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A standard normal variable, also known as the standard score or *z-score*, is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations from the mean. In the case of the standard normal distribution, the variable is normally distributed with a mean (μ) of 0 and a standard deviation (σ) of 1. The standard normal distribution is denoted as Z \~ N(0,1). It is foundational in statistics for hypothesis testing, confidence interval estimation, and many inferential procedures. --- Important Properties of a Standard Normal Variable: 1. Bell-Shaped Curve: The standard normal distribution is symmetric and bell-shaped. The highest point of the curve corresponds to the mean (z = 0), and the curve approaches the x-axis but never touches it, extending infinitely in both directions. 2. Mean, Median, and Mode are Equal: For a standard normal distribution, the mean (μ), median, and mode are all equal and located at 0. 3. Standard Deviation Equals 1: The dispersion of the distribution is determined by the standard deviation, which in the standard normal distribution is 1. This implies that 68% of the data lies between z = –1 and z = 1. 4. Total Area Under the Curve Equals 1: The total probability or area under the standard normal curve is 1. This means that any segment under the curve represents a probability that falls within the \[0,1] interval. 5. Symmetry: The curve is perfectly symmetric around the mean. Thus, the area to the left of z = 0 equals the area to the right of z = 0, i.e., 0.5. 6. Empirical Rule: According to this rule: * About 68% of data falls within one standard deviation (z = –1 to z = 1). * About 95% falls within two standard deviations (z = –2 to ____ ________ _________ __________ ________ _________ _____.
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