A sample has a mean of $m=71$ and a standard deviation of $s=3$. In this sample, which of the following statements is true?

• A. The variance of the sample is $71$.
• B. The most frequently occurring value in the sample is $71$.
• C. The average value of the sample data is $71$.
• D. The sum of all sample values is $3$.

Answer: (C)

Verified step by step guidance

1. Understand the definitions of the given statistics: the sample mean (\( m \)) is the average of all sample values, and the sample standard deviation (\( s \)) measures the spread of the data around the mean.
2. Recall that the variance (\( s^2 \)) is the square of the standard deviation, so it is calculated as \( s^2 = s \times s \). Given \( s = 3 \), the variance is \( 3^2 = 9 \), not 71.
3. Recognize that the most frequently occurring value in a data set is called the mode, which is not necessarily equal to the mean (\( m = 71 \)) unless specified. Since no mode is given, we cannot assume it is 71.
4. The sample mean (\( m = 71 \)) represents the average value of the sample data, so the statement 'The average value of the sample data is 71' is true by definition.
5. The sum of all sample values is calculated by multiplying the mean by the sample size (\( n \)), i.e., \( \text{sum} = m \times n \). Since the sample size is not provided, the sum cannot be 3.