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T Chart How-To¶
This walkthrough builds a complete T chart from twenty measured intervals between rare failures. It ticks one box that is not a default, Fit exponential distribution only, and it does so on purpose: with the shape fixed at 1 every line on the finished chart is a plain multiple of the average interval, so you can check the whole chart with a calculator. The last section shows what clearing the box changes.
The data is on this page rather than in a file to download. Press Copy for Excel, then paste it into a blank worksheet.
The data¶
Twenty unplanned stoppages on one pump, in the order they happened. Each value is the number of days the pump ran since the previous stoppage.
| Days Between |
|---|
| 3.5 |
| 1.5 |
| 9.5 |
| 0.7 |
| 6.9 |
| 4.1 |
| 0.5 |
| 6.4 |
| 0.3 |
| 5.1 |
| 0.7 |
| 0.9 |
| 5.0 |
| 15.8 |
| 1.2 |
| 2.3 |
| 8.9 |
| 89.4 |
| 7.7 |
| 4.5 |
Event 18 is the one to watch. Every other interval is under 16 days; that one is 89.4. Note also that these are measurements, not counts: 0.3 of a day is seven hours, and a G chart would have thrown that fraction away.
Steps¶
-
Put the data in Excel
Press Copy for Excel above the table. In Excel, open a blank worksheet, click cell A1, and press Ctrl+V. The header lands in A1 and the twenty intervals in A2:A21.
-
Start the chart
From the Excel ribbon: QXL Stat Tools Tab > Control Charts > Rare Event Charts > T Chart (Time Between Events).
-
Give it the cells
In the Data Selection window set Selected Range: to A1:A21. Leave Data in Columns selected and First Row/Column is Header checked. Press Next >.
-
Check the data column
The T Chart dialog opens on the Data tab. Check Days Between in Data Columns (required). A Continuous column is what this chart wants, and there is no subgroup group on the dialog: one row is one event.
-
Tick the exponential box
On the Limits & Splits tab, tick Fit exponential distribution only.
Why this walkthrough changes a default
Cleared, which is how the dialog opens, Quantum XL fits both Weibull parameters by an iterative solve. Ticked, the shape is fixed at 1 and the scale is simply the average of your intervals, so every number below can be reproduced by hand. Ticking it also clears and greys out Historical Weibull β (shape):, since the shape is no longer being estimated.
-
Finish
Press Finish. The chart is written to a new worksheet named TChart, with no space in the tab name, and the Control Chart analysis pane opens on the right.
What you should see¶
One panel with twenty points, its Y axis titled Time Between Events.
| Value | |
|---|---|
| Weibull α (scale) | 8.745 |
| Weibull β (shape) | 1 |
| CenterLine | 6.06 |
| UCL | 57.78 |
| LCL | 0.01 |
Every one of those numbers is a multiple of the average interval. The twenty intervals average 8.745 days, and with the shape fixed at 1 that average is the scale. Then:
| Line | Multiple of the scale | Value |
|---|---|---|
| Lower control limit | 0.00135 | 0.01 |
| Center line | 0.693, which is \(\ln 2\) | 6.06 |
| Upper control limit | 6.608 | 57.78 |
The center line is a median, not the average. It is 6.06 days where the average is 8.745. Half the intervals should fall below it, and the chart draws that rather than the mean, because the fitted distribution is right-skewed and the median is the middle of it.
The band is wildly asymmetric, and the arithmetic above says why. The upper limit sits 51.7 days above the center line; the lower limit sits 6.05 days below it, and is only 0.01 days off zero. A waiting time cannot be negative and has no ceiling, so the multipliers are 0.00135 on one side and 6.608 on the other. That is the same pair of probabilities a 3-sigma normal band covers, put through a skewed distribution.
Event 18 is out of control, and it is good news. It plots at 89.4 days, above the upper limit of 57.78, and is drawn in a different color. It is the only marked point. On this chart a point above the upper limit is a wait longer than the model expects, which means failures became rarer: the pump ran nearly three months where the fitted rate expects about six days. Nothing can be marked at the bottom, because the lower limit is a hundredth of a day and no interval here comes near it.
Only one test is running
A T chart opens with Rule 1 only. Rules 2, 3 and 4 are present and unchecked, and there is no Rule 9 on this dialog at all: the zero-count test belongs to the G Chart. A T chart with no marked points means nothing broke that single test. See the Rules & Display section of the Options page.
Things to try next¶
- Clear the exponential box and compare. Modify the analysis, clear Fit exponential distribution only, and finish again. Quantum XL now fits both parameters, and the shape it returns will not be exactly 1, so every line moves. Read the shape off the summary table: above 1 says the longer the pump has run the more likely the next stoppage becomes, below 1 says the opposite. That reading is the reason the general fit is the default, and it is not available from the exponential model, which assumes the rate never changes.
- Switch the zone lines on and see how uneven they are. Tick Show 1-sigma / 2-sigma zone lines on the Rules & Display tab. With the exponential fit the upper 1-sigma line lands at 16.10 days and the upper 2-sigma line at 33.08, while the lower two are at 1.51 and 0.20. A thirds rule would have put the upper 1-sigma line at 23.30, seven days from where it actually goes.
- Supply the scale instead of estimating it. With the exponential box still ticked, type
8into Historical Weibull α (scale):. The center line moves to 5.55 and the upper limit to 52.86, and event 18 is still the only marked point. This is the setting for a known long-run rate you do not want a twenty-event record to overrule. - Turn on the pattern tests. Tick rules 2, 3 and 4 and rebuild. Nothing new is marked on this data, so the single Rule 1 signal was not hiding a second pattern.
- Feed it date-times instead of intervals. Put the moment of each stoppage in the column, set its type to DateTime, and pick a Time Unit beside the column: Days, Hours, Minutes or Seconds. Quantum XL differences consecutive stamps and divides by the unit, keeping the fraction, and the Y axis takes the unit's name. Twenty-one stamps give twenty intervals, because the first only establishes the starting point.