Chi-Square Example

• Computations

• Effect Size

• Writing up the chi-square results

Goodness of Fit test

• Acme Toy Company prints baseball cards.

• The company claims that 30% of the cards are rookies, 60% veterans, and 10% are All-Stars.

• The cards are sold in packages of 100.

• Suppose a randomly-selected package of cards has 50 rookies, 45 veterans, and 5 All-Stars. Is this consistent with Acme's claim?

• 2 = 19.583, p = .0001

• Cramer’s V = 0.191993464 (small to moderate)

Rookies Veterans All-Stars

Claim 30% 60% 10%

Observed 50 45 5

Expected 30 60 10

(O-E)2 / E 13.3333 3.75 2.5

Chi-Square 19.5833

df (=k-1) 2

p-value 0.0001

Chi-Square ExampleMindset Training – Middle School At-Risk Math students

• Students identified as being at risk to fail math in 7th grade.

• All received special tutor sessions• Some received training in using a

“Growth Mindset”

Observed Repeated Passed

Tutor only 33 30 63

Mindset 7 27 34

30 67 97

Expected Repeated Passed

Tutor only 19.4845361 43.5154639 63

Mindset 10.5154639 23.4845361 34

30 67 97

(O-E)2 / E Repeated Passed

Tutor only 9.37501227 4.19776669

Mindset 1.17526784 0.52623933

Chi-Square 15.27429

p-value 0.0001

• 47.6% passed in Tutor only group• 79.4% passed in Mindset group• 2(1) = 15.27, p = .0001

Effect size in Chi-Square

• Phi (2x2 tables) and Cramer’s V (larger tables)

• Both range from 0 (no relationship) to 1 (perfect relationship).

• Interpreted as approximations to the Pearson Product-Moment Correlation Coefficient r

n

2

1)-(cor 1)-(r ofsmaller *df where*

2

dfn

V

Sample size affects 2

but not measures of effect size.Observed Repeated Passed Observed Repeated Passed

Tutor only 33 30 63 Tutor only 330 300 630

Mindset 7 27 34 Mindset 70 270 340

30 67 97 400 570 970

Expected Repeated Passed Expected Repeated Passed

Tutor only 19.4845361 43.5154639 63 Tutor only 194.845361 435.154639 630

Mindset 10.5154639 23.4845361 34 Mindset 105.154639 234.845361 340

30 67 97 300 670 970

(O-E)2 / E Repeated Passed (O-E)2 / E Repeated Passed

Tutor only 9.37501227 4.19776669 Tutor only 93.7501227 41.9776669

Mindset 1.17526784 0.52623933 Mindset 11.7526784 5.26239331

Chi-Square 15.27429 Chi-Square 152.742861

p-value 0.0001 p-value <.00005

Phi 0.157467 Phi 0.15746687

Chi-Square Write-up

Teachers at an urban middle-school wondered whether teaching at-risk students about Dweck’sGrowth Mindset might have an impact on their success. At mid-year, 97 students were identified as having some risk of not passing math. All were scheduled to receive extra math tutoring sessions during the second half of the year. In one of three tutoring groups, students were also taught that their ability to do math was not fixed – that it could grow. At the end of the year, 47.6% of the students who received only tutoring passed math, while 79.4% of those who also received the Mindset Training had passed their math class. The difference in pass rates was significant (2(1) = 15.27, p = .0001). The size of the relationship was small (phi = .157). The teachers decided to add the Mindset Training to all sections of math in the coming year.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

Tutor only Mindset

Fig. 1: Pass Rate

What Now?The good news – especially if you just recognized yourself as being someone who holds the Fixed Mindset worldview – is that it is possible to change from one to the other. In fact, I think this should be taught to all children.

What’s more, Dweck has shown that people can learn to adopt the latter belief and make dramatic strides in performance. […]

The most dramatic proof comes from a recent study by Dweck and Lisa Sorich Blackwell of low-achieving seventh graders. All students participated in sessions on study skills, the brain and the like; in addition, one group attended a neutral session on memory while the other learned that intelligence, like a muscle, grows stronger through exercise. Training students to adopt a growth mind-set about intelligence had a catalytic effect on motivation and math grades; students in the control group showed no improvement despite all the other interventions.

“Study skills and learning skills are inert until they’re powered by an active ingredient,” Dweck explains. Students may know how to study, but won’t want to if they believe their efforts are futile. “If you target that belief, you can see more benefit than you have any reason to hope for.”

From: Fixed vs Growth Mindset: Which one are you?http://michaelgr.com/2007/04/15/fixed-mindset-vs-growth-mindset-which-one-are-you/