Is a distribution still considered right-skewed if the majority of responses are zero? i have a distribution in which the majority of cases take the value of zero and then there are a few (perhaps 10%) with values of 1,2 or 3. would this distribution still count as right skewed even though it is truncated at zero?  and if so could i log transform it to "normalise" the distribution of the variable for analysis? Thanks!
 A: This is certainly possible. The most common definition for a distribution to be right skewed is that the skewness
$$ \gamma_1 := E\bigg(\Big(\frac{X-\mu}{\sigma}\Big)^3\bigg) $$
be positive.
For instance, the Poisson distribution with parameter $\lambda$ has skewness $\frac{1}{\sqrt{\lambda}}>0$, so it is always right skewed. And for sufficiently small $\lambda$, a majority of the mass could be at zero. If $\lambda<\ln 2$, more than half the mass is at zero.
The same can hold for zero inflated distributions.

Regarding your second question, it is usually not necessary to transform data to be "more" normal (although this is a common misconception), especially not discrete data. You may want to ask a separate question on this topic. If you do so, please explain why you believe your data should be transformed to normality.
A: A note from this 11th grade stats class states:

For a right skewed distribution, the mean is typically greater than the median. Also notice that the tail of the distribution on the right hand (positive) side is longer than on the left hand side.

As noted in the comments of this post, the above statement is a generalization, and not a definition (though the nonparametric skew does, by definition, require the mean to be greater than the median to establish positive skewness).  
