Which of the following is not a characteristic of the $\chi ^2$ distribution?

• A. It is always positively skewed, especially for small degrees of freedom.
• B. As the degrees of freedom increase, the distribution becomes more symmetric.
• C. It can take on negative values.
• D. Its mean is equal to its degrees of freedom.

Answer: (C)

Verified step by step guidance

1. Understand the nature of the chi-square distribution: it is a continuous probability distribution that arises from the sum of the squares of independent standard normal random variables.
2. Recall that the chi-square distribution is defined only for non-negative values because it is based on squared terms, so it cannot take on negative values.
3. Review the shape characteristics: the distribution is positively skewed, especially when the degrees of freedom (df) are small, and it becomes more symmetric as df increases.
4. Remember the mean of the chi-square distribution is equal to its degrees of freedom, i.e., \(\text{mean} = df\).
5. Identify the statement that contradicts these properties: since the chi-square distribution cannot be negative, the statement 'It can take on negative values' is not a characteristic of the chi-square distribution.