Methods and formulas for T² Chart

Data in subgroups

When data are in subgroups, T² is calculated as follows:

where:

is the mean vector of (mean of x_jk values), which is calculated as follows:

S = sample covariance matrix

The sample covariance matrix, S, is calculated as follows:

where:

where:

, the sample variance for the j^th characteristic in the k^th sample, is calculated as follows:

where:

where:

, the covariance, =

The average of the S matrices is an unbiased estimate of the variance when the process is in control. n must be greater than p and there must be no strong correlations between the variables so that the sample covariance matrix is not singular.

When the data are in subgroups, the chart displays a missing value for any subgroup that is an individual observation.

Individual observations

When data are individual observations, T² is calculated as follows:

where:

where:

Notation

Term	Description
n	sample size
	sample mean vector
xijk	the ith observation on the jth characteristic in the kth sample
m	number of samples

Minitab plots the T² statistic on a control chart. If a plotted point exceeds the control limits, the process is out of control at that point. See the table and sample equations for reference on Minitab calculations.

The following data come from a cleaning solution development process. The amounts of sodium citrate and glycerine affect the potency of the solution.

1. Calculate the subgroup means for each variable, X₁ and X₂. In this case, each subgroup had four samples.
2. If you have individual observations, Minitab uses them instead of the subgroup means in all calculations.
3. Calculate the subgroup variances, S_{1 2} and S_{2 2}.
4. Calculate the subgroup covariances, S_{1 2 k}.
5. Calculate the means of the subgroup means, the means of subgroup variances, and the mean of the covariances.
6. Designate the sample covariance matrix S and the mean vector.
7. Calculate T², which is given by:

   

Minitab plots T² on the T² chart and compares it to the control limits to determine if individual points are out of control.

The center line for the T² chart is KX. The calculation of K and X depends on the maximum sample size and whether Minitab estimates the covariance matrix from the data.

Data in subgroups

When data are in subgroups, KX is calculated as follows:

Given covariance matrix

Estimated covariance matrix

Individual observations

When data are individual observations, KX is calculated as follows:

Given covariance matrix

Estimated covariance matrix

where:

Notation

Term	Description
P	number of variables
M	number of subgroups
N	sample size
	the inverse cumulative F distribution with u numerator degrees of freedom and v denominator degrees of freedom
	the inverse cumulative beta distribution with first shape parameter α and second shape parameter β

Data in subgroups

The upper control limit if you do not specify parameters is:

The upper control limit if you specify parameters is:

Individual observations

The upper control limit if you do not specify parameters is:

where:

See Woodall et al.¹ for more information.

The upper control limit if you specify parameters is:

Notation

Decomposed T² statistic:

where:

where:

x_i^{(p − 1)}is the decomposed mean vector

S_xx is the (p – 1) × (p – 1) principal submatrix of S

T²_{p|1,..., p−1} is an approximation that differs for phases and whether you have subgroups or individual observations:

Phase 1 for data in subgroups:

Phase 2 for data in subgroups:

Phase 1 for individual observations:

Phase 2 for individual observations:

Minitab calculates phase 1 control limits when you don't specify parameter estimates and phase 2 control limits when you do.

See Mason et al.² for more information on the Decomposed T² statistic.

Notation

Box-Cox formula

If you use a Box-Cox transformation, Minitab transforms the original data values (Y_i) according to the following formula:

where λ is the parameter for the transformation. Minitab then creates a control chart of the transformed data values (W_i). To learn how Minitab chooses the optimal value for λ, go to Methods and formulas for Box-Cox Transformation.

Common λ values

The following table shows some commonly used λ values and their transformations.

λ	Transformation
2
0.5
0
−0.5
−1