent.
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H0 is rejected (variables are dependent), if
H0 is accepted (variables are independent), if
Conclusion based on p-value
,050 p
0,05 p
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Example
• Is percent of female employees the same for clerks and managers?
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data
Numeric orstring
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Here!
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rowNext, here
column
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check
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Then go
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check
check32© V.Čekanavičius, G.Murauskas
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JOBCAT Employment Category * GENDER Gender Crosstabulation
206 157 363
56.7% 43.3% 100.0%
95.4% 68.0% 81.2%
10 74 84
11.9% 88.1% 100.0%
4.6% 32.0% 18.8%
216 231 447
48.3% 51.7% 100.0%
100.0% 100.0% 100.0%
Count
% within JOBCAT Employment Category
% within GENDER Gender
Count
% within JOBCAT Employment Category
% within GENDER Gender
Count
% within JOBCAT Employment Category
% within GENDER Gender
1 Clerical
3 Manager
JOBCAT EmploymentCategory
Total
f Female m Male
GENDER Gender
Total
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Chi-Square Tests
54.935b 1 .000
53.154 1 .000
61.256 1 .000
.000 .000
447
Pearson Chi-Square
Continuity Correctiona
Likelihood Ratio
Fisher's Exact Test
N of Valid Cases
Value dfAsymp. Sig.
(2-sided)Exact Sig.(2-sided)
Exact Sig.(1-sided)
Computed only for a 2x2 tablea.
0 cells (.0%) have expected count less than 5. The minimum expected count is 40.59.b.
p-value
p < 0.05, therefore, corresponding percents differ statistically significantly.
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Conclusion
• Applying chi-square test we got that among clerks there is statistically significantly greater percent of women (56,7%), than among managers (11,9 %), p<0,01.
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SPSS Special case• One hundred children watched violence-prone
shows and 100 watched nonviolent programs. After two weeks of observation each child was classified as either agressive or nonagressive. 63 watched violent shows and were agressive, 37 watched violent shows and were nonagressive, 30 nonviolent and agressive and 70 nonviolent and nonagressive.
• Are TV and behavior related?
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Numeric orstring
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Weight by‘kiek’
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Leave alone!Po to čia!Next, here!
Statistics and Cells are delt in the same way as before 39
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ELGESYS * TV Crosstabulation
30 63 93
32.3% 67.7% 100.0%
30.0% 63.0% 46.5%
70 37 107
65.4% 34.6% 100.0%
70.0% 37.0% 53.5%
100 100 200
50.0% 50.0% 100.0%
100.0% 100.0% 100.0%
Count
% within ELGESYS
% within TV
Count
% within ELGESYS
% within TV
Count
% within ELGESYS
% within TV
agres
neagr
ELGESYS
Total
nesmurt smurt
TV
Total
violent TV watchers are more agressive
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Chi-Square Tests
21.887b 1 .000
20.581 1 .000
22.314 1 .000
.000 .000
200
Pearson Chi-Square
Continuity Correctiona
Likelihood Ratio
Fisher's Exact Test
N of Valid Cases
Value dfAsymp. Sig.
(2-sided)Exact Sig.(2-sided)
Exact Sig.(1-sided)
Computed only for a 2x2 tablea.
0 cells (.0%) have expected count less than 5. The minimum expected count is 46.50.b.
stat. significantly
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Conclusion
• Applying chi-square test we established that among watchers of violent TV are greater percent of agressive children (63%) then among non-watchers (30 %), p<0,01.
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Mc Nemar test
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Mc Nemar test
Most freaquently is applied when we have dichotomuous data for the same respondents.
• Buyers and non-buyers before and after advertisment.
• Voters and non-voters before and after TV debates.
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Data One two-valued categorical variable
observed in two related populations Or in one population twice.
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Duomenys
dc
ba
After
Before
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Statistical hypothesis
H0 : no impact of advertisment
H1 : significant impact
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H0 is rejectest (impact stat. significant), if
H0 is accepted (impact not significant), if
Here 0.05 is the level of significancy.
Conclusion based on p - value
05.0 p
05.0 p
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• Voters were twice asked about their support for candidate before and after TV debates:
• Before and after debates vote support candidate 200
• Before debates support, after - not 30• Before do not support, after support 60• Before and after do not support100• Does dabates influenced voters preferences?
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Weight by
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Here!
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variables
here
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check
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pries * po Crosstabulation
Count
po
TotalNe užpries
Ne 100 60 160už 30 200 230
Total 130 260 390
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Number of supporters increased
statistically significantly.
Chi-Square Tests
Value Exact Sig. (2-sided)
McNemar Test .002a
N of Valid Cases 390
a. Binomial distribution used.
p-value
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Nonparametric tests
• Are also called rank tests• Normality of variables is not required;• Fits small samples;• More difficult to interpret; test is nonparametric test but not
a rank test
Typical hypothesis
• H0 : distributions of X and Y are equal• H1 : distributions of X and Y differ
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Mann - Whitney test
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Mann-Whitney test
1. Analogue of Student‘s test for independent samples;
2. Means are not compared;
3. Compares distributions;
4. The lager mean rank shows which variable is stochastically larger.
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Data1. Two independent interval or rank
samples.
2. Sample sizes can be dfferent.
3. Rank variable has at least 5 different outcomes.
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Statistical hypothesis
H0 : distributions are equal,
H1 : distributions differ.
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H0 is rejected (distributions differ) if
H0 is accepted (distributions do not differ) if
Here is the level of significance
Conclusion based on p - value
α p
α p
α
© V. Čekanavičius, G. Murauskas 65
Example
• We investigate respondents, who are older than 40 years.
• Do classical music is equally appreciated by men and women?
• Values: 1-like it very much, 2-like it,….,5- hate it.
SPSS
• After suitable select cases (age >40)
• Analyze -> Nonparametric Tests -> Legacy Dialogs ->2 independent samples
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• Males chose greater marks -> they like classical music less.
Ranks sex Respondent's
Sex N Mean Rank
Sum of Ranks
classicl Classical Music
1 Male 321 412,07 132273,50
2 Female 462 378,06 174662,50
Total 783
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• Statistically significantly, p =0,033<0,05Test Statisticsa
classicl Classical Music
Mann-Whitney U 67709,500
Wilcoxon W 174662,500
Z -2,134
Asymp. Sig. (2-tailed)
,033
a. Grouping Variable: sex Respondent's Sex
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Wilcoxon test
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Wilcoxon test
1. Analogue of Students paired samples test;
2. Means are not compared;
3. Compares distributions;
4. The lager mean difference rank shows which variable is stochastically larger.
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Data1. Two dependent (paired) interval or
rank samples.
2. Rank variable has at least 5 different outcomes.
3. Usually the same respondent measured twice.
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Statistical hypothesis
H0 : distributions are equal,
H1 : distributions differ.
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H0 is rejected (distributions differ) if
H0 is accepted (distributions do not differ) if
Here is the level of significance
Conclusion based on p - value
α p
α p
α
© V. Čekanavičius, G. Murauskas 75
Example
• If respondents, older than 50 years, like classical music more than jazz?
• Each respondent rated both music styles by using the following scale: 1- like it very much,......7 – hate it very much.
SPSS
• After suitable select cases (age >50)
• Analyze -> Nonparametric Tests -> Legacy Dialogs ->2 related samples
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Ranks
138a 157.43 21725.00
198b 176.22 34891.00
161c
497
Negative Ranks
Positive Ranks
Ties
Total
JAZZ - CLASSICN Mean Rank Sum of Ranks
JAZZ Jazz Music < CLASSICL Classical Musica.
JAZZ Jazz Music > CLASSICL Classical Musicb.
CLASSICL Classical Music = JAZZ Jazz Musicc.
Ranks for differences
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Test Statisticsb
-3.782a
.000
Z
Asymp. Sig. (2-tailed)
JAZZ JazzMusic -
CLASSICL Classical
Music
Based on negative ranks.a.
Wilcoxon Signed Ranks Testb.
p-reikšmė
Difference is statistically significant.
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Spearman correlation
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Spearman correlation test
1. Analogue of Pearson’s correlation.
2. Has the same interpretation.
3. Calculates Pearson’s correlation between ranks;
4. Can be used for already ranked data.
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Data1. Two dependent interval or ranked
variables.
2. Rank variable has at least 5 different outcomes.
3. In a special case data can be ranked.
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Statistical hypothesis
H0 : variables do not correlate.
H1 : variables correlate.
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H0 is rejected (variables correlate statistically significantly) if
H0 is accepted (variables do not correlate) if
Here is the level of significance
Conclusion based on p - value
α p
α p
α
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Example• Respondents older than 50years.• Do the data support a statement that
the more respondent likes musicals, the more he/she likes classical music.
Analyze -> Correlate -> Bivariate
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Un-check
Check
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SPSS
Variables correlate statistically significantly.
Correlation is positive, but weak.
Correlations
classicl jazz
Spearman's rho classicl Correlation Coefficient
1,000 ,205**
Sig. (2-tailed) . ,000
N 504 497
jazz Correlation Coefficient
,205** 1,000
Sig. (2-tailed) ,000 .
N 497 514
**. Correlation is significant at the 0.01 level (2-tailed).
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Spearman correlation test for ranked data
1. Two teachers ranked their students:
2. First teacher: A, B, C, D, E, F, G, H, I,J, K, L.
3. Second teacher: B, C, A, D, H,E, F, G, K, I,J, L.
4. Do their rankings correlate?
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Statistical hypothesis
H0 : variables do not correlate.
H1 : variables correlate.
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SPSS
•First: A,B,C,D,E,F, G,H,I,J,K,L
•Second: B, C, A,D, H,E, F,G,K,I,J,L.
This variable is auxiliary
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Correlations
1.000 .916**
. .000
12 12
.916** 1.000
.000 .
12 12
Correlation Coefficient
Sig. (2-tailed)
N
Correlation Coefficient
Sig. (2-tailed)
N
MOKYT1
MOKYT2
Spearman's rhoMOKYT1 MOKYT2
Correlation is significant at the .01 level (2-tailed).**.
Correlation is very strong, significant and positive
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Kruskal - Wallis test
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Kruskal-Wallis test
1. Mann-Whitney test extended to more than 2 samples.
2. Interpretation is the same as fo M-W test.
3. The larger mean rank corresponds to larger scores.
4. Gives no information on which variables differ.
5. Is also called ANOVA for rank data.
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Data1. Two or more independent interval or
rank samples.
2. Each rank variable has at least 5 different outcomes.
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Statistical hypothesis
H0 : all distributions are the same
H1 : some distributions differ.
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H0 is rejected (some distributions differ st. significantly), if
H0 is accepted (all distributions are equal), if
Here is the level of significance.
Conclusion with p - value
α p
α p
α
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Example• We investigate respondents with at
leasy 13years of formal education.• Do all races equally like rap music?• Rank variable rap: 1-like it very much,
….,5-hate it.
SPSS
• After: select cases ->if -> educ >13
• Analyze -> Nonparametric Tests -> Legacy Dialogs -> K independent Samples
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SPSS
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Here
SPSS
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Ranks
617 372.20
65 254.05
34 309.59
716
RACE Racewof Respondent1 white
2 black
3 other
Total
RAP Rap MusicN Mean Rank
Blacks like best (coding).
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Test Statisticsa,b
23.311
2
.000
Chi-Square
df
Asymp. Sig.
RAP RapMusic
Kruskal Wallis Testa.
Grouping Variable: RACE Racew of Respondentb.
p-reikšmė
The scores statistically significantly
depend on the respondents race.
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Friedman test
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Friedman test1. Generalization of Wilcoxon‘s test for
more samples than 2.
2. For 2 samples, Wilcoxon‘s test is more powerful.
3. Easy to interpret.
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Interpretation of ranks
1. Lat us assume that respondent evaluated performances of three actors (larger score – better perfomance): 10 for actor A , 6 for actor B, 8 for actor C.
2. Scores are ranked. Ranks: 3 for A, 1 for B, 2 for C.
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Data1. Two or more dependent interval or
rank samples.
2. Each rank variable has at least 5 different outcomes.
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Statistical hypothesis
H0 : all distributions are the same
H1 : some distributions differ.
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H0 is rejected (some distributions differ st. significantly), if
H0 is accepted (all distributions are equal), if
Here is the level of significance.
Conclusion with p - value
α p
α p
α
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Example• We investigate respondents with formal
education longer than 15years.• Do musicals, classical music and rap
music are equally popular?• Rank variable rap: 1-like it very much,
….,5-hate it.
SPSS
• After: select cases ->if -> educ >15
• Analyze -> Nonparametric Tests -> Legacy Dialogs -> K related Samples
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Ranks
1.87
2.05
2.08
CLASSICL Classical Music
MUSICALS BroadwayMusicals
BIGBAND Bigband Music
Mean Rank
Classical music got lowest scores
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Test Statisticsa
343
14.286
2
.001
N
Chi-Square
df
Asymp. Sig.
Friedman Testa.
p-reikšmė
Not all styles are equally popular.
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Friedman‘s test special case
• Five experts ranked three sorts of bear: A,B and C.
• First: B, C, A (i.e. the best is B bear)• Second: B, C, A • Third: A or C, B• Fourth: A, B,C• Fifth: B, A,C• Do all sorts are equally popular?
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ranks!
sorts
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Ranks
2.10
1.60
2.30
A
B
C
Mean Rank
Most popular is sort B
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Test Statisticsa
5
1.368
2
.504
N
Chi-Square
df
Asymp. Sig.
Friedman Testa.
Differences are st. Insignificant.