91Ó°ÊÓ

When conducting a chi-square test, understanding expected frequency is essential. This refers to the frequency you would expect to find in a data set if the null hypothesis were true. It acts like a theoretical benchmark, representing perfect balance in your test circumstances.
For example, when flipping a fair coin 100 times, we anticipate it landing on heads 50 times and tails 50 times. This is because the likelihood of either outcome is the same: 50%. The expected frequency here is calculated by multiplying the probability of the outcome by the total number of trials, which in this case is 0.5 * 100. Therefore, the expected frequency for both heads and tails is 50.
Identifying expected frequencies allows comparisons against actual results, a crucial step in determining the chi-square statistic.

The null hypothesis is a fundamental starting point in statistical tests. It's essentially an assumption that there is no significant difference or effect. When dealing with a chi-square test, the null hypothesis suggests there is no association or difference between the observed and expected frequencies.
In the scenario where we flip a coin 100 times and get values of 58 heads and 42 tails, the null hypothesis affirms that the variations are purely by chance, indicating that the coin is fair. It's a neutral assumption, claiming that if you repeated the test, you'd still expect to see no systematic deviation from chance.
By calculating the chi-square statistic and comparing it to a threshold found on the chi-square distribution table, we can decide whether to reject this null hypothesis.

The chi-square formula is central to evaluating deviations between observed and expected frequencies. It helps quantify how much your data departs from expectations under the null hypothesis.
The formula is expressed as: \[\chi^2 = \sum \frac{(O - E)^2}{E}\]where:- \(O\) represents observed frequencies,- \(E\) denotes expected frequencies.Every category's deviation is squared to eliminate negative discrepancies and divided by the respective expected frequency to normalize the differences. These normalized differences are then summed up.
Applying this to our coin flip example, the calculations for heads and tails are:

• For heads: \(\frac{(58 - 50)^2}{50} = 1.28\)
• For tails: \(\frac{(42 - 50)^2}{50} = 1.28\)

The chi-square value 2.56 (1.28 + 1.28) becomes a statistical measure of how likely differences are to have occurred by chance.

Observed frequency refers to the actual data obtained from your experiment or study. It's what truly happened when you gathered your data, contrasting the anticipated perfect balance.
In the context of flipping a coin 100 times, your observed frequencies were 58 heads and 42 tails. These numbers showcase real occurrences, not model predictions. Observed frequencies are the key players in the chi-square test, representing the deviations you wish to investigate against the expected frequencies.
These frequencies allow you to determine how much the real results differ from what you theorized under the null hypothesis, driving the calculations for the chi-square statistic.