# 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,… , yk. Listed below are the steps Minitab uses to compute ANOM results for data with a binomial distribution.

1. Computes the k proportions:
• pi = yi / n (i = 1, 2, …, k)
2. Computes the overall proportion, or the average of the proportions:
• p̅ = Σk i=1 pi / 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 + hα s * Sqrt((k - 1)/ k)
• LDL = p̅ - hα s * Sqrt((k - 1)/ k)

where hα = 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.