p chart

Each point is a proportion: the items that failed in a period divided by the items checked in it. The centre line is the overall proportion across all periods. The limits are three standard errors either side of it, and because the standard error depends on how many items were checked, the limits step in and out from one period to the next.

That stepping is the point of the chart. A failure rate of 15% means much more when it comes from 2,000 items than from 20, and the p chart builds that into its limits so you do not have to judge it by eye.

Use it for pass or fail data where each item either has the problem or does not, and the number checked varies: claims returned, appointments missed, orders sent late, forms with an error. Each item counts once, however many things are wrong with it.

If you count every error on an item rather than whether the item failed, you have defect counts, and a c or u chart is the right one. If the number checked is the same every period, an np chart plots the count directly and is easier to explain. And if the proportion is very small, with most periods showing no failures at all, the chart says little; chart the time or the number of items between failures instead.

The centre line is the total failures divided by the total checked, across all periods. For each period, the standard error is the square root of the centre line times one minus the centre line, divided by that period's number checked. The limits are the centre line plus and minus three of those standard errors. A lower limit below zero is set to zero.

A benefits team tracked, each week for 16 weeks, how many claims it processed and how many it had to return to the claimant because information was missing. The figures are illustrative, built to show the method.

Across all 16 weeks, 12.4% of claims were returned. Because the weekly volume varied between 187 and 256 claims, the upper limit moved between 18.6% and 19.7%. Week 11 returned 52 of 228 claims, 22.8%, well above its limit of 19.0%: a special cause. No other week is outside its limits, and there is no long run on either side of the centre line.

The useful conversation is about week 11. In a team like this the cause is usually a change upstream: a new version of a form, a guidance note that went out, a partner sending claims in a different way. Find it while people still remember the week. The other weeks, which range from 10.7% to 12.9%, are the same process doing what it does; asking why week 3 was higher than week 2 would be asking about noise.

Three steps: the overall proportion, the standard error for each week, and the limits. Each is stated in words first, then written out, then worked for week 11.

One row per period with the number checked and the number that failed. Add a column for the proportion. Work out the centre line once, then the upper and lower limits on each row from that row's number checked. Draw the proportion as a line, the centre as a flat line, and the limits as stepped lines, not smoothed ones.

It is the baseline in Measure for any failure rate, and in Control it shows whether a fix to the failure rate has held. When a project aims to cut a rate, the p chart is also how you show the change was real: a run of points below the old centre line, or points below the old lower limit.