What Is a Bell Curve in Math and Science?

The term bell curve is used to describe the mathematical concept called normal distribution, sometimes referred to as Gaussian distribution. A bell-shaped curve refers to the shape created when a line is plotted using the data points for an item that meets the criteria of normal distribution.

The center of a bell curve contains the greatest number of a value and therefore is the highest point on the arc of the line. This point is referred to as the mean, but in simple terms, it is the highest number of occurrences of an element (in statistical terms, the mode).

Normal Distribution

The important thing to note about a normal distribution is that 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, compared with other distributions. Also, the bell curve signifies the data is symmetrical, meaning you 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 you have measured the amount of deviation in the data. This is 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 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.

Bell Curve Probability and Standard Deviation

To understand the probability factors of a normal distribution, you need to understand the following rules:

1. The total area under the bell curve is equal to 1 (100%).
2. About 68% of the area under the bell-shaped curve falls within one standard deviation.
3. About 95% of the area under the bell curve falls within two standard deviations.
4. About 99.7% of the area under the bell-shaped curve falls within three standard deviations.

Items 2, 3, and 4 above are sometimes referred to as the empirical rule or the 68–95–99.7 rule. Once you determine the data is normally distributed (a bell-shaped curve) and calculate the mean and standard deviation, you can determine the probability that a single data point will fall within a given range of possibilities.

Bell-Shaped Curve Examples

A solid bell-shaped curve example is the roll of two dice. The distribution is centered around the number seven and the probability decreases as you move away from the center.

Here is the percent chance of the various outcomes when you roll two dice:

• Two: (1/36) 2.78%
• Three: (2/36) 5.56%
• Four: (3/36) 8.33%
• Five: (4/36) 11.11%
• Six: (5/36) 13.89%
• Seven: (6/36) 16.67% = most likely outcome
• Eight: (5/36) 13.89%
• Nine: (4/36) 11.11%
• Ten: (3/36) 8.33%
• Eleven: (2/36) 5.56%
• Twelve: (1/36) 2.78%

Normal distributions have many convenient properties, so in many cases, especially in physics and astronomy, random variations with unknown distributions are often assumed to be normal to allow for probability calculations. Although this can be a dangerous assumption, it is a good approximation due to a surprising result known as the central limit theorem.

This theorem states that the mean of any set of variants with any distribution having a finite mean and variance tends to occur in a normal distribution. Many common attributes such as test scores or height follow roughly normal distributions, with few members at the high and low ends and many in the middle.

When You Shouldn't Use the Bell Curve

Some data sets don't follow a normal distribution pattern and shouldn't be forced to fit a bell curve. A classic example would be student grades, which often have two modes. Other data types that don't follow the bell-shaped curve include income, population growth, and mechanical failures.