Mean, mode and median equal for a random variable Suppose for a random variable mean, mode and median are equal. What is the intuitive meaning of this ?
Notice that the mean is the average of the values the random variable can take, while median is just the middlemost value. Thus if the mean is larger than the median then the distribution is heavy-tailed, that is it assigns higher probability mass to large enough values, and so the the distribution must be positively skewed. Similarly If the mean is less than the median then the distribution is negatively skewed intuitively (although there may may be exceptions).

If mean=median, then obviously the distribution is symmetric. Moreover if mean=mode=median then the distribution is both symmetric and unimodal.
Notice that the mean is the average of the values the random variable can take, while median is just the middlemost value. Thus if the mean is larger than the median then the distribution is heavy-tailed, that is it assigns higher probability mass to large enough values, and so the the distribution must be positively skewed. Similarly If the mean is less than the median then the distribution is negatively skewed intuitively (although there may may be exceptions).

If mean=median, then obviously the distribution is symmetric. Moreover if mean=mode=median then the distribution is both symmetric and unimodal.