This thesis is concerned with a design of cumulative scored control chart to control the process mean with a normally distributed quality characteristic. The average run length (ARL) is obtained from the average sample number of sequential probability ratio test (SPRT) on trinomial distribution. Using the analogy between cumulative scored control chart and SPRT for trinomial observations, a procedure is presented to determine four control chart parameters; lower and upper scoring boundaries, action limit, and elastic barrier. The parameters are determined by minimizing the ARL when the process average is out of control with prespecified ARL when the process average is in control. Using Markov chain methods, modified ARL is obtained and it's effects to control chart parameters are also examined.