Question

6 from unit 1, lesson 12 the standard deviation for a data set is 0. what can you conclude about the data?

Explanation:

Step1: Recall standard - deviation formula

The formula for the standard deviation $\sigma=\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\mu)^{2}}{n}}$, where $x_{i}$ are data - points, $\mu$ is the mean, and $n$ is the number of data - points.

Step2: Analyze when standard deviation is 0

If $\sigma = 0$, then $\sqrt{\frac{\sum_{i = 1}^{n}(x_{i}-\mu)^{2}}{n}}=0$. Squaring both sides gives $\frac{\sum_{i = 1}^{n}(x_{i}-\mu)^{2}}{n}=0$. Since $n>0$, then $\sum_{i = 1}^{n}(x_{i}-\mu)^{2}=0$. The sum of non - negative terms $(x_{i}-\mu)^{2}$ is 0 only when $x_{i}-\mu = 0$ for all $i = 1,2,\cdots,n$. This means $x_{1}=x_{2}=\cdots=x_{n}=\mu$.

Answer:

All the data values in the data set are equal.