Dear student
No, it is not necessary that mean, mean and mode of the data should be near to each other.
If the mean, median, and mode are approximately equal to each other, the distribution can be assumed to be approximately symmetrical. If the mean > median > mode, the distribution will be skewed to the left or negatively skewed. If the mean < median < mode, the distribution will be skewed to the right or positively skewed.
Regards