Methods and formulas for T Chart

Each plotted point, x_i, represents the amount of time between successive events.

Center line (CL)

The center line is the 50^th percentile of the distribution.

Lower control limit (LCL)

Upper control limit (UCL)

Notation

Term	Description
Φ	cumulative distribution function of the standard normal distribution
INV	inverse cumulative distribution function
Kz	distance from the center line expressed in units of the standard deviation of a normal distribution; the value used for Test 1

The T chart analyzes the durations between subsequent events. If the data are entered as dates or times of events, then Minitab first converts the data to number of days between subsequent events. If all durations are greater than zero, then Minitab estimates the distribution parameters using the maximum likelihood method described in Parametric Distribution Analysis.

If one or more durations = 0

A duration of zero indicates that two events occurred simultaneously. If one or more durations is equal to zero, then Minitab uses an alternative method to estimate the parameters.

Let y_i = the duration (in days) between event i and event i – 1. Let x_i be defined as follows:

where Rank(x_i) is the rank, from smallest to largest, of x_i in x.

For each y_i = 0, exclude y_i and x_i from the remaining calculations.

Weibull

To estimate the parameters for the Weibull distribution, first transform both y and x by taking the natural log of each. Then use simple linear regression to fit the model y = β₀ + β₁x. The scale for the Weibull distribution is estimated as exp(β₀) and the shape is estimated as 1 / β₁.

Exponential

The scale for the exponential distribution is estimated as the regression coefficient, β₁, in the simple linear regression equation, y = β₁x. Note that the intercept (β₀) is not fit in this model.