The normal distribution is a continuous probability distribution that forms a bell-shaped curve. It is symmetrical around its mean, denoted by mu, which determines the center of the curve. The standard deviation, denoted by sigma, determines how spread out the curve is. The normal distribution is completely defined by these two parameters. The mathematical formula for the normal distribution is shown at the bottom of the graph. This distribution is fundamental in statistics and appears frequently in natural phenomena.
The standard normal distribution is a special case where the mean is 0 and the standard deviation is 1. We can transform any normal distribution into the standard normal distribution using Z-scores. A Z-score tells us how many standard deviations a data point is from the mean. For example, a Z-score of 2 means the value is 2 standard deviations above the mean. The standard normal distribution has important probability properties: 68% of the data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. This is known as the empirical rule or the 68-95-99.7 rule.
The shape of a normal distribution is determined by its parameters. The mean, μ, controls the center of the distribution. Changing the mean shifts the entire curve horizontally without affecting its shape. Here we see curves with means at -2, 0, and 2. The standard deviation, σ, controls the spread or width of the distribution. A larger standard deviation creates a wider, flatter curve, while a smaller standard deviation creates a narrower, taller curve. Notice how the purple curve with σ=2 is wider and flatter, while the orange curve with σ=0.5 is narrower and taller. Regardless of the parameters, the total area under any normal curve always equals 1, representing 100% probability.
The normal distribution, also known as the Gaussian distribution, is a probability distribution that forms the famous bell curve. It's defined by two parameters: the mean, which determines the center of the curve, and the standard deviation, which controls its width. The mathematical formula shown here gives us the probability density at any point x. The Central Limit Theorem explains why the normal distribution appears so frequently in nature and data analysis. It states that when we take many independent random samples from any distribution and calculate their means, the distribution of these sample means will approach a normal distribution. In our animation, we start with a uniform distribution that's completely flat. As we increase the sample size, watch how the distribution of sample means gradually transforms into the bell-shaped normal curve predicted by the theorem. This fundamental principle underlies statistical inference, measurement error analysis, and many natural phenomena.
Let's summarize what we've learned about the normal distribution. The normal distribution is a symmetric, bell-shaped curve completely defined by two parameters: the mean and standard deviation. The standard normal distribution has a mean of zero and standard deviation of one, with Z-scores telling us how many standard deviations a value is from the mean. According to the empirical rule, 68% of data falls within one standard deviation of the mean, 95% within two, and 99.7% within three. Changing the mean shifts the curve horizontally, while changing the standard deviation affects its width and height. The Central Limit Theorem explains why the normal distribution appears so frequently in nature and data analysis. The normal distribution has countless applications across fields including statistics, quality control, finance, natural sciences, social sciences, and engineering. Its mathematical properties make it an essential tool for modeling random phenomena and conducting statistical inference.