1. Calculating the 1-way χ2 and 2-way χ2 Statistics
Using statistics in small-scale language education research
Jean Turner
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2. 1-way χ2
The fabricated Halloween data are used in this discussion—questionnaire data from 300 students regarding whether they plan to attend their school Halloween party. The first step in statistical logic is…
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3. State the formal research hypotheses.
Null: There is no statistically significant pattern of response to the questionnaire. (There is no prevailing tendency.)
Alternative: There is a statistically significant pattern of response to the questionnaire.
(There is a prevailing tendency.)
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4. Set alpha.
Alpha is set at .01 because people may make decisions about allocation of resources based on the findings.
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5. Propose the statistic to use in the analysis.
Hmmm….
There’s a single nominal variable.
The data are frequency counts.
The researcher wants to know if there’s a statistically significant pattern of response to the question.
Sounds like the 1-way χ2 is appropriate.
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6. Collect the data. Here’s a summary table of the data.
Number of respondents who indicated “yes”
Number of respondents who indicated “no”
Number of participants who indicated
“I haven’t decided yet”
Number of participants who chose not to respond to the question 61 65 84 90
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7. Check the assumptions. The independent variable is nominal. OK
The data are frequency counts. OK
Each piece of information is independent from every other (no influenced votes!) This seems to be OK.
Each individual is included in only one level of a variable. OK
df > 1, so all expected frequencies must be greater than or equal to 5. (Three hundred people participated; expected frequency is 300/4 = 75. We’re OK with this assumption too.)
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8. Calculate the observed value of the statistic.
Cell fo fe
fo ‒ fe
(fo ‒ fe)2
(fo ‒ fe)2 / fe A 61 75 B 65 C 84 D 90
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9. Cell fo fe fo ‒ fe (fo ‒ fe)2 (fo ‒ fe)2 / fe A 61 75 ‒14 B 65 ‒10 C84 +9 D 90
+15
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10. Cell fo fe fo ‒ fe (fo ‒ fe)2 (fo ‒ fe)2 / fe A 61 75 ‒14 196 B 65 ‒10
100 C 84 +9 81 D 90
+15
225
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11. Cell fo fe fo ‒ fe (fo ‒ fe)2 (fo ‒ fe)2 / fe A 61 75 ‒14 196
196/75 = 2.613 B 65
‒10
100
100/75 = 1.33 C 84 +9 81
81/75 = 1.08 D 90
+15
225
225/75 = 3
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12. The sum of the last column is χ2observed.
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13. Calculating 2-way χ2
Research question: Is there a statistically significant relationship between teachers’ degree of scoring agreement depending on how well they know their students?
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14. State the formal research hypotheses.
Null hypothesis: There is no statistically significant relationship between the variables.
Alternative hypothesis: There is a statistically significant relationship between the variables.
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15. Set alpha.
Alpha is set at .01 because the researcher believes important decisions may be made on the basis of the findings.
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16. Propose the statistic to use in the analysis.
There are two nominal variables.
The data are frequency counts.
The researcher wants to know if there’s a statistically significant relationship between the two variables.
Sounds like the 2-way χ2 statistic is appropriate.
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17. Check the assumptions.
The independent variable and the moderator variable are nominal. OK
The data are frequency counts. OK
Each piece of information is independent of every other. (Depends on how the data were collected.) I think this is OK.
Each individual is included in only one level of a variable. OK
df > 1, so all expected frequencies must be greater than or equal to 5. Must be calculated and checked.
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18. Calculate the observed value.
Two raters agree
T who knows Ss rates higher
T who knows Ss rates lower
1st semester Ss 13
(cell A) 6
(cell B) 5
(cell C)
2nd semester Ss 11
(cell D) 17
(cell E)
(cell F)
3rd semester Ss
(cell G) 10
(cell H) 19
(cell I)
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19. T who knows Ss rates higher T who knows Ss rates lower
Two raters agree
T who knows Ss rates higher
T who knows Ss rates lower
Marginals for the rows
1st semester Ss 13
(cell A) 6
(cell B) 5
(cell C) 24
2nd semester Ss 11
(cell D) 17
(cell E)
(cell F) 41
3rd semester Ss
(cell G) 10
(cell H) 19
(cell I) 35
Marginals for the columns 30 33 37
N = 100
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20. Cell fo fe fo ‒ fe (fo ‒ fe)2 (fo ‒ fe)2/ fe A 13 B 6 C 5 D 11 E 17 FG H 10 I 19
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21. Cell fo fe fo ‒ fe (fo ‒ fe)2 (fo ‒ fe)2/ fe A 13 (24)(30) / 100 = B 6
(24)(33) / 100 = C 5
(24)(37) / 100 = D 11
(41)(30) / 100 = E 17
(41)(33) / 100 = F
(41)(37) / 100 = G
(35)(30) / 100 = H 10
(35)(33) / 100 = I 19
(35)(37) / 100 =
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22. Cell fo fe fo ‒ fe (fo ‒ fe)2 (fo ‒ fe)2/ fe A 13
(24)(30) / 100 = 7.20 B 6
(24)(33) / 100 = 7.92 C 5
(24)(37) / 100 = 8.88 D 11
(41)(30) / 100 = 12.30 E 17
(41)(33) / 100 = 13.53 F
(41)(37) / 100 = 15.17 G
(35)(30) / 100 = 10.50 H 10
(35)(33) / 100 = 11.55 I 19
(35)(37) / 100 = 12.95
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23. Cell fo fe fo ‒ fe (fo ‒ fe)2 (fo ‒ fe)2/ fe A 13
(24)(30) / 100 = 7.20
13 – 7.20 = 5.80 B 6
(24)(33) / 100 = 7.92
6 – 7.92 = ‒1.92 C 5
(24)(37) / 100 = 8.88
5 – 8.88 = ‒3.88 D 11
(41)(30) / 100 = 12.30
11 – = ‒1.30 E 17
(41)(33) / 100 = 13.53
17 – = 3.47 F
(41)(37) / 100 = 15.17
13 – = ‒2.17 G
(35)(30) / 100 = 10.50
6 – = ‒4.50 H 10
(35)(33) / 100 = 11.55
10 – = ‒1.55 I 19
(35)(37) / 100 = 12.95
19 – = 6.05
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24. Cell fo fe fo ‒ fe (fo ‒ fe)2 (fo ‒ fe)2/ fe A 13 (24)(30) / 100 =7.20
13 – 7.20 = 5.80
33.64 B 6
(24)(33) / 100 = 7.92
6 – 7.92 = ‒1.92
3.69 C 5
(24)(37) / 100 = 8.88
5 – 8.88 = ‒3.88
15.05 D 11
(41)(30) / 100 = 12.30
11 – = ‒1.30
1.69 E 17
(41)(33) / 100 = 13.53
17 – = 3.47
12.04 F
(41)(37) / 100 = 15.17
13 – = ‒2.17
4.71 G
(35)(30) / 100 = 10.50
6 – = ‒4.50
20.25 H 10
(35)(33) / 100 = 11.55
10 – = ‒1.55
2.40 I 19
(35)(37) / 100 = 12.95
19 – = 6.05
36.60
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25. Cell fo fe
fo ‒ fe
(fo ‒ fe)2
(fo ‒ fe)2/ fe A 13
(24)(30) / 100 =7.20
13 – 7.20 = 5.80
33.64
33.64 / 7.20 = 4.67 B 6
(24)(33) / 100 = 7.92
6 – 7.92 = ‒1.92
3.69
3.69 / 7.92 = .47 C 5
(24)(37) / 100 = 8.88
5 – 8.88 = ‒3.88
15.05
15.05 / 8.88 = 1.69 D 11
(41)(30) / 100 = 12.30
11 – = ‒1.30
1.69
1.69 / = .14 E 17
(41)(33) / 100 = 13.53
17 – = 3.47
12.04
12.04 / = .89 F
(41)(37) / 100 = 15.17
13 – = ‒2.17
4.71
4.71 / = .31 G
(35)(30) / 100 = 10.50
6 – = -4.50
20.25
20.25 / = 1.93 H 10
(35)(33) / 100 = 11.55
10 – = ‒1.55
2.40
2.40 / = .21 I 19
(35)(37) / 100 = 12.95
19 – = 6.05
36.60
36.60 / = 2.83
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26. Cell fo fe
fo ‒ fe
(fo ‒ fe)2
(fo ‒ fe)2/ fe A 13
(24)(30) / 100 =7.20
13 – 7.20 = 5.80
33.64
33.64 / 7.20 = 4.67 B 6
(24)(33) / 100 = 7.92
6 – 7.92 = ‒1.92
3.69
3.69 / 7.92 = .47 C 5
(24)(37) / 100 = 8.88
5 – 8.88 = -3.88
15.05
15.05 / 8.88 = 1.69 D 11
(41)(30) / 100 = 12.30
11 – = ‒1.30
1.69
1.69 / = .14 E 17
(41)(33) / 100 = 13.53
17 – = 3.47
12.04
12.04 / = .89 F
(41)(37) / 100 = 15.17
13 – = ‒2.17
4.71
4.71 / = .31 G
(35)(30) / 100 = 10.50
6 – = ‒4.50
20.25
20.25 / = 1.93 H 10
(35)(33) / 100 = 11.55
10 – = ‒1.55
2.40
2.40 / = .21 I 19
(35)(37) / 100 = 12.95
19 – = 6.05
36.60
36.60 / = 2.83
χ2 = ∑(fo ‒ fe)2/ fe = 13.14
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27. Using R to calculate 2-way χ2
Enter the values for the first variable.
x = c(1,1,1,1,1,3,2,2,1,1,1,1,2,1,2,3,2,2,3,1,3,2,2,2,1,3,2,1,3, 2,1,2,2,3,1,1,3,2,3,3,2,2,2,2,3,2,3,2,3,2,3,2,2,1,3,3,2,1,1,2,2,2, 2,3,3,3,3,3,3,2,3,3,3,3,2,2,1,2,2,1,2,2,1,3,3,2,2,2,2,3,3,3,3,2,3, 2,3,1,1,3)
Enter the values for the second variable.
y = c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, 2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2, 3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3, 3,3,3,3)
chisq.test (x,y)
χ2 = , df = 4, p-value =
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