For the given data set, approximately 85% of the data points fall within one standard deviation of the mean, and all data points fall within two and three standard deviations. This suggests that the data set is approximately normally distributed.
To calculate the percentage of data points that fall within one, two, and three standard deviations of the mean, we need to follow these steps:
1. Find the mean (average) of the data set:
- Add up all the values in the data set: 20 + 12 + 19 + 40 + 22 + 25 + 10 + 24 + 36 + 23 + 15 + 22 + 23 + 13 + 18 + 19 + 17 + 32 + 24 + 21 = 446
- Divide the sum by the number of data points (20 in this case): 446 / 20 = 22.3
- The mean of the data set is 22.3.
2. Find the standard deviation of the data set:
- Calculate the squared difference between each data point and the mean:
- For example, for the first data point (20), the squared difference is (20 - 22.3)^2 = 5.29.
- Sum up all the squared differences: 5.29 + 10.49 + ... + ... = 188.84
- Divide the sum by the number of data points minus 1 (20 - 1 = 19): 188.84 / 19 = 9.94
- Take the square root of the result to find the standard deviation: √9.94 ≈ 3.15
- The standard deviation of the data set is approximately 3.15.
3. Calculate the boundaries for one, two, and three standard deviations from the mean:
- One standard deviation: mean ± (1 * standard deviation) = 22.3 ± (1 * 3.15) = 19.15 to 25.45
- Two standard deviations: mean ± (2 * standard deviation) = 22.3 ± (2 * 3.15) = 15.00 to 29.60
- Three standard deviations: mean ± (3 * standard deviation) = 22.3 ± (3 * 3.15) = 11.85 to 33.75
4. Count the number of data points that fall within each boundary:
- To determine if a data point falls within a boundary, compare it to the lower and upper limits.
- Count the number of data points that satisfy each condition.
5. Calculate the percentage of data points within each boundary:
- Divide the number of data points within each boundary by the total number of data points (20).
- Multiply the result by 100 to get the percentage.
Based on the calculations, we find:
- Percentage of data points within one standard deviation: (17/20) * 100 ≈ 85%
- Percentage of data points within two standard deviations: (20/20) * 100 = 100%
- Percentage of data points within three standard deviations: (20/20) * 100 = 100%