To determine the correct formula to standardize a value, we need to recall the process of standardization, which transforms data to have a mean of zero and a standard deviation of one. This process is especially useful in statistics and machine learning.
Standardizing a value [tex]\(x\)[/tex] involves the following steps:
1. Subtract the mean ([tex]\(\mu\)[/tex]) of the data set from the original value ([tex]\(x\)[/tex]).
2. Divide the result by the standard deviation ([tex]\(\sigma\)[/tex]) of the data set.
So, the formula for standardization is:
[tex]\[ \text{Standardized Value} = \frac{x - \mu}{\sigma} \][/tex]
Let us evaluate each given option against this formula:
1. [tex]\( \frac{\text{Original value} + \text{Mean}}{\text{Standard deviation}} \)[/tex]
- This formula incorrectly adds the mean to the original value instead of subtracting it. Therefore, it is not correct.
2. [tex]\( \text{Standard deviation} \times (\text{Original value} - \text{Mean}) \)[/tex]
- This formula suggests multiplying the difference between the original value and the mean by the standard deviation, which is incorrect.
3. [tex]\( \frac{\text{Original value} - \text{Mean}}{\text{Standard deviation}} \)[/tex]
- This formula correctly subtracts the mean from the original value and then divides by the standard deviation. This is the correct standardization formula.
4. [tex]\( \frac{\text{Standard deviation}}{(\text{Original value} - \text{Mean})} \)[/tex]
- This formula suggests dividing the standard deviation by the difference between the original value and the mean, which is incorrect.
After evaluating the options, we determine:
The correct formula to standardize a value is:
[tex]\[ \frac{\text{Original value} - \text{Mean}}{\text{Standard deviation}} \][/tex]
Therefore, the correct index among the given choices is the third one.
