Final answer:
To calculate the population variance and standard deviation of the data set 18, 7, 19, 7, 25, 23, we find the mean, compute squared differences, sum them up, and divide by the number of data points. The variance is 50.58, and the standard deviation is approximately 7.1, both rounded to one decimal place.
Step-by-step explanation:
To find the population variance and standard deviation for the given population data set, we will use the following steps:
- First, calculate the mean of the population data set.
- Next, subtract the mean from each data point and square the result.
- Then, sum up all the squared differences obtained in step 2.
- Finally, to get the population variance, divide the sum of squared differences by the number of data points (N).
- The population standard deviation is the square root of the population variance.
The data points given are: 18, 7, 19, 7, 25, 23.
Mean = (18 + 7 + 19 + 7 + 25 + 23) / 6 = 99 / 6 = 16.5
Squared differences (from the mean):
- (18 - 16.5)^2 = 2.25
- (7 - 16.5)^2 = 90.25
- (19 - 16.5)^2 = 6.25
- (7 - 16.5)^2 = 90.25
- (25 - 16.5)^2 = 72.25
- (23 - 16.5)^2 = 42.25
Sum of squared differences = 2.25 + 90.25 + 6.25 + 90.25 + 72.25 + 42.25 = 303.5
Population variance = 303.5 / 6 = 50.58 (rounded to one decimal place)
Population standard deviation = √50.58 ≈ 7.1 (rounded to one decimal place)