Which ONE of the following options is the CORRECT formula for Chi-Square ($\chi^2$) test?
Given, $O_1, O_2, \ldots, O_n$ are the observed values, $E_1, E_2, \ldots, E_n$ are the corresponding expected values, $\bar{O}$ is the mean of observed values, $n$ is the number of observations, $\mu$ is the mean of $E_i, i = 1, \ldots, n$, $\sigma_o$ is the standard deviation of observed values, and $\sigma_E$ is the standard deviation of expected values $E_i, i = 1, \ldots, n$.

Question

Which one of the following gases is primarily responsible for the formation of photochemical smog?

Answer 1

A useful way to check this without memorizing the formula outright is to look at what each option is actually built from and match it against what a 'goodness of fit' statistic needs to do. A goodness-of-fit measure needs to compare each individual observed value to its own corresponding expected value, for every one of the $n$ categories, and it needs to penalize larger mismatches more heavily (via squaring) while accounting for scale (by dividing by the expected value).

Option (B) only uses the observed values and their overall mean $\bar{O}$, never touching $E_i$ at all, so it cannot be measuring agreement between observed and expected data; it is structurally the sample standard deviation formula. Option (C) compares $\bar{O}$ to $\mu$, losing all the per-category detail. Option (D) reduces everything to a ratio of two whole-dataset variances, describing the test statistic used to compare two variances (the F-test).

Only the summation form $\sum_{i=1}^n \frac{(O_i-E_i)^2}{E_i}$ preserves the per-category comparison that the Chi-Square test is built on.

\[\boxed{\chi^2 = \sum_{i=1}^{n}\frac{(O_i-E_i)^2}{E_i}}\]

Answer 2

A useful way to check this without memorizing the formula outright is to look at what each option is actually built from and match it against what a 'goodness of fit' statistic needs to do. A goodness-of-fit measure needs to compare each individual observed value to its own corresponding expected value, for every one of the $n$ categories, and it needs to penalize larger mismatches more heavily (via squaring) while accounting for scale (by dividing by the expected value).

Option (B) only uses the observed values and their overall mean $\bar{O}$, never touching $E_i$ at all, so it cannot be measuring agreement between observed and expected data; it is structurally the sample standard deviation formula. Option (C) compares $\bar{O}$ to $\mu$, losing all the per-category detail. Option (D) reduces everything to a ratio of two whole-dataset variances, describing the test statistic used to compare two variances (the F-test).

Only the summation form $\sum_{i=1}^n \frac{(O_i-E_i)^2}{E_i}$ preserves the per-category comparison that the Chi-Square test is built on.

\[\boxed{\chi^2 = \sum_{i=1}^{n}\frac{(O_i-E_i)^2}{E_i}}\]

Show Hint

The Correct Option is A

Solution and Explanation

Questions Asked in GATE ES exam