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In the context of the chi-squared test of independence, the term 'observed frequency' refers to the actual counts or occurrences in the different categories of a contingency table. When we perform a chi-squared test, we are essentially investigating whether there is an association or independence between two categorical variables. The observed frequency, denoted as \( N_{ij} \), corresponds to the number of occurrences you observe in the data for each cell of your contingency table. In simpler terms, it's the actual count you would record when you classify data points into categories. Understanding observed frequencies is crucial because it provides the raw data that we compare against our expectations to determine if any significant relationship exists between the variables. It's the starting point in the analysis, and its accuracy is vital because subsequent calculations rely significantly on these numbers.

'Expected frequency' refers to what you would expect the frequency count to be in each cell of a contingency table if there were no association between the variables. In the chi-squared test, this is calculated under the assumption of independence between the categories. Knowing how to calculate the expected frequencies \( E_{ij} \) is key, as it represents the hypothesized counts if our categorical variables were independent. The expected frequency for each cell is generally calculated using:\[ E_{ij} = \frac{(Row \, total) \times (Column \, total)}{Overall \, total} \]This formula distributes the total counts into expected counts proportionate to the marginal totals of the rows and columns. Hence, expected frequencies help measure the discrepancy between what we observe and what we hypothesize, which is critical for understanding the strength of the independence or association.

In statistics, computational efficiency deals with how resources, such as time and computing power, are utilized to perform calculations. When conducting a chi-squared test of independence, efficiency can mean the difference between smoothly handling large datasets or being bogged down by computational challenges.The formula \[ \chi^{2} = \sum_{i=1}^{l} \sum_{j=1}^{J} \left( \frac{N_{ij}^{2}}{\dot{E}_{ij}} \right)-n \] for computing the chi-squared statistic is more efficient than the standard formula because it simplifies the calculation process by reducing the number of steps. It omits unnecessary arithmetic operations like subtraction and squaring, which typically add extra computational load, especially with large tables or datasets.By directly computing the ratio \( \frac{N_{ij}^{2}}{E_{ij}} \), we combine operations, minimizing processing time, and making it more efficient. Efficiency is particularly beneficial when tests are run iteratively, where such streamlining can save a significant amount of compute power and time, making it practical for larger or more complex analyses. Simplified calculations not only save resources but reduce potential errors, critical for ensuring accuracy in statistical analyses.