Introduction:
A cumulative frequency table is a statistical tool used to organize data in a systematic manner. It displays the cumulative frequencies for each value or interval in a dataset. This table is beneficial in determining various statistical measures. In this case, the cumulative frequency table can help in determining the median.

Explanation:
- Mean: The mean is the average of a set of numbers. It is calculated by summing up all the values in the dataset and dividing the sum by the total number of values. The construction of a cumulative frequency table does not directly provide the necessary information to calculate the mean. It only displays the cumulative frequencies of each value or interval. Therefore, the cumulative frequency table is not useful in determining the mean.

- Median: The median is the middle value in an ordered set of data. To find the median, the data must be sorted in ascending or descending order. The construction of a cumulative frequency table involves arranging the data in ascending order and calculating the cumulative frequencies. This process automatically organizes the data, making it easier to identify the middle value or values. By examining the cumulative frequency table, one can determine the median more easily and accurately. Thus, the cumulative frequency table is useful in determining the median.

- Mode: The mode is the value or values that occur most frequently in a dataset. The construction of a cumulative frequency table does not directly provide the necessary information to calculate the mode. However, by examining the cumulative frequencies, one can identify the values with the highest frequencies more easily. This can help in determining the mode, but it is not as straightforward as determining the median using the cumulative frequency table. Therefore, the cumulative frequency table can be somewhat useful in determining the mode but not as useful as in determining the median.

Conclusion:
In conclusion, the construction of a cumulative frequency table is primarily useful in determining the median. While it can provide some assistance in determining the mode, it does not directly provide the necessary information to calculate the mean. The cumulative frequency table organizes the data, making it easier to identify the median, which is the middle value in the dataset.