Normal Curve

The normal distribution is one of the most important probability distributions in statistics in that many statistical analyses build on the assumption that the data follow the normal distribution, and in that many physical and biological phenomena in real life can be approximated by the normal distribution.

A normal distribution is specified by two parameters: mean µ and standard deviation σ. If a random variable X follows the normal distribution with mean µ and standard deviation σ, it is often denoted by X ∼ N (µ, σ2). The normal distribution has a bell shape as shown below, which is called the normal curve.

Technically, the normal curve is given by the following formula:

The variable x takes any real value. The mean µ specifies the central location of the distribution, and the standard deviation σ determines to what extent the distribution is spread out. The curve has a maximum height at x = µ and is symmetric about µ. In probabilistic terms, the normal curve represents the probability density function of the normal distribution; the height of the curve at each point of x denotes the corresponding probability density. The most important characteristic of a probability density function is that an area under the curve on a specified interval taken on the x-axis represents a probability that the random variable takes values within that interval. In the case of the normal distribution, as shown in Figure 1, the probability that X is within one standard deviation from the mean is approximately 68.2%, and the probability that X is within two standard deviations from the mean is approximately 95.4%.

Figure 1 The Normal Curve.

Especially if µ = 0 and σ = 1, the distribution is called the standard normal distribution, which is frequently used as the reference distribution in statistical testing. A probability table for the standard normal distribution is usually provided in standard statistics textbooks. Any random variable that follows an arbitrary normal distribution can be transformed to the standard normal by subtracting the mean and then dividing by the standard deviation. In other words, if X ∼ N (µ, σ2), then (X-µ)/σ ∼ N (0, 1). Thus, probability values for any normal distribution can be referred to in a probability table for the standard normal distribution.

The central limit theorem exemplifies the theoretical importance of the normal distribution. Suppose that you take a random sample of size n from an arbitrary distribution and compute the mean (or sum) of the sampled values. The central limit theorem states that the distribution of the sample mean (or sum) tends to be approximated by a normal distribution as the sample size n increases, no matter what the [Page 734]original distribution is, as long as it has finite mean and variance. For example, when you flip a coin n times and count the number of heads, the resulting distribution is the binomial distribution with probability .5 and the number of trials n. As n increases, however, the binomial distribution becomes closer to the normal distribution with mean n/2 and standard deviation √n/2.

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