Final answer:
The total area under a normal distribution curve represents a 100% probability, indicating the entire range of outcomes for a variable. This area is always equal to one for any normal distribution, including the standard normal distribution with a mean of zero and a standard deviation of one.
Step-by-step explanation:
The total area under the normal distribution curve is equal to a probability of 1. In statistics, this represents the entire range of possible outcomes for a normally distributed variable. Normal distributions are fundamental in statistics and are used to represent real-world variables that tend to cluster around the mean.
Understanding the Normal Distribution
The normal distribution is a continuous probability distribution commonly known as the bell curve due to its shape. A specific case of the normal distribution is the standard normal distribution, which has a mean (μ) of zero and a standard deviation (σ) of one.
Probabilities in a normal distribution are found by calculating areas under the curve. To find the probability that a value falls within a certain range, you use the cumulative distribution function (cdf). For example, if you want to determine the cumulative probability to the left of a z-score of 1.28, you'd find that the area corresponds to approximately the probability of 0.9.
Probabilities can also be represented by statements such as P(X < x), which indicates the likelihood that a random variable X will take on a value less than x. The area to the left of a point on a normal distribution equates to the probability that X is less than that value.