Question

the dot plot below shows the lengths of 10 new afrobeats songs, rounded to the nearest half minute. the mean of the data set is 4. what is the standard deviation of the data set, rounded to the nearest hundredth? standard deviation = \sqrt{\frac{sum of squared distances from mean}{total number of data points}} song length <--|----|----|----|----|----|----|----| 2 2.5 3 3.5 4 4.5 5 minutes type the number in the box. round your answer to the nearest hundredth.

Explanation:

Step1: Count data - point frequencies

There is 1 song of length 2.5, 2 songs of length 3, 0 songs of length 3.5, 3 songs of length 4, 1 song of length 4.5, 3 songs of length 5.

Step2: Calculate squared - distances from the mean

For \(x = 2.5\): \((2.5 - 4)^2=(- 1.5)^2 = 2.25\)
For \(x = 3\): \((3 - 4)^2=(-1)^2 = 1\)
For \(x = 4\): \((4 - 4)^2=0^2 = 0\)
For \(x = 4.5\): \((4.5 - 4)^2=(0.5)^2 = 0.25\)
For \(x = 5\): \((5 - 4)^2=(1)^2 = 1\)

Step3: Calculate sum of squared - distances weighted by frequencies

\[

$$\begin{align*} &\text{Sum}=(1\times2.25)+(2\times1)+(0\times0)+(3\times0)+(1\times0.25)+(3\times1)\\ &=2.25 + 2+0 + 0+0.25 + 3\\ &=7.5 \end{align*}$$

\]

Step4: Calculate standard deviation

The total number of data - points \(n = 10\).
The standard deviation \(s=\sqrt{\frac{7.5}{10}}=\sqrt{0.75}\approx0.87\)

Answer:

0.87