While nominally the aircraft altimeter measures altitude, what it really measures is air pressure, so the needle shows the pressure difference between the altimeter setting source and the aircraft. An altimeter setting source is just a station (usually an aerodrome) that gives out altimeter settings. When you call a station for their "altimeter setting" they tell you a number that instructs you how to set the altimeter to display station elevation at that station. If the air pressure where you are is the same as at the station, your altimeter will read station elevation. For every one inch of mercury ("inches of mercury" is a measure of atmospheric pressure) difference between the pressure at the altimeter setting source and at the airplane, the altimeter displays an altitude gain of one thousand feet.
If the air is at standard temperature, then a column of air a thousand feet deep exerts just enough pressure to equal one inch of mercury. But if the air is colder than standard, it's denser, so a column of a thousand feet of air actually weighs more. Or, equivalently, the column of air sufficient to produce a one inch of mercury change in pressure is less than a thousand feet thick. So an airplane with an altimeter that claims it has climbed a thousand feet over the aerodrome is actually less than a thousand feet over the aerodrome.
I've written about the above before, but yesterday I was in the position of simultaneously knowing the laser-measured, linear height above sea level of the cloud base (5536') and the altitude displayed on a correctly set altimeter at that altitude (5700'). Was the altimeter wrong? No, the altimeter was correctly displaying what the altimeter is legally required to display. This cold weather discrepancy is one that pilots are trained to be aware of, and to compensate for.
There are formulae for those who like that sort of thing, and also a table for those who would rather not have to work everything out from first principles. Neither way is convenient to do in situ, so you work out the corrections ahead of time and scrawl them on a photocopy of the the approach plate. I'm guessing that if you have modern avionics you can tell it the location and temperature of the altimeter setting source and it will work out and display all your new minima for you, right on the EFIS.
Imagine that for safety I was supposed to be at 5536' asl, perhaps because there was a hill on approach. In cold weather, I would want to know how high to fly by my altimeter in order to know that I was really that high. The elevation of the altimeter setting source is 2255' and it was -10 on the ground. So I go to the table in Figure 9.1. (You're welcome to use the formula instead. Let me know how that works out for you.) Minus ten is the second row. You can see that's really not all that cold by chart standards. It is, after all, only October. The height above the elevation of the altimeter setting source is 5536-2255=3281'. I think of that as 3000 + 200 + 81. The correction at -10 are:
81': .4 x 20 = 8'
The example uses the 2000' correction to work out the per-foot correction, but if you look carefully at the -10 row, you'll see it's a 10% correction all the way to 2000, and that's close enough for me.
Three hundred eighteen feet is about double the correction I actually observed, and admittedly not what I was expecting when I started to blog this. Reflecting back I recall that there was a temperature inversion that day, so when we cruising at 8500' it was only zero outside. The table is very conservative, rounding up at every opportunity and assumes a decrease in temperature with altitude. So while it's not a very good example, it is real life. I was thinking of reworking the numbers to make it work out to match the table, but I'm too honest.
Cloud bases on a METAR are given as height above the aerodrome. That value may be measured with a laser ceilometer, reported with reference to the height of a nearby mountain, calculated based on the temperature-dewpoint spread, or transcribed from a PIREP. The METAR never indicates which of these methods has been used, but to properly estimate ones chances of seeing the field at minimums, the ceiling should be compared to the uncorrected minima for the first two case and to the cold-corrected minima for the second. That is, a pilot breaking through a ceiling measured at 5500' asl with a ceilometer could be indicating over 5800' at the relatively mild temperature of -10. With very cold temperatures and high minima, there would be a considerable discrepancy between accurately reported and pilot-observed ceilings. I wonder if Nav Canada has a policy about this, if the FSS applies a correction to PIREPs, or if the METAR/ATIS ceilings fluctuate between pilot reports and observations relative to mountain heights.