Difference between Normal distribution and standard normal distribution
视频信息
答案文本
视频字幕
Welcome to our exploration of Normal distributions. Today we'll understand the key difference between a general Normal distribution and the Standard Normal distribution. A Normal distribution is defined by two parameters: the mean mu and the standard deviation sigma. The Standard Normal distribution is a special case where mu equals zero and sigma equals one.
A general Normal distribution is denoted as N of mu sigma squared, where mu is the mean and sigma is the standard deviation. The mean determines the center of the distribution, while the standard deviation controls how spread out the data is. Let's see how changing these parameters affects the shape and position of the curve.
The Standard Normal distribution, denoted as N of zero one, is a special case where the mean is exactly zero and the standard deviation is exactly one. This distribution is also called the Z-distribution and serves as a reference point for all normal distributions. Notice how the curve is perfectly centered at zero, with standard deviation markers at plus and minus one, and plus and minus two.
The key connection between normal and standard normal distributions is the Z-score transformation. Any normal distribution can be converted to the standard normal using the formula Z equals X minus mu divided by sigma. This process is called standardization. For example, if we have N of 2 comma 4, a value of X equals 4 becomes Z equals 1 in the standard normal distribution.
To summarize the key differences: A Normal distribution can have any mean and any positive standard deviation, giving us infinite possibilities. In contrast, the Standard Normal distribution has a fixed mean of zero and standard deviation of one, making it a unique reference point. The Standard Normal distribution serves as the foundation for probability calculations and statistical inference in many applications.