Suppose you have a dataset {x} = {x1, x2, x3, x4} consisting of 4 items. You know that x1 = 5 and x2 = 10 and that after standardization ^x1 = 0 and ^x2 = 0.5. Find mean {x} and st... Suppose you have a dataset {x} = {x1, x2, x3, x4} consisting of 4 items. You know that x1 = 5 and x2 = 10 and that after standardization ^x1 = 0 and ^x2 = 0.5. Find mean {x} and std {x}. Find x3 and x4 given that x3 ≤ x4. Read more

Steps to Solve

1. Calculate the Mean

To find the mean ($\mu$) of the dataset, sum all the values and divide by the number of values.

$$ \mu = \frac{x_1 + x_2 + x_3 + x_4}{n} $$

where $n$ is the total number of values in the dataset.

2. Calculate the Standard Deviation

Next, calculate the standard deviation ($\sigma$). First, find the variance by averaging the squared differences from the mean.

$$ \sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + (x_3 - \mu)^2 + (x_4 - \mu)^2}{n} $$

Then, take the square root of the variance to get the standard deviation.

$$ \sigma = \sqrt{\sigma^2} $$

3. Standardization of the Values

To standardize the values, use the formula:

$$ z_i = \frac{x_i - \mu}{\sigma} $$

This converts the original values into standardized scores (z-scores).

4. Derive (x_3) and (x_4)

If you know the standardized values for (x_3) and (x_4) (let's denote them as (z_3) and (z_4)), you can find (x_3) and (x_4) using the inverse of standardization:

$$ x_i = z_i \cdot \sigma + \mu $$

for (i = 3, 4).