Background
Currently, there are four company speicals related to battle stats, which can be found in torn wiki:

Furniture Store: 3* Heavy Lifting Special: About 4.5E worth of strength gains in your current gym per job point.
Mining Corporation: 5* Rock Salt Special: About 4.5E worth of defense gains in your current gym per job point.
Television Network: 10* Press Pass Special: (about .25% to a random stat.)
Fitness Center: 5* Roid Rage Special: About 4.5E worth of strength gains in your current gym per job point.

Besides, Logistics Management: 1* Efficiency: exchange for speed

I used to save my job points to wait for opening gym with higher training ratio. However, recently one of my friends told me that he found he always gained nearly the same stats no matter which gym he stayed in.
In a futher discussion, we found there are also posts discussing about the interesting problems, players have confirmed these ground truths[5* Mining special - Rock Salt]:

1. The stats exchanged by job points are not affected by your active gym.
2. The stats exchanged by job points are not affected by current happy.

We can also infer that:

1. The stats exchanged by job points are not affected by any other perks from private island, education or book reading.

The problem left for us to answer is that: how can we estimate the stats gained by using job points?
Should I use them as early as possible? Or should I firstly accumulate large stats due to the removal of gym cap last year in Patch#250:

Applied this same system to the 4 stat-boosting Company specials so they can also utilize exponential growth

That’s where we started. I collected these data from my friends with defense in a range between 10k to 2b.

Methodology
To compare the efficiency of the stats exchanging, we first need a uniform criteria.
And here is the assumption:

1. Set George’s as active gym, which has 7.3 training ratio for all stats and takes 10 energy for each train.
2. Set default happiness to 5025.(According to Vladar’s training formula[Training Formula V2.0], the happiness means a lot to low stats, we will see its effect in the following)
3. No other perks.

Now we could calculate the energy worth for job points for different stats.
Let us see an example:

You used 1 job points on Rock Salt increasing your defense by 53.62 to 14,765.63

In this case, we use 1 job points to increase 53.62 defense from 14712.01.
In order to to calculate how much energy is worthy for these stats. We have a simple algorithm for it:

1. Base on Valdar’s training simulator, we first fill in parameters and train one time.
   
   
2. it is obviously 53.62 > 29.53, then we do the next train.
3. After conducting the second train, we gain more stats than that exchanged by job points. So we calculate the percentage: (53.62-29.53)*10/29.54=8.15. That means we need 10+8.15=18.15E to gain 53.62 defense.
4. Finally, we divide the point used for exchanging: 1JP=18.15/1=18.15E.

Result
WIth the algorithm, we conducted the analysis on other logs and the result is shown as below:
Since the battle stats have a great range, we use log10 to shrink it. What can we find from the graph?

1. When you have low stats, the stats exchange ratio can be extremely high.
2. After 50m stats, the exchange ratio is going down and comes close to a constant value(nearly 1:4.53)

You may wonder what will happen if we set happy as basic 100? Here is the result graph:
The ratio for low stats becomes so high. It's natrual if you are familiar with valdar's formula: happiness is extremely important for low stats training.

Corner Case
1. What will happen to different stats? As is shown in Valdar's forumula, different stats have different modifier parameters. Besides, are there any differences between company specials?
Thanks to data provided by colaman32[2091413], now we can calculate exchange ratio for different stats and company specials. Here are brief conclusion:

• Different stats do not show differences in the exchange ratio.
• At least for Logistics Management and Mining Corporation, we can assume they apply the same formula. We can further guess all company specials work in the same way.

TL;DR
Now we can answer our questions at the beginning.
The exchange ratio for company specials in stats keeps growing low as your battle stats grow. We also gain some useful conclusion:

• As a newbie with few cash and nearly zero stat, the company specials can help them grow actually fast. (I do not advertise for my company, it's full, lmao)
• If you are in the company with specials to exchange stats, just use it as early as possible. No need to wait for higher stats. It's worthy and helpful.

Besides, I also notice Army provides with the exchange for strength and defense. I simply calculate it base on some data collected from forum, and find out: the exchange ratio for Army is about half of companies for now. It should be affected by the rebuilding in Patch#250.

Acknowledgement
Much thanks to contr4l_[2893026] and zaiyiding[2722372], we have discussed together for this question. And lots of friends who provide with the detailed logs. Without their helps, I could not write out this tutorial.


There are still problems remaining for us to answer:
(1) Is there anyway to summarize accurate formula for job points exchanging?
(2) Due to the rough analysis, I could not guarantee the precision absolutely. Will the conclusion still work over larger datasets?
(3) Would the exchange ratio be affected by user's gym progress? Though active gym does not affect the ratio, what does a newbie could gain by the exchange?

If you have any suggestions or questions, let us discuss about that :D

Update History
09/08/23
1. Add LM to company specials list.
2. Update extra 170 data points. (167 coming from a gradient usage of job points in Mine Corp and 3 comes from colaman32[2091413]).