Bell Curve, Normal Distribution Defined
By Deb Russell
Definition: The term bell curve is used to describe the mathematical concept called normal distribution, sometimes referred to as Gaussian distribution. ‘Bell curve’ refers to the shape that is created when a line is plotted using the data points for an item that meets the criteria of ‘normal distribution’. The center contains the greatest number of a value and therefore would be the highest point on the arc of the line. This point is referred to the mean, but in simple terms it is the highest number of occurences of a element. ( statistical terms, the mode). The important things to note about a normal distribution is the curve is concentrated in the center and decreases on either side. This is significant in that the data has less of a tendency to produce unusually extreme values, called outliers, as compared to other distributions. Also the bell curve signifies that the data is symetrical and thus we can create reasonable expectations as to the possibility that an outcome will lie within a range to the left or right of the center, once we can measure the amount of deviation contained in the data . These are measured in terms of standard deviations. A bell curve graph depends on two factors, the mean and the standard deviation. The mean identifies the position of the center and the standard deviation determines the the height and width of the bell. For example , a large standard deviation creates a bell that is short and wide while a small standard deviation creates a tall and narrow curve.
To understand the probability factors of a normal distribution you need to understand the following ‘rules’:
1. The total area under the curve is equal to 1 (100%)
2. About 68% of the area under the curve falls within 1 standard deviation.
3. About 95% of the area under the curve falls within 2 standard deviations.
4 About 99.7% of the area under the curve falls within 3 standard devations.
Items 2,3 and 4 are sometimes referred to as the ‘empirical rule’ or the 68-95-99.7 rule. In terms of probability, once we determine that the data is normally distributed ( bell curved) and we calculate the mean and standard deviation, we are able to determine the probability that a single data point will fall within a given range of possibilities.
A good example of a bell curve or normal distribution is the roll of two dice. The distribution is centered around the number 7 and the probability decreases as you move away from the center.
Here are the % chance of the various outcomes when you roll two dice.
2 – 2.78 % 8 – 13.89 %
3 – 5.56 % 9 – 11.11 %
4 – 8.33 % 10- 8.33 %
5 – 11.11% 11- 5.56 %
6 – 13.89% 12- 2.78 %
7 – 16.67%
Normal distributions have many convenient properties, so in many cases, especially in physics and astronomy, random variates with unknown distributions are often assumed to be normal to allow for probability calculations. Although this can be a dangerous assumption, it is often a good approximation due to a surprising result known as the central limit theorem. This theorem states that the mean of any set of variates with any distribution having a finite mean and variance tends to the normal distribution. Many common attributes such as test scores, height, etc., follow roughly normal distributions, with few members at the high and low ends and many in the middle.