The standard normal distribution –
## Core Concept
The standard normal distribution, also known as the Gaussian distribution or bell curve, is a probability distribution with a mean of 0 and a standard deviation of 1. It is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. In graph form, the standard normal distribution will have 100% of the data falling within three standard deviations of the mean.
## Why the Correct Answer is Right
The correct answer, , represents the standard normal distribution. This is because it has a mean (ΞΌ) of 0 and a standard deviation (Ο) of 1, which are the defining characteristics of a standard normal distribution. The formula for the standard normal distribution is (Z = frac{X - mu}{sigma}), which simplifies to (Z = X) when (mu = 0) and (sigma = 1). This distribution is widely used in statistics for comparing different data sets and for calculating probabilities.
## Why Each Wrong Option is Incorrect
- **Option A:** - This option does not represent a standard normal distribution because its mean is not 0, or its standard deviation is not 1, or both.
- **Option B:** - Similarly, this option does not fit the criteria for a standard normal distribution for the same reasons as Option A.
- **Option D:** - This option also does not represent a standard normal distribution as it likely has a different mean and/or standard deviation.
## Clinical Pearl / High-Yield Fact
A key fact to remember is that about 68% of the data falls within 1 standard deviation of the mean in a standard normal distribution, about 95% falls within 2 standard deviations, and about 99.7% falls within 3 standard deviations. This is often referred to as the 68-95-99.7 rule.
## Correct Answer: .