Final answer:
- Percentage within one standard deviation = (14 / 20) * 100 ≈ 70%
- Percentage within two standard deviations = (20 / 20) * 100 = 100%
- Percentage within three standard deviations = (20 / 20) * 100 = 100%
And, 70% of the data points fall within one standard deviation, while all data points fall within two and three standard deviations. The data set appears to be fairly normally distributed.
Step-by-step explanation:
To calculate the percentage of data points that fall within one, two, and three standard deviations of the mean, we need to first calculate the mean and standard deviation of the data set.
Then, we can determine how many data points are within each standard deviation range and calculate the percentage.
Step 1: Calculate the mean of the data set:
Mean = (20 + 12 + 19 + 40 + 22 + 25 + 10 + 24 + 36 + 23 + 15 + 22 + 23 + 13 + 18 + 19 + 17 + 32 + 24 + 21) / 20
Mean = 21.5
Step 2: Calculate the standard deviation of the data set:
Standard Deviation = sqrt(((20 - 21.5)^2 + (12 - 21.5)^2 + (19 - 21.5)^2 + (40 - 21.5)^2 + (22 - 21.5)^2 + (25 - 21.5)^2 + (10 - 21.5)^2 + (24 - 21.5)^2 + (36 - 21.5)^2 + (23 - 21.5)^2 + (15 - 21.5)^2 + (22 - 21.5)^2 + (23 - 21.5)^2 + (13 - 21.5)^2 + (18 - 21.5)^2 + (19 - 21.5)^2 + (17 - 21.5)^2 + (32 - 21.5)^2 + (24 - 21.5)^2 + (21 - 21.5)^2) / 20)
Standard Deviation ≈ 6.51
Step 3: Calculate the percentage of data points within each standard deviation range:
- Within one standard deviation: Count the number of data points that fall within the range of (mean - standard deviation) to (mean + standard deviation). In this case, it is 14 data points.
- Within two standard deviations: Count the number of data points that fall within the range of (mean - 2 * standard deviation) to (mean + 2 * standard deviation). In this case, it is 20 data points.
- Within three standard deviations: Count the number of data points that fall within the range of (mean - 3 * standard deviation) to (mean + 3 * standard deviation). In this case, it is 20 data points.
Step 4: Calculate the percentage:
- Percentage within one standard deviation = (14 / 20) * 100 ≈ 70%
- Percentage within two standard deviations = (20 / 20) * 100 = 100%
- Percentage within three standard deviations = (20 / 20) * 100 = 100%
Based on these calculations, 70% of the data points fall within one standard deviation, while all data points fall within two and three standard deviations. The data set appears to be fairly normally distributed.