Final answer:
The median of the data set is 10.1 mph. The mean of the data set is 10.26 mph. The population standard deviation of the data set is approximately 0.5339 mph.
Step-by-step explanation:
The median of a data set is the middle value when the data is arranged in ascending or descending order. To find the median of the given data set, we need to arrange the wind speeds in ascending order:
- 9.6
- 9.7
- 9.9
- 9.9
- 10.0
- 10.1
- 10.1
- 10.3
- 10.7
- 10.8
- 11.0
- 11.9
Since we have 12 values in the data set, the median will be the average of the 6th and 7th values, which are both 10.1. Therefore, the median of the data set is 10.1 mph.
The mean or average of a data set is found by summing all the values and dividing by the number of values. For the given data set, the sum of the wind speeds is 123.1 mph (9.6 + 9.7 + 9.9 + 9.9 + 10.0 + 10.1 + 10.1 + 10.3 + 10.7 + 10.8 + 11.0 + 11.9) and there are 12 values. Dividing the sum by 12, the mean of the data set is 10.26 mph.
The population standard deviation is a measure of the spread or dispersion of the data. To calculate it, we need to subtract the mean from each value, square the result, sum them all, divide by the number of values, and take the square root. Using the given wind speeds:
- (9.6 - 10.26)^2 = 0.0576
- (9.7 - 10.26)^2 = 0.3136
- (9.9 - 10.26)^2 = 0.0964
- (9.9 - 10.26)^2 = 0.0964
- (10.0 - 10.26)^2 = 0.0676
- (10.1 - 10.26)^2 = 0.0256
- (10.1 - 10.26)^2 = 0.0256
- (10.3 - 10.26)^2 = 0.0016
- (10.7 - 10.26)^2 = 0.0196
- (10.8 - 10.26)^2 = 0.0324
- (11.0 - 10.26)^2 = 0.0544
- (11.9 - 10.26)^2 = 2.7264
Summing these values gives us 3.4368. Dividing by 12, we get 0.2864. Finally, taking the square root, the population standard deviation of the data set is approximately 0.5339 mph.