If $X_1, X_2, X_3...... X_n$, are $n$ observations in a sample then which of following is/are TRUE?

(A) The mean $\bar X$ has $n$ degree of freedom
(B) The mean $\bar X$ has $(n-1)$ degree of freedom
(C) The standard deviation of the sample has $(n-1)$ degree of freedom.
(D) The standard deviation of the sample has $n$ degree of freedom.

Choose the correct answer from the options given below:

(A) and (C) only

The correct answer is Option (1) → (A) and (C) only **

For a sample of $n$ observations:

The sample mean $\bar{x}$ is calculated using all $n$ data values freely.

Therefore, the degrees of freedom for the mean = $n$.

The sample standard deviation uses one constraint (the mean), so only $n-1$ values are free to vary.

Therefore, the degrees of freedom for standard deviation = $n-1$.

The correct statements are (A) and (C).