find the mean.variance.and standard deviation for the following data 6,12,6,8,10,8,12,8,14,16
a nice calculator can be found at
http://easycalculation.com/statistics/standard-deviation.php
It will verify the answers which you have calculated on your own >wink wink<
To find the mean, variance, and standard deviation for the given data set {6, 12, 6, 8, 10, 8, 12, 8, 14, 16}, follow these steps:
Step 1: Find the Mean (Average)
To find the mean of a set of numbers, you need to sum all the numbers and then divide the sum by the count of numbers in the set. In this case, sum all the numbers and then divide the sum by 10 (since there are 10 numbers in the set).
Mean = (6 + 12 + 6 + 8 + 10 + 8 + 12 + 8 + 14 + 16) / 10 = 100 / 10 = 10
Therefore, the mean of the given data set is 10.
Step 2: Find the Variance
To find the variance, you need to find the squared difference of each data point from the mean, sum these squared differences, and then divide by the count minus 1.
Variance = Σ[(xᵢ - mean)²] / (n - 1)
where Σ represents the sum, xᵢ represents each observation, mean represents the mean of all observations, and n represents the count of numbers in the set.
First, subtract the mean from each data point, square the differences, and sum them:
Variance = [(6 - 10)² + (12 - 10)² + (6 - 10)² + (8 - 10)² + (10 - 10)² + (8 - 10)² + (12 - 10)² + (8 - 10)² + (14 - 10)² + (16 - 10)²] / (10 - 1)
= [(-4)² + 2² + (-4)² + (-2)² + 0² + (-2)² + 2² + (-2)² + 4² + 6²] / 9
= [16 + 4 + 16 + 4 + 0 + 4 + 4 + 4 + 16 + 36] / 9
= 108 / 9
= 12
Therefore, the variance of the given data set is 12.
Step 3: Find the Standard Deviation
The standard deviation is the square root of the variance. We take the square root to ensure that the final value is in the same units as the original data.
Standard Deviation = √(Variance)
Standard Deviation = √(12) ≈ 3.464
Therefore, the standard deviation of the given data set is approximately 3.464.