Is It Statistically Significant

How to Get the P-Value in a Chi-Square Test

The P-value in a Chi-Square test tells you the probability of observing your results (or something more extreme) if the null hypothesis is true. A smaller P-value suggests the observed differences are unlikely due to chance.

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Steps to Calculate the P-Value

1. Calculate the Chi-Square Statistic (( \chi^2 )):

   Use the formula:

   χ² = Σ [(O - E)² / E]
   

   where:

   • ( O ): Observed value.
   • ( E ): Expected value.

2. Determine Degrees of Freedom (df):

   For one variable:

   df = (number of categories - 1)
   

   For contingency tables:

   df = (rows - 1) × (columns - 1)
   

3. Use Software or a Chi-Square Table to Find the P-Value:

   • Input ( \chi^2 ) and df into statistical software or look up the P-value in a Chi-Square table.
   • Compare the P-value to the significance level (( \alpha )).

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Example 1: Red and Blue Candies

• Observed Counts (O): Red = 40, Blue = 20.
• Expected Counts (E): Red = 30, Blue = 30.

1. Calculate ( \chi^2 ):

   χ² = [(40 - 30)² / 30] + [(20 - 30)² / 30]
      = [10² / 30] + [(-10)² / 30]
      = (100 / 30) + (100 / 30)
      = 6.67
   

2. Determine df:

   df = (number of categories - 1)
      = 2 - 1
      = 1
   

3. Use software to calculate the P-value:

   • For ( \chi^2 = 6.67 ) and ( df = 1 ), ( P = 0.0098 ).

4. Compare P to ( \alpha ) (e.g., 0.05):

   P = 0.0098 < 0.05
   

   Conclusion: The result is statistically significant.

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Example 2: Contingency Table

Testing candy preference by age group:

	Red Candies	Blue Candies	Total
Kids	50	30	80
Adults	20	40	60
Total	70	70	140

1. Calculate expected values for each cell:

   Expected = (row total × column total) / grand total
   

   Example for Kids-Red:

   Expected = (80 × 70) / 140 = 40
   

2. Compute ( \chi^2 ):

   χ² = Σ [(O - E)² / E]
      = [(50 - 40)² / 40] + [(30 - 40)² / 40] + ...
      = 2.5
   

3. Determine df:

   df = (rows - 1) × (columns - 1)
      = (2 - 1) × (2 - 1)
      = 1
   

4. Calculate P-value:

   • For ( \chi^2 = 2.5 ) and ( df = 1 ), ( P = 0.113 ).

5. Compare P to ( \alpha ):

   P = 0.113 > 0.05
   

   Conclusion: The result is not statistically significant.

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Summary:

• Formulas:
  • ( \chi^2 = Σ [(O - E)² / E] )
  • ( df = (categories - 1) ) or ( (rows - 1) × (columns - 1) ).
• Use software (e.g., Python, R) or tables to find the P-value.
• Compare P-value to ( \alpha ) to decide if the result is significant.