Question

If all frequencies of classes are same, the value of chi-square ($\chi^2$) is:

• A. 0.0
• B. 1.0
• C. 2.0
• D. 0.5

Hint:

Chi-square measures the discrepancy between observed and expected values. If all classes have identical observed frequencies, they match the uniform expectation perfectly, making the discrepancy (and thus $\chi^2$) zero.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The chi-square ($\chi^2$) goodness-of-fit test measures how much an observed frequency distribution differs from an expected theoretical distribution.
Key Formula or Approach:
The formula for the chi-square statistic is: \[ \chi^2 = \sum \frac{(O_i - E_i)^2}{E_i} \] where $O_i$ represents the observed frequencies and $E_i$ represents the expected frequencies.

Step 2: Detailed Explanation:
Let there be $k$ classes, each with the same observed frequency, say $O_i = C$ for all $i = 1, 2, \dots, k$.
Under the null hypothesis of equal class probabilities (uniform distribution), the expected frequency for each class is: \[ E_i = \frac{\text{Total Frequency}}{k} = \frac{k \times C}{k} = C \]
Since $O_i = C$ and $E_i = C$ for all classes: \[ O_i - E_i = C - C = 0 \]
Substituting this difference into the chi-square formula: \[ \chi^2 = \sum_{i=1}^k \frac{(0)^2}{C} = 0 \]
Therefore, when the observed frequencies across all classes are identical, there is perfect agreement with the uniform expected distribution, resulting in a chi-square value of exactly 0.

Step 3: Final Answer:
The value of chi-square is 0.0.