A standard normal distribution, often denoted as Z, is a specific type of normal distribution. It's characterized by two key parameters:
Mean (μ): The mean of the standard normal distribution is always 0. This means the distribution is centered around zero on the x-axis.
Standard Deviation (σ): The standard deviation of the standard normal distribution is always 1. This indicates the spread or dispersion of the data around the mean.
Key Properties:
Symmetry: The standard normal distribution is perfectly symmetrical around its mean (0).
Unimodal: It has a single peak at the mean.
Total Area: The total area under the standard normal curve is equal to 1, representing the total probability.
Empirical Rule (68-95-99.7 Rule): Approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. This is a key characteristic of normal distributions in general, but is especially important for the standard normal.
Probability Calculation: The area under the curve between any two points on the x-axis represents the probability of a value falling within that range. These probabilities are often found using a Z-table or statistical software. The values in the Z-table represent the cumulative probability to the left of a given z-score.
Z-score: A Z-score represents the number of standard deviations a given data point is away from the mean. It's calculated as: Z = (X - μ) / σ, where X is the data point, μ is the mean, and σ is the standard deviation. The standard normal distribution is the distribution of z-scores.
Applications:
The standard normal distribution is fundamental in statistics because:
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