The Normal Distribution and the Central Limit Theorem

The normal distribution

The normal distribution is a bell-shaped curve. It’s defined by two parameters—the mean and the standard deviation. The value of the mean determines the position where the bell curve is centred (the value of its highest point). The distribution is symmetric about this point, with the width of the bell—the length of its tails—determined by the value of the standard deviation. Large values of the standard deviation produce short, wide bells with long tails, and small values generate tall, narrow curves with short tails. The bell-shaped curve occurs frequently and is very well-known as a result. This is because things often vary when they’re influenced by many small effects. For example, human heights typically have an approximately normal distribution because height is determined by the effects of many genes as well as environmental influences during development. As with all probability distributions, the sum of the area under the bell curve is one. Known proportions of the curve lie within certain distances from the center of the bell curve. The mean defines the center, and the distance from the center is measured in standard deviations. The idealized standard normal distribution (given in the figure below) is defined with a mean of zero and a standard deviation of one. Over two-thirds of the area under the curve (67.8%) lies within ±1 SD, 95% within ±2 SDs, and 99.8% within ±3 SDs.