Question

the graph shows a distribution of data. what is the standard deviation of the data? 0.5 1.5 2.0 2.5

Explanation:

Step1: Recall the property of normal distribution

In a normal distribution, about \(68\%\) of the data lies within \(1\) standard deviation (\(\sigma\)) of the mean (\(\mu\)). That is \(P(\mu - \sigma<X<\mu+\sigma)\approx0.68\).

Step2: Identify mean and a value within one - standard - deviation range

From the graph, the mean \(\mu = 2\). A value within one - standard - deviation range: let's take \(x=\mu+\sigma\). We know that when \(x = 2.5\), if \(\mu = 2\), then using the formula \(x=\mu+\sigma\), we substitute \(x = 2.5\) and \(\mu=2\) into it.

Step3: Solve for \(\sigma\)

Substitute into \(x=\mu+\sigma\), we get \(2.5=2+\sigma\).
Subtract \(2\) from both sides of the equation: \(\sigma=2.5 - 2=0.5\).

Answer:

\(0.5\)