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Let $\{X_{i}, i\in J\}$ be a family of locally dependent non-negativeinteger-valued random variables with finite expectation and variance. Weconsider the sum $W=\sum_{i\in J}X_i$ and establish general error upper boundsfor the total variation distance $d_{TV}(W, Y^{d})$, where $Y^{d}$ is thediscretized normal distribution. The major ingredient of the proof is toapproximate $W$ by a three-parametric intermediate random variable $M$ based onStein's method. As applications, we study in detail four well-known examples,which are counting vertices of all edges point inward, birthday problem,counting monochromatic edges in uniformly colored graphs, and triangles in theErd\H{o}s-R\'{e}nyi random graph. Through delicate analysis and computations weobtain sharper upper error bounds than existing results.

Discretized Normal Approximation of Sums of Locally Dependent Random Variables via Stein's Method