Methods and formulas for binomial data in Analysis of Means

Use a binomial distribution to represent data that consists of the number or proportion of observations that have a particular attribute. Data include samples of equal size (n) from k populations. The number of observations having the attribute of interest in each of k samples are denoted as y₁, y₂,… , y_k. Listed below are the steps Minitab uses to compute ANOM results for data with a binomial distribution.

1. Computes the k proportions:
   • p_i = y_i / n (i = 1, 2, …, k)
2. Computes the overall proportion, or the average of the proportions:
   • p̅ = Σ^k _i=1 p_i / k
3. Compute an estimate of the standard deviation of the proportion:
   • s = Sqrt [p̅(1 -p̅) / n

   where n = number of observations.

4. Determine the decision lines at significance α:
   • UDL = p̅ + h_α s * Sqrt((k - 1)/ k)
   • LDL = p̅ - h_α s * Sqrt((k - 1)/ k)

   where h_α = inverse cumulative probability of α2 for the standard normal distribution, where α2 = 1 -α / (2 * k).

   If the number of rows in your response column (k) equal 2, then α2 = 1 -α / 2

5. Plots the proportions with the decision lines and the center line.