Some weather related mental calcuations I do while flying, listening to ATIS/AWOS/ASOS ( ...uhm, weather):

Freezing level: half of temperature (in Celsius - which is in the METAR) then add three zeros to the result, For example: temp on the surface is 6C degrees, half of it is 3, plus three zeros makes it three thousand feet AGL. So you can expect 0 Celsius degrees at 3000' AGL - that's because the adiabatic lapse rate is about 2 Celsius degrees per thousand feet. By the way, water freezes as 0 C degrees, most of the time.

Ceiling: Temperature minus dew point, then the result times 4 and then add two more zeros to the result. Example: Temp 6 C, dew point 1 C degree. So 6-1=5, then 5*4=20. Now add two more zeros to it to get 2000. So you can expect the cloud base around 2000 feet AGL.

Visibility: If temperature minus dew point difference is 4 celsius degrees or closer, then expect fog, visible moisture. This rule holds true for altitude as well. So if you look at a sounding for an airport and you see the temperature and dew point converging, you can expect clouds at that altitude where the difference is so close. The soundings you can find either at https://www.windy.com, then right click on any airport and select the soundings. The sounding is a cross section of the air, with all kinds of very important information, like winds aloft, temperature, dew point and so on. Another place to find the sounding information is: https://rucsoundings.noaa.gov/. Every ATIS/ASOS/AWOS or other weather information reports the surface temperature and dew point.

Crosswind calculation is just a multiplication problem. So the angle of the wind with respect to the direction of flight / runway is what matters and that is a simple sine function value of that angle. There is no mystery here, these numbers can be memorized or just write them down somewhere and refer to it when needed. So here are the numbers:

So that was the hard part. The rest is just multiplication. Notice there is a column called: "number I use". That is the number I associate with the degree (wind angle) and use for mental calculation, because it is simpler than say 0.93. So the idea here is to get an estimate of the cross wind component. The number I use is really the percent of the wind I mentally think of. So 80 percent of the 12kts wind to me is 9.6kts so with turbulence and wind shear I think of it ast 10kts. Is it accurate, no, but close enough to know if I can handle that crosswind or not - 10 miles out while getting the wether.

So suppose your runway is 36 and the wind is coming from 330 at 8 kts. Well we have a 30 degree between the direction of runway and the wind direction. So the 30 degree sine value is 0.5 - meantally it is a "half". So half of the 8 knots wind is 4 kts, which is the crosswind component of the total wind. (8 * 0.5 = 4)

How about runway 7 and wind is out of the north east from 020 at 12 kts. Now the difference between the wind is 50 degrees, and its sine values is 0.76 which number I use is: 80. At this point I have to mention that the difference between the number I use and the actual (rounded) sine value is 0.04. So 4 hundredth of a 12kts wind is, only 0.48kts, about half a knot crosswind, which won't make a big difference in the landing. When there is a 12kts wind, usually there is turbulence associated with it - quite frankly, I could never nail down that 0.48kts difference. Back to the example: So 12*8 = 9.6 kts crosswind. This can be calculated for wind direction in flight too. The information comes from a winds aloft report along the route and the intended course. The calculation is the same, and can be done in your head in no time.

OK, one more for fun: how about a tail wind? Suppose the heading is 360 and the wind is coming out of the south east 140 at 15kts? The angle is 40 degrees tail wind, which angle I associate with 60. So 60% of 15kts is 9kts. So the crosswind component is 9 kts. Spoiler alert: the tail wind is 12kts.

Tailwind:How did I get the tail wind component? Well, any wind at an agle can be broken down to two parts: a crossind and a head/tail wind component. That could be drawn up as a right angle triangle. The sine value calculates the triangle's opposite side's length, or whatever wind comes against the side of the airplane. The cosine value calculates the adjacent side of the triangle, or wind, which ever is in/opposite of the direction of the flight. To make it short, the cosine value I derived from the same table as above, except now I count backwards on the table for my crosswind angle. So I imagine that the "wind Angle" 80 degree is now as the 10 degree and its "number I use" (20), then the 20 degree falls on what is in the table 70 and "number I use" 90, and so on I count up untill I count to the 40 degrees which I have in this example, and that number falls on the 50 degree in the table above and the "number I use" show 80. So, I take the 80% of the 15kts tail wind and that gives me 12kts. Neat, eh ? The reason the sine value table can be used for the cosine values is because there is a 90 degree shift between the since and cosine functions, so that's why this works - but that's .... math.

This is a two step process for mental calculation. Suppose you fly a Cessna 152 at 100kts at 070 heading with a winds aloft coming from 020 at 12 kts. So how much wind correction should you use to track a straigth course? Well, we calculated, above, the crosswind component to be 9.6kts, so let's just say, that it is 9kts. The airplane is traveling at 100 kts an hour. So how about, 60 minutes divided by 100 kts (60/100 = 0.6) gives us 0.6. So, now multiply your crosswind componnent, 9 kts by 0.6 and you get 5.4. (9 * 0.6 = 5.4). And that is the wind correction angle. So travelling at 100 kts and you have a 50 degree crosswind at 12 knots, your wind correction angle should be about 5 and a half degrees into the wind. In practice, a Cessna 152 DG is so small that 5.4 degrees is very hard to maintain, not to mention turbulence, so over all the 5.4 degree trend is what really matters.

Here is another example: travelling at 120kts, wind at 20 degrees at 15kts. What wind correction should we use?. 20degrees sine number I use is = 30. So, 15 kts * 0.3 = 4.5. So the cross wind is 4.5 kts. Now the speed value is 60 minutes / 120 kts = 0.5 (mentally it is "half"), so half of 4.5 is about 2.2 degrees. So your wind correction angle should be 2.2 degrees into the wind at 120kts speed with a 15kts wind from a 20 degree angle

Headwind: technically, the headwind component can be calculated too. (this is just a repeat of the above explanation with the tail wind) The headwind component would be the cosine value. So that is not a big deal either, because, the relationship between a sine and a cosine function is a 90 degree shift. So the sine value table can be read backwards, would be the cosine values. So looking at the sine value table last one is 80 degrees. Let's pretend that to be 10. So, to find the cosine value of 30 degrees then we just read backwards on that list until you get to 30 degrees which falls on the 60 degree sine() value and the number I use: 0.85 . Let's stick with the example we already calculated: Runway 7 wind 020 at 12 kts. We have already calculated that the crosswind component is 9.6 kts. What is the headwind component? So the angle between the runway and the wind is 50 degrees, so we look at the cosine value of 50 which is 0.64. To get the cosine, value of 50 just count backwards on the table above, as 10, 20, 30, 40, 50 degrees. That cosine 50 degrees value falls on our sine table 40 and the number I use value: 60. Ok, so 60% of the 12 kts wind is= 7.2 kts. And that is the headwind component.

Of course these are rough estimates, but they still provide a good idea of what to expect and where. I use these mental calculations in flight, then when I am listening to the ATIS/ASOS/AWOS. With practice, they are easy and can be done without any computers.