Methods and formulas for one-way designs with normal data in Analysis of Means

Analysis of means is a procedure to determine if individual factor level means are different from the grand mean (mean of all observations in a factor). Listed below are the steps Minitab uses to compute ANOM results for a one-way model:

1. Computes the mean at each factor level, y̅_i. (i = 1, …, r).
2. Computes the grand mean of all observations, y….
3. Computes s_p, an estimate of the standard deviation of observation.
4. Determines the value h_α, which is the corresponding value to the significance level chosen for the test and is used in the upper and lower decision lines.
5. Computes the upper and lower decision limits (UDL and LDL).
6. Plots the means of each factor level with the upper and lower reference lines, and the center line at the grand mean.

Formula

The average of the observations at each factor level. Minitab plots the mean for each factor level on the graph.

Notation

Formula

The average of all observations across factor levels. Minitab uses the grand mean as the center line on the graph.

Notation

Notation

Term	Description
yij	observations at the ith factor level
	mean of observations at the ith factor level
ni	number of observations at the ith factor level

An estimate of the variation across all factor levels. The pooled standard deviation is used to calculate the decision limits.

Formula

Notation

The decision limits indicate whether factor level means are different from the grand mean. Points that lie outside the upper decision limit (UDL) or lower decision limit (LDL) are statistically different from the grand mean.

The calculation of the upper and lower decision limits varies based on the number of levels in the factor and the number of observations at each level.

Two-level factor with an equal number of observations at each level

• UDL = y.. + h_α s_p* Sqrt(1/ n_T)
• LDL = y.. - h_α s_p* Sqrt(1/ n_T)

where h_α = absolute value (t(a / 2, n_T - 2)), s_p = pooled standard deviation, and n_T = total number of observations.

Factor with more than 2 levels with an equal number of observations at each level

• UDL = y.. + h_α s_p* Sqrt[(r-1) / (rn₁)]
• LDL = y.. - h_α s_p* Sqrt[(r-1) / (rn₁)]

where r = number of levels in the factor and n₁ = number of observations at each level.

The degrees of freedom are (n₁- 1) * r.

For values of alpha outside the range of 0.001 and 0.1, the decision limits are:

• UDL = y.. + h_α s_p* Sqrt[(n_T - n₁) / (n_T* n₁)]
• LDL = y.. - h_α s_p* Sqrt[(n_T - n₁) / (n_T* n₁)]

where h_α = absolute value (t(α2, df) and α2 = (1- (1- a )** (1 / r)) / 2 and df = n_T - r.

To obtain the h_α for values of α between 0.001 and 0.1 see Nelson¹.

Factor with 2 or more levels with an unequal number of observations at each level

• UDL_i = y.. + h_α s_p* Sqrt[(n_T - n_i) / (n_T* n_i)]
• LDL_i = y.. - h_α s_p* Sqrt[(n_T - n_i) / (n_T* n_i)]