Question

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 \(E_i, i = 1, \ldots, n\).

• A. \(\chi^2 = \displaystyle\sum_{i=1}^{n} \dfrac{(O_i - E_i)^2}{E_i}\)
• B. \(\chi^2 = \sqrt{\dfrac{\sum_{i=1}^{n} (O_i - \bar{O})}{n-1}}\)
• C. \(\chi^2 = \dfrac{\bar{O} - \mu}{\sigma_o \sqrt{n}}\)
• D. \(\chi^2 = \dfrac{(\sigma_o)^2}{(\sigma_E)^2}\)

Hint:

The Chi-Square statistic sums the squared difference between each observed and expected value, divided by that expected value, across all n categories.

Answer 1

The Correct Option is A

Step 1: Recall what the Chi-Square test measures.
The Chi-Square goodness-of-fit test quantifies how far a set of observed values \(O_i\) deviates from a set of expected (theoretical) values \(E_i\), across \(n\) categories or observations.

Step 2: Recall the standard definition.
The standard statistical definition of the Chi-Square statistic is \[\chi^2 = \sum_{i=1}^{n} \frac{(O_i - E_i)^2}{E_i}\] Each term squares the deviation of an observed value from its expected value (so that positive and negative deviations do not cancel out) and then divides by the expected value (so that deviations are scaled relative to the size of what was expected).

Step 3: Eliminate the incorrect options.
Option (B) resembles the formula for sample standard deviation, not Chi-Square, and it does not even involve the expected values \(E_i\) at all. Option (C) resembles a z-score / standard error type expression comparing means, which is unrelated to the Chi-Square construction. Option (D) is simply a ratio of two variances, which describes an F-statistic, not a Chi-Square statistic.

Step 4: Conclude.
Only option (A) matches the correct, standard Chi-Square formula.