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{\text{sum of squared distances from mean}}{\text{total number of data points}}} )

song length
( dot plot with number line labeled minutes, ticks at 2, 2.5, 3, 3.5, 4, 4.5, 5; dots: 1 at 2.5, 2 at 3, 3 at 4, 1 at 4.5, 3 at 5 )

type the number in the box. round your answer to the nearest hundredth.

Explanation:

Step1: Identify data points from dot plot

From the dot plot:

• 2.5: 1 dot
• 3: 2 dots
• 4: 3 dots
• 4.5: 1 dot
• 5: 3 dots

So data points: \(2.5, 3, 3, 4, 4, 4, 4.5, 5, 5, 5\)

Step2: Calculate squared distances from mean (mean = 4)

• For \(2.5\): \((2.5 - 4)^2 = (-1.5)^2 = 2.25\)
• For \(3\) (two times): \((3 - 4)^2 = (-1)^2 = 1\) (each), total for two: \(2\times1 = 2\)
• For \(4\) (three times): \((4 - 4)^2 = 0^2 = 0\) (each), total for three: \(3\times0 = 0\)
• For \(4.5\): \((4.5 - 4)^2 = (0.5)^2 = 0.25\)
• For \(5\) (three times): \((5 - 4)^2 = (1)^2 = 1\) (each), total for three: \(3\times1 = 3\)

Step3: Sum of squared distances

Sum = \(2.25 + 2 + 0 + 0.25 + 3 = 7.5\)

Step4: Calculate standard deviation

Standard deviation formula: \(\sqrt{\frac{\text{sum of squared distances}}{\text{total number of data points}}}\)
Total data points = 10
So, \(\sqrt{\frac{7.5}{10}} = \sqrt{0.75} \approx 0.87\) (rounded to nearest hundredth)

Answer:

0.87