What is the total area under the normal curve

General
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what is the total area under the normal curve

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What is the total area under the normal curve?

Answer:

The total area under the normal curve, also known as the Gaussian or bell curve, is equal to 1. This value represents the fact that the normal distribution is a probability distribution and the area under the curve corresponds to the total probability, which must always sum to 1.

Understanding the Normal Curve:

  1. Definition and Characteristics:

    • The normal distribution is a continuous probability distribution that is symmetrical around its mean.
    • It is characterized by its mean (\mu) and standard deviation (\sigma).
    • The curve follows the equation:
    f(x) = \frac{1}{{\sigma \sqrt{2\pi}}} e^{- \frac{(x - \mu)^2}{2\sigma^2}}

  2. Properties of the Normal Distribution:

    • Mean, Median, and Mode are equal and located at the center of the distribution.
    • The curve is bell-shaped and extends indefinitely in both directions.
    • It is described by the Empirical Rule, which states:
      • Approximately 68% of the data falls within one standard deviation (\mu \pm \sigma) of the mean.
      • Approximately 95% of the data falls within two standard deviations (\mu \pm 2\sigma) of the mean.
      • Approximately 99.7% of the data falls within three standard deviations (\mu \pm 3\sigma) of the mean.

  3. Integral and Total Area:

    • The total area under the normal curve is found using integration over its entire range (from -\infty to \infty):
      \int_{-\infty}^{\infty} \frac{1}{{\sigma \sqrt{2\pi}}} e^{- \frac{(x - \mu)^2}{2\sigma^2}} \, dx = 1
    • This integral shows that the total probability (area under the curve) is equal to 1, affirming that it accounts for all possible outcomes.

Visual Representation:

The normal distribution curve can be visualized as a smooth, bell-shaped graph with the following features:

  • The peak represents the mean (\mu).
  • As you move away from the mean, the probability density decreases symmetrically.
  • The tails of the curve approach, but never touch, the horizontal axis.

In summary, the total area under the normal curve is 1, reflecting the fact that it encompasses all possible probabilities for the distribution of values around its mean. This fundamental property is crucial for statistical analyses and applications involving normally distributed data.