It Doesn’t Clearly Make Sense to Me: The “Sample Mean” in Statistics.

A friend of mine says…

A concept in statistics that does not "clearly" makes sense to me is why in a "sample mean," to determine a specific frequency, that "n-1" is the mean divided by the figures making up that sample, whereas in a "population mean" the divided variable remains as just the variable of "n."

I understand that in a population mean we determine that the mean is divided by the variable "n" to equal "one number," and then to subtract the "one number" by each individual arithmetic number, squared, of the mean divided by the variable "n" to determine a specific frequency.

For example, if in a population mean, the numbered equal answer to determine the mean were 5, and the arithmetic numbers were 4, 5, and 6, and the obvious mean were 3,

...with the equation looking like

(4+5+6/3 = 15/3 factored as 5),

...and to determine the frequency the equation would look like

x= [(4-5)^2+(5-5)^2+(6-5)^2]/ 3 (n),

...why is it that in a sample mean the divided variable in the equation

x= [(4-5)^2+(5-5)^2+(6-5)^2] would be divided by (n-1)?

I understand the correct steps and which equation to use dependent on the question asked, but why (n-1)?

My response…

Here's the skinny! If you calculate the standard deviation of a whole population, your answer is correct no doubt about that.

On the other hand, if you calculate the standard deviation of a sample, you cannot be sure your answer is correct because you don't have all the data. For this reason, mathematicians have found that using a slightly lower number in the denominator (n-1), will raise the standard deviation slightly to compensate for the fact that you don't have all the data. Does that help at all?

My friend’s response…

Wow...Why couldn't they have just said that? They had to use the term "UNBIASED ESTIMATE" instead of the "possibility of an incorrect answer." This is what I gather. Because a whole population can be calculated by standard deviation, the denominator of some figure "n" can be used. However, a sample may not possess enough data for an accurate answer to an equation, therefore instead of calculating the wrong answer and in order to determine if the answer can be checked as correct, they substitute the denominator by a lower number, which is "n-1." Got it, Got it. Thanks a lot.