In this paper, we show series expansion of the ratio on integral of normal probability density function with itself. By investigating approximations to the ratio, we recommend a simple and better approximation formula. Stirling� formula is used to study normal approximations to probability mass function of Poisson, binomial and negative binomial variables. We find that these approximations have similar characteristics. Combining with Chebyshev�s sum inequality, we give some bounds on ratios of sum of probabilities of Poisson, binomial and negative binomial variables with their normal approximations.