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The main reason a project manager will want to use the PERT Formula is to have an estimate for cost or duration that is accurate. When project managers - especially inexperienced ones - estimate duration, sometimes they choose an amount of time that is too short or too long. The same scenario also happens with cost.
At times, if a formula isn't used for determining the estimated cost, a project manager may budget too little monies for the project. However, initial estimations aren't the only times when the PERT Formula is useful. It is also useful with project crashing and fast-tracking projects.
To learn how to manually calculate using the PERT Formula, check out Deanna Reynolds' article, Project 2007: Estimating Task Duration Using the PERT Formula.