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 x - axis labeled minutes, ticks at 2, 2.5, 3, 3.5, 4, 4.5, 5; 1 dot at 2.5, 2 dots at 3, 3 dots at 4, 1 dot at 4.5, 3 dots at 5

Explanation:

Step1: Count data points

• 2.5: 1 point
• 3: 2 points
• 4: 3 points
• 4.5: 1 point
• 5: 3 points

Step2: Calculate squared distances

• For 2.5: $(2.5 - 4)^2 = (-1.5)^2 = 2.25$ (1 point: $1\times2.25 = 2.25$)
• For 3: $(3 - 4)^2 = (-1)^2 = 1$ (2 points: $2\times1 = 2$)
• For 4: $(4 - 4)^2 = 0$ (3 points: $3\times0 = 0$)
• For 4.5: $(4.5 - 4)^2 = (0.5)^2 = 0.25$ (1 point: $1\times0.25 = 0.25$)
• For 5: $(5 - 4)^2 = (1)^2 = 1$ (3 points: $3\times1 = 3$)

Step3: Sum squared distances

Sum = $2.25 + 2 + 0 + 0.25 + 3 = 7.5$

Step4: Apply standard deviation formula

Standard deviation = $\sqrt{\frac{7.5}{10}} = \sqrt{0.75} \approx 0.87$

Answer:

0.87