The question might seem basic, but why do juries in the United States consist of 12 members?
Was there an experimental determination of this number?
Would the addition or removal of a juror operate more effectively?
The origin of the jury is a complex mix of Saxon, Danish and Norman custom which morphed and melded along with English Common Law, which is the basis of the law in all ex-British colonies including the USA.
Danish towns in the north and east of England had hereditary “law men”, often 12 in number who decided legal disputes. In parallel the West Saxons (Wessex) in the south and west charged 12 theigns in each area with keeping the peace. When the Normans conquered (who were also of Viking origin) they adopted and adapted the existing legal structures.
In the 12th century, Henry II established that a jury of 12 should decide land disputes. Meanwhile other juries of various sizes were formed to investigate crimes and bring charges - this is the origin of the Grand Juries that still live on in some jurisdictions in the USA although they have been replaced by a judge in other common law jurisdictions (and some states in the USA).
Ultimately the jury that brought charges and the jury that decided guilt were split and the size was standardised. However, it is incorrect to say that it is always 12: some jurisdictions have different numbers for different purposes. For example, rule 48 of the US Federal rules for civil procedure sets the number at not less than 6 and not more than 12.
The jury system continues to evolve with various jurisdictions adopting different numbers, majority verdicts and judge only trials.
There is nothing special or “scientific” about 12: it is what it is because it is what it is.
It isn't 12, for federal trial courts it's at least 6 and no more than 12. https://www.law.cornell.edu/rules/frcp/rule_48 For grand juries it's more https://www.uscourts.gov/services-forms/jury-service/types-juries. In az it's 6,8, or 12 depending on the matter being considered. https://law.justia.com/codes/arizona/2005/title21/00102.html. I can't go through all the states but it's not always 12.
It would not suprise me if the historical establishment of 12 jurors was rooted in Judeo-Christian tradition, based upon the Biblical establishment of the 12 judges of the twelve tribes of Israel in the Old Testament Book of Judges, to wit the New Testament Book of Revelations declares that the 12 Apostles of Jesus will sit upon 12 thrones judging the 12 tribes of Israel. The purpose of both references was to show how sin and evil can bring bondage into your life, yet showing God’s faithfulness to deliver His people when they turn to Him. I cannot help but think the number 12 seated in judgment over the affairs of an accused is rooted in Biblical text.
I don't have any legal schooling or knowledge, however the first thought that springs to my mind is that the removal or addition of a jury-member would create an odd number. This in turn might create less hung juries.
So to answer your last question, would the addition or removal of an jury member make the jury more effective, I think it might.