I recently got into a debate with someone who stated that infantry fighting in loose formation will achieve greater accuracy than fighting in a traditional line formation（if they have the same training and are both infantry). He said that the deciding factor is the amount of smoke generated from the musket affects soldier less because they are standing further away from each other. Hence increase the accuracy of the men hitting the target. Is that so? If the accuracy do increase, by how much?
You might be interested in this previous post discussing musket accuracy. The theoretical performance of even a low-quality musket tended to be vastly higher than what was typically achieved on the battlefield due to fear, smoke, noise, and general chaos. As a result there are hints that hints that soldiers could shoot more accurately in a small skirmish involving just a couple dozen men and fewer distractions than in a pitched battle involving thousands or tens of thousands. This was certainly the opinion of Ardant du Pict in his late 19th century Battle Studies though he was writing during when breach loading rifles had become the typical infantry weapon.
The fire of skirmishers is then the most deadly used in war, because the few men who remain cool enough to aim are not otherwise annoyed while employed as skirmishers. They will perform better as they are better hidden, and better trained in firing.
The accuracy of fire giving advantages only in isolated fire, we may consider that accurate weapons will tend to make fighting by skirmishers more frequent and more decisive.
For the rest, experience authorizes the statement that the use of skirmishers is compulsory in war. To-day all troops seriously engaged become in an instant groups of skirmishers and the only possible precise fire is from hidden snipers.
keep in mind though that this doesn't necissarily mean that skirmishing was any more deadly or inflicted higher casualties.
However, let us not have illusions as to the efficacy of the fire of skirmishers. In spite of the use of accurate and long range weapons, in spite of all training that can be given the soldier, this fire never has more than a relative effect, which should not be exaggerated.
The fire of skirmishers is generally against skirmishers. A body of troops indeed does not let itself be fired on by skirmishers without returning a similar fire. And it is absurd to expect skirmishers to direct their fire on a body protected by skirmishers. To demand of troops firing individually, almost abandoned to themselves, that they do not answer the shots directed at them, by near skirmishers, but aim at a distant body, which is not harming them, is to ask an impossible unselfishness.
As skirmishers men are very scattered. To watch the adjustment of ranges is difficult. Men are practically left alone. Those who remain cool may try to adjust their range, but it is first necessary to see where your shots fall, then, if the terrain permits this and it will rarely do so, to distinguish them from shots fired at the same time by your neighbors. Also these men will be more disturbed, will fire faster and less accurately, as the fight is more bitter, the enemy stauncher; and perturbation is more contagious than coolness.
The target is a line of skirmishers, a target offering so little breadth and above all depth, that outside of point blank fire, an exact knowledge of the range is necessary to secure effect. This is impossible, for the range varies at each instant with the movements of the skirmishers. 49
Thus, with skirmishers against skirmishers, there are scattered shots at scattered targets. Our fire of skirmishers, marching, on the target range, proves this, although each man knows exactly the range and has time and the coolness to set his sights. It is impossible for skirmishers in movement to set sights beyond four hundred meters, and this is pretty extreme, even though the weapon is actually accurate beyond this.