Question

In statistics, another common measure of spread is Standard Deviation which uses the squares of the errors instead of absolute values of errors. First, you determine the error by taking each data element, one by one and subtracting the mean. Next you square each of those values individually. Variance (of the sample) is the average squared error (usually called the mean squared error or MSE) of data from the mean. Average the squared errors (add all the squares and divide by the number of elements) to get the Variance. Because we squared the data, it is not in the correct units, so you need to take the square root of the Variance to get Standard Deviation.

Standard Deviation (of the sample) is the square root of the average squared error of data (Variance) from the mean (sometimes called RMS error).
Given a x = [ -1, -1, 2 , 4 ] (average is 1 and median is .05)

Which of the below answers provides the Standard Deviation for x?

Answer 1

The standard deviation for the dataset x is approximately 2.45, calculated by finding the mean of the squared deviations from the mean (variance) and then taking the square root of the variance.

Step-by-step explanation:

Calculating Standard Deviation

To calculate the standard deviation of the dataset x = [ -1, -1, 2, 4 ], we must first compute the variance. The variance is the mean of the squared deviations from the mean. Here are the steps to follow:

1. Calculate the mean of the dataset, which is 1.
2. Compute the deviations by subtracting the mean from each data point: (-1-1), (-1-1), (2-1), (4-1).
3. Square each deviation: (2)^2, (2)^2, (1)^2, (3)^2.
4. Add the squared deviations: 4 + 4 + 1 + 9 = 18.
5. Since we have a sample, divide by the number of data points minus one, n-1. In this case, it is 4-1=3. So, the variance (s²) is 18/3 = 6.
6. Take the square root of the variance to get the sample standard deviation (s), which is √6 ≈ 2.45.

The standard deviation of the dataset x is approximately 2.45.