To determine the median of a grouped data set, we need to find the midpoint of the data. The median is the value that separates the data into two equal halves, with an equal number of data points on either side.

Given that the mode of the grouped data is 10, it means that the value 10 occurs most frequently in the data set. However, the mode does not provide any information about the position or order of the data points, so it does not directly help us in finding the median.

We are also given that the mean of the grouped data is 4. The mean is the average of all the data points and is calculated by summing all the values and dividing by the total number of values. Therefore, we have:

Mean = Sum of all values / Total number of values

From this information, we cannot directly determine the position of the median either.

To find the median, we need to consider the intervals and the frequencies of the grouped data. The frequency tells us how many times a particular value occurs in the data set.

Since we do not have the complete grouped data or the frequency distribution, it is not possible to determine the exact value of the median. However, based on the given information, we can make an educated guess.

If the mode is 10 and the mean is 4, it is likely that the data is positively skewed, meaning that there are a few larger values that are pulling the mean higher. In this case, the median is expected to be lower than the mean.

Since the mode is 10, it is reasonable to assume that the median is closer to 10 than to any other value. Among the given options, the only value that is closer to 10 than to any other value is 6. Therefore, the correct answer is option C, 6.

However, it is important to note that without the complete data or the frequency distribution, we cannot determine the exact value of the median. The given information only allows us to make an educated guess.