Question

determine whether each statement makes sense or does not make sense, and explain your reasoning.
im working with data sets with different means and the same standard deviation.
choose the correct answer below.
a. the statement makes sense. the mean of a data set is completely unrelated to the standard deviation of a data set. thus data sets can have different means and the same standard deviation.
b. the statement does not make sense. if data sets have different means, then they must have different standard deviations.
c. the statement makes sense. data sets can have different means and the same standard deviation. for example, if two data sets contained the values {1,3} and {2,4}, the means for each would be 2 and 3 respectively, but they both have the same standard deviation of approximately 1.41421.
d. the statement does not make sense. data sets can never have the same standard deviation unless they are the same data set, in which case their means would be the same.

Explanation:

The mean is the average of a data set, calculated as $\frac{\sum_{i = 1}^{n}x_{i}}{n}$. The standard deviation measures the spread of data around the mean, calculated as $\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}}$ (for population) or $\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n - 1}}$ (for sample). As shown in option C, different data sets can have different means but the same spread (standard deviation). For example, for data set $\{1,3\}$, mean $\bar{x}_1=\frac{1 + 3}{2}=2$, standard deviation $s_1=\sqrt{\frac{(1 - 2)^{2}+(3 - 2)^{2}}{2 - 1}}=\sqrt{2}\approx1.41421$. For data set $\{2,4\}$, mean $\bar{x}_2=\frac{2+4}{2}=3$, standard deviation $s_2=\sqrt{\frac{(2 - 3)^{2}+(4 - 3)^{2}}{2 - 1}}=\sqrt{2}\approx1.41421$.

Answer:

C. The statement makes sense. Data sets can have different means and the same standard deviation. For example, if two data sets contained the values {1,3} and {2,4}, the means for each would be 2 and 3 respectively, but they both have the same standard deviation of approximately 1.41421.