What Is a Bell Curve?
A bell curve is a graph used to visualize the distribution of a set of chosen values across a specified group that tend to have central, normal values that peak, with low and high extremes tapering off relatively symmetrically on either side.
It is a graph of a normal (Gaussian) distribution, with a large, rounded peak tapering away at each end. It is assumed that during any measurement, values will follow a normal distribution, with an equal number of measurements above and below the mean value.
How Does a Bell Curve Work?
The bell curve, or normal distribution, occurs throughout statistics. Like snowflakes, there are an infinite number of normal distributions. It is determined by the mean and standard deviation of the distribution of data points. The formula for the normal equation is as follows:
- y is the value of the random variable
- x is the normal random variable
- μ is the mean (the center or the midpoint of the distribution)
- σ is a standard deviation (noted by the Greek letter, sigma)
- π is 3.14159 (pie)
- e is 2.71828 (the constant)
To solve for the above equation, the standard deviation must be calculated. The value of the standard deviation is related to the spread of the distribution. Its formula is as follows:
x1 = the data point we are solving for in the set
N = the total number of data points
For example, to calculate the standard deviation of the heights of a group of nine people:
Calculate the mean from each individual height. In this case, the mean height of the group is 75 inches.
Subtract the mean from each data point.
Square the absolute value before adding them together.
Divide this amount by the number of data points (9).
Take the square root to determine the standard deviation.
The square root term is present to normalize the formula. It determines the total area under the curve. The entire area = 1, corresponding to 100%. As the value of σ increases, the normal distribution becomes more spread out. Specifically, the peak of the distribution is not as high, and the tails of the distribution become fatter.
In this example, the sum of the squares, or ď x = 784
The variance, or x / 9 = 87.1
The standard deviation, or σ = √x = 9.3
A bell curve indicates that about 68% of the data lies within one standard deviation, about 95% of the data lies within two standard deviations, and about 99.7% of the data lies within three standard deviations. This relationship is known as the 68-95-99.7 rule (or the empirical rule). The rule is primarily used to calculate the confidence interval of normal probability distribution, a representation of the distribution of random variables whose real distribution is unknown.
Thus, 68% of the 9 individuals were 75 inches tall plus or minus 9.3 inches (one standard deviation away from the mean), 95% were 75 inches plus or minus 18.6 inches (two standard deviations away from the mean), and 99.7% were 75 inches plus or minus 27.9 inches (3 standard deviations away from the mean).
Bell Curve and Investing
The assumption of a normal distribution is fundamental in many financial pricing models used to predict future returns of a security. Normal probability distribution is assumed in financial models; however, the returns of many securities tend to demonstrate a non-normal distribution.
For example, some distributions are skewed with a kurtosis that differs from that of a normal distribution. Kurtosis corresponds to a broadening of the peak and “thickening” of the tails. With kurtosis, the distribution can be described as having “fat tails” or outliers.
Why You Need to Know About Bell Curves
The bell curve has many practical applications in statistics, science, game theory and finance. For example:
- Statistics. Bell curves model real-world data such as classroom test results and employee performance reviews in an organization.
- Science. The bell curve graph is useful for repeated measurement of equipment and measuring characteristics in biology, chemistry and physics.
- Game theory. The bell curve has relevance when assessing the statistical outcome of flipping coins several times or drawing playing cards from a deck.
- Finance. Financial analysts typically rely on the normal probability distribution in analyzing the returns of stocks.
The bell curve has one mode, and it coincides with the mean and median. This mode is the center of the bell curve, and it is the highest point. When a bell curve is folded in half vertically, two equal parts are created along the vertical axis. The two parts are mirror images of each other and, therefore, are symmetrical.
- Approximately 68% of the data lies within one standard deviation.
- Approximately 95% of the data lies within two standard deviations.
- Approximately 99.7% of the data lies within three standard deviations.
The smallest possible value standard deviation can reach is zero. Where there are at least two numbers in the data set that are not exactly equal, standard deviation must be greater than zero. A low standard deviation indicates data are clustered around the mean, forming a thin bell curve, whereas a high standard deviation indicates data are more spread out, thus forming a wide bell curve. Both high or low standard deviations indicate that data points are respectively above or below the mean.