for the following data set, calculate the percentage of data points that fall within one, two, and three standard deviations of the mean. does the data appear to be normally distributed? show your work 20,12,19,40,22,25,10,24,36,23,15,22,23,13,18,19,17,32,24,21
To determine the percentage of data points that fall within different standard deviations of the mean, we first need to calculate the mean and standard deviation.
1. Calculate the mean:
Mean = (20 + 12 + 19 + 40 + 22 + 25 + 10 + 24 + 36 + 23 + 15 + 22 + 23 + 13 + 18 + 19 + 17 + 32 + 24 + 21) / 20
= 397 / 20
= 19.85
2. Calculate the standard deviation:
- Calculate the squared deviation for each data point:
(20 - 19.85)^2 = 0.0225
(12 - 19.85)^2 = 61.5225
(19 - 19.85)^2 = 0.7225
(40 - 19.85)^2 = 431.2225
(22 - 19.85)^2 = 4.7025
(25 - 19.85)^2 = 27.3025
(10 - 19.85)^2 = 96.1025
(24 - 19.85)^2 = 17.9025
(36 - 19.85)^2 = 266.9025
(23 - 19.85)^2 = 10.3025
(15 - 19.85)^2 = 23.1025
(22 - 19.85)^2 = 4.7025
(23 - 19.85)^2 = 10.3025
(13 - 19.85)^2 = 44.8225
(18 - 19.85)^2 = 3.4225
(19 - 19.85)^2 = 0.7225
(17 - 19.85)^2 = 8.1025
(32 - 19.85)^2 = 148.8225
(24 - 19.85)^2 = 17.9025
(21 - 19.85)^2 = 1.3525
- Calculate the variance:
Variance = (0.0225 + 61.5225 + 0.7225 + 431.2225 + 4.7025 + 27.3025 + 96.1025 + 17.9025 + 266.9025 + 10.3025 + 23.1025 + 4.7025 + 10.3025 + 44.8225 + 3.4225 + 0.7225 + 8.1025 + 148.8225 + 17.9025 + 1.3525) / 20
= 1176.73 / 20
= 58.8365
- Calculate the standard deviation:
Standard deviation = √(58.8365)
= 7.67 (approx.)
3. Determine the number of data points within one, two, and three standard deviations of the mean:
a. Within one standard deviation:
Lower limit = Mean - (1 * Standard deviation) = 19.85 - (1 * 7.67) = 12.18
Upper limit = Mean + (1 * Standard deviation) = 19.85 + (1 * 7.67) = 27.52
Number of data points within one standard deviation = 0 (none of the data points fall within this range)
b. Within two standard deviations:
Lower limit = Mean - (2 * Standard deviation) = 19.85 - (2 * 7.67) = 4.51
Upper limit = Mean + (2 * Standard deviation) = 19.85 + (2 * 7.67) = 35.19
Number of data points within two standard deviations = 20 (all data points fall within this range)
c. Within three standard deviations:
Lower limit = Mean - (3 * Standard deviation) = 19.85 - (3 * 7.67) = -2.16
Upper limit = Mean + (3 * Standard deviation) = 19.85 + (3 * 7.67) = 41.86
Number of data points within three standard deviations = 20 (all data points fall within this range)
4. Calculate the percentage of data points within one, two, and three standard deviations of the mean:
Percentage within one standard deviation = (0 / 20) * 100 = 0%
Percentage within two standard deviations = (20 / 20) * 100 = 100%
Percentage within three standard deviations = (20 / 20) * 100 = 100%
From the calculations, it can be observed that none of the data points fall within one standard deviation of the mean, while all of the data points fall within two and three standard deviations of the mean. This distribution suggests that the data may not be normally distributed.