Chi Squared Goodness of Fit

Goal: Check if Colors of GEMS are Unifomly distributed

Outline

1. Step 1: Took 5 packets of gems.
2. Step 2: Arranged the colors
3. Step 3: Stated my hypothesis
4. Step 4: Calculated Chi2
5. Step 5: Reject / Fail to Reject the Null Hypothesis
6. Conclusion

Step 1: Took 5 packets of gems.

• Each packet had 9 gems
• Resisted from eating them (this was the hardest part)

import pandas as pd
from scipy.stats import chi2,chisquare

Step 2: Arrange the colors

• Sorted and counted how many of each color appeared
• This gives us the observed data from our sample

observed_gems = pd.DataFrame([{
    "Blue": 8,
    "Light Pink": 9,
    "Green": 5,
    "Purple": 6,
    "Yellow": 2,
    "Dark Pink": 7,
    "Orange": 8
}])

observed_gems

	Blue	Light Pink	Green	Purple	Yellow	Dark Pink	Orange
0	8	9	5	6	2	7	8

observed_gems = observed_gems.transpose()

observed_gems

	0
Blue	8
Light Pink	9
Green	5
Purple	6
Yellow	2
Dark Pink	7
Orange	8

observed_gems.columns = ["Observed"]
observed_gems

	Observed
Blue	8
Light Pink	9
Green	5
Purple	6
Yellow	2
Dark Pink	7
Orange	8

Step 3: Stating My Hypothesis

• H0 (Null Hypothesis): All colors are equally likely
• H1 (Alternative Hypothesis): At least one color is not equally likely

#Uniform distribution
obs_sum = observed_gems["Observed"].sum()
obs_count = observed_gems["Observed"].count()

# Since uniform distribution
expected_gems = pd.DataFrame([{
    "Blue": obs_sum/obs_count ,
    "Light Pink": obs_sum/obs_count ,
    "Green": obs_sum/obs_count ,
    "Purple": obs_sum/obs_count ,
    "Yellow": obs_sum/obs_count ,
    "Dark Pink": obs_sum/obs_count ,
    "Orange": obs_sum/obs_count
}])

expected_gems

	Blue	Light Pink	Green	Purple	Yellow	Dark Pink	Orange
0	6.428571	6.428571	6.428571	6.428571	6.428571	6.428571	6.428571

expected_gems=expected_gems.transpose()

expected_gems = expected_gems.rename(columns={0: "Expected"})
expected_gems

	Expected
Blue	6.428571
Light Pink	6.428571
Green	6.428571
Purple	6.428571
Yellow	6.428571
Dark Pink	6.428571
Orange	6.428571

Step 4: Calculate Chi2

• Found the expected frequency for each color assuming equal probability
• Compared observed vs expected values
• Calculated:
  • Chi-Square Statistic

calc_table = pd.merge(observed_gems, expected_gems, left_index=True, right_index=True)

calc_table

	Observed	Expected
Blue	8	6.428571
Light Pink	9	6.428571
Green	5	6.428571
Purple	6	6.428571
Yellow	2	6.428571
Dark Pink	7	6.428571
Orange	8	6.428571

ChiSq_calc,P_value_calc = chisquare(f_obs=calc_table["Observed"], f_exp=calc_table["Expected"])

ChiSq_calc,P_value_calc

(5.2444444444444445, 0.5128652198163051)

alpha = 0.05
dof = len(observed_gems) - 1
ChiSq_critical = chi2.ppf(1-alpha, df=dof)
print(ChiSq_critical)

12.591587243743977

Step 5: Reject / Fail to Reject the Null Hypothesis

• Calculated:
  • Degrees of Freedom
  • Critical value
  • p-value

Chi-Square Statistic Method

• ChiSq_calc > ChiSq_critical: Reject the null hypothesis
  → The result is unlikely to occur if H₀ is true.
• ChiSq_calc ≤ ChiSq_critical: Fail to reject the null hypothesis
  → The result is not unusual enough to reject H₀.

p-value Method

• p_value < α: Reject the null hypothesis
  → The probability of getting this kind of result is too low assuming H₀ is true.
• p_value ≥ α: Fail to reject the null hypothesis
  → The probability of getting this result is high enough that it doesn't seem unusual.

if ChiSq_calc > ChiSq_critical:
    print("Reject the null hypothesis")
else:
    print("Fail to reject the null hypothesis")

Fail to reject the null hypothesis

if P_value_calc < alpha:
    print("Reject the null hypothesis")
else:
    print("Fail to reject the null hypothesis")

Fail to reject the null hypothesis

Conclusion

• In this sample, the Chi-Square value was not large enough to reject the null hypothesis.
• So, we fail to reject H₀ — meaning we don't have strong evidence that the gem colors are unequally distributed.
• It doesn't prove that all colors are perfectly equal in the entire population — just that based on this small sample of 5 packets (45 gems), we can't say otherwise.