The exact answer to the question is straightforward: the total area under the curve in a distribution is 1 or 100%.
This fundamental concept applies to all continuous probability distributions, such as the normal distribution (often depicted as a bell curve). The area under the curve represents the total probability of all possible outcomes occurring. Since the sum of probabilities for all possible outcomes must be equal to 1 (or 100%), the total area is normalized to this value.
As referenced, the total area under the curve is 1 or 100%.
Understanding the Area Under the Curve
The area under the curve is a graphical representation of probability. For any range of values along the horizontal axis (the variable's values), the area under the curve between those values represents the probability that the variable will fall within that specific range.
- Total Area: Represents the probability of any outcome occurring, which is always 1 or 100%.
- Partial Area: Represents the probability of outcomes within a specific interval along the distribution.
Why is the Total Area 1 or 100%?
In probability theory, the sum of all possible probabilities must equal 1 (or 100%). Since a probability distribution curve encompasses all possible values of the variable, the area under the entire curve corresponds to the probability of any value occurring within its range. Therefore, the total area is standardized to 1.
This normalization makes the area directly interpretable as a probability.
Z-Scores and Associated Probabilities
Within a standard normal distribution (a specific type of distribution curve with a mean of 0 and a standard deviation of 1), z-scores are used to locate specific points.
As stated in the reference: "Every z score has an associated p value that tells you the probability of all values below or above that z score occurring. This is the area under the curve left or right of that z score."
- A z-score indicates how many standard deviations a particular value is away from the mean.
- A p-value associated with a z-score is the area under the curve to the left or right of that z-score. This area represents the probability of observing a value as extreme as or more extreme than the one corresponding to the z-score.
For example:
- The area under the curve to the left of a z-score of 0 is 0.5 (or 50%), representing the probability of a value being less than the mean.
- The total area under the curve between two z-scores represents the probability of a value falling within that range.
| Z-score | Area to the Left (Approx.) | Area to the Right (Approx.) |
|---|---|---|
| -2 | 0.0228 | 0.9772 |
| 0 | 0.5000 | 0.5000 |
| 2 | 0.9772 | 0.0228 |
Note: The sum of the area to the left and the area to the right for any z-score always equals 1.
Key Takeaway
The concept that the total area under a probability distribution curve equals 1 is foundational. It ensures that the curve accurately represents probabilities, where any part of the area corresponds to the probability of the variable falling within a specific range.
