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 minutes

Explanation:

Step1: Determine the data - set values

The data - set from the dot - plot has 1 value of 2.5, 2 values of 3, 3 values of 4, 1 value of 4.5, and 3 values of 5.

Step2: Calculate the squared distances from the mean for each value

For \(x = 2.5\), \((2.5 - 4)^2=(-1.5)^2 = 2.25\)
For \(x = 3\), \((3 - 4)^2=(-1)^2 = 1\) (and since there are 2 values of 3, the total contribution is \(2\times1 = 2\))
For \(x = 4\), \((4 - 4)^2=0^2 = 0\) (and since there are 3 values of 4, the total contribution is \(3\times0 = 0\))
For \(x = 4.5\), \((4.5 - 4)^2=(0.5)^2 = 0.25\)
For \(x = 5\), \((5 - 4)^2=(1)^2 = 1\) (and since there are 3 values of 5, the total contribution is \(3\times1 = 3\))

Step3: Calculate the sum of squared distances

The sum of squared distances \(S=2.25+2 + 0+0.25+3=7.5\)

Step4: Calculate the standard deviation

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

Answer:

0.87