Answer :
To solve this problem, let's analyze the characteristics of a standard normal distribution. A standard normal distribution is a special case of the normal distribution, which has specific parameters:
1. Mean (μ): The average or the central value of the distribution.
2. Standard Deviation (σ): A measure of the spread or dispersion of the distribution.
In a standard normal distribution, the values of the parameters are defined as follows:
- The mean (μ) is always equal to 0.
- The standard deviation (σ) is always equal to 1.
Now, let's understand the other variables mentioned:
- x: This represents a data point or a value on the horizontal axis of the distribution. It varies depending on the distribution and does not have a fixed value.
- z: This represents the z-score, which is the number of standard deviations a data point (x) is from the mean (μ). For a standard normal distribution, z can have any real number value, as it measures how far and in which direction a point deviates from the mean.
Given these definitions, we can determine which variable always equals 0:
- μ (Mean): Always 0 in a standard normal distribution.
- σ (Standard Deviation): Always 1 in a standard normal distribution, not 0.
- x (Data Point): Variable and does not always equal 0.
- z (Z-score): Variable and does not always equal 0.
Hence, the correct choice is:
[tex]\[ \mu \][/tex]
The mean (μ) of a standard normal distribution always equals 0.
1. Mean (μ): The average or the central value of the distribution.
2. Standard Deviation (σ): A measure of the spread or dispersion of the distribution.
In a standard normal distribution, the values of the parameters are defined as follows:
- The mean (μ) is always equal to 0.
- The standard deviation (σ) is always equal to 1.
Now, let's understand the other variables mentioned:
- x: This represents a data point or a value on the horizontal axis of the distribution. It varies depending on the distribution and does not have a fixed value.
- z: This represents the z-score, which is the number of standard deviations a data point (x) is from the mean (μ). For a standard normal distribution, z can have any real number value, as it measures how far and in which direction a point deviates from the mean.
Given these definitions, we can determine which variable always equals 0:
- μ (Mean): Always 0 in a standard normal distribution.
- σ (Standard Deviation): Always 1 in a standard normal distribution, not 0.
- x (Data Point): Variable and does not always equal 0.
- z (Z-score): Variable and does not always equal 0.
Hence, the correct choice is:
[tex]\[ \mu \][/tex]
The mean (μ) of a standard normal distribution always equals 0.
