Imagine you have 4 numbers and the mean of them is 5.

a , b , c , d  mean is  5. so you must have 4 numbers that the sum of them is equal to 20.

Now I want to suggest these 4 numbers freely. for the first one I say 5

5 + b + c + d = 20 

for next number i suggest 2

5 + 2 + c + d  = 20

for the next number i suggest 0

5 + 2 + 0 + d = 20

now for the fourth number (d) I have not the freedom to suggest a number anymore, because the fourth one (d) must be 13. 

so you have freedom to choose 3 of them minus 1 of them.

so n-1 is the degree of freedom for measuring the mean of a sample form a population.

I think the reaons is, that you don't have the mean of your sample. Instead you use the sample mean. This reduces your degrees of freedom by 1.

Maybe this article is helpful:

http://numoraclerecipes.blogspot.de/2007/12/students-t-distribution-degrees-of.html

The degrees of freedom depend on the number of parameters you are estimating. Thus, from an n-sized sample you have n-1 degrees of freedom if, as it usually happens, you need to estimate the population mean through the sample mean.

Nevertheless, an n-sized sample will lead you to n-k degrees of freedom if you need to estimate k parameters (as it happens in regression models).

@Ana Lopez-Menendez. by the way what is the physical significance of degree of freedom in this context.

The concept of degrees of freedom is often used in mechanics, engineering... refering to the number of parameters than can be fixed for a certain purpose.

Here you have a couple of easy examples:

- If you take a bus with 6 free seats but two of them are reserved for handicapped people, then you have 4 degrees of freedom

- If you go shopping with a fixed budget and you buy presents for your 3 childrens you have 2 degrees of freedom (once you buy for two of them, the remaining quantity is given...).

Imagine you have 4 numbers and the mean of them is 5.

a , b , c , d  mean is  5. so you must have 4 numbers that the sum of them is equal to 20.

Now I want to suggest these 4 numbers freely. for the first one I say 5

5 + b + c + d = 20 

for next number i suggest 2

5 + 2 + c + d  = 20

for the next number i suggest 0

5 + 2 + 0 + d = 20

now for the fourth number (d) I have not the freedom to suggest a number anymore, because the fourth one (d) must be 13. 

so you have freedom to choose 3 of them minus 1 of them.

so n-1 is the degree of freedom for measuring the mean of a sample form a population.

Could anyone explain why the population mean has N degree of freedom, and the sample mean has n-1 degree of freedom?

I agree with Mr. Hassan Jafari. In the data processing, freedom degree is the number of independent data, but always, there is one dependent data which can obtain from other data. So , freedom degree=n-1.

becuse one of them, Sigma(Xi-mean) is definite and it is zero. remember we have n degree of freedom if we have n data. n-1 (Xi-mean)  and Sigma(Xi-mean)

In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary. So, if we have 10 subjects in a group making a sum of 100, the value of the nine subjects can vary and the value of tenth subject is decided depending on the values of nine subjects. So we have freedom to select the value of (10-1)=9 subjects. That's why n-1 degree of freedom. Hope it helps

A relatively simple and intuitive angle to think about this question is: t-statistics contains the term of sample variance (Bessel corrected). To obtain the sample variance, sample mean needs to be calculated, consuming 1 degree of freedom.

Considering that sample variance is a measure of deviation from mean, it makes sense to calculate sample mean as an intermediate step.

There are 11 members in a football team. The coach can suggest these 11 players for goal keeper freely. Coach can continue suggesting the remaining players from player number 2 to no 10. freely. However, to assign the last remaining player , the coach has no other option or freedom except directly placing in no. 11.