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:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (C) only
The degrees of freedom of a statistic are the number of observations that can vary freely: n minus the number of constraints on them.
The mean \(\bar X\) is computed from the n observations with no restriction on them, so all n are free.
The standard deviation is computed from the deviations \(X_i - \bar X\), which must satisfy one constraint:
\( \displaystyle \sum_{i=1}^{n} (X_i - \bar X) = 0 \)
Once \(n - 1\) deviations are chosen, the last one is fixed.
So the mean has n degrees of freedom and the standard deviation has n − 1.
Checking the statements
(A) The mean \(\bar X\) has \(n\) degree of freedom — Correct. No constraint, so all n values are free.
(B) The mean \(\bar X\) has \((n-1)\) degree of freedom — Incorrect. The mean has n, not \(n - 1\).
(C) The standard deviation of the sample has \((n-1)\) degree of freedom. — Correct. The deviations add up to 0, which removes one.
(D) The standard deviation of the sample has \(n\) degree of freedom. — Incorrect. It has \(n - 1\), because of the constraint on the deviations.