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 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

From dot plot: 2.5 (1), 3 (2), 4 (3), 4.5 (1), 5 (3). Total \( n = 10 \).

Step2: Calculate squared distances

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

Step3: Sum squared distances

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

Step4: Apply standard deviation formula

\( \text{Standard Deviation} = \sqrt{\frac{7.5}{10}} = \sqrt{0.75} \approx 0.87 \)

Answer:

\( 0.87 \)