Bell Curve

What is the Bell Curve?

When the line of the variable of the normal distribution is plotted on the graph then that line takes the shape of the bell and the same is known as the bell curve where highest or top point of the curve represents most probable event out of all the data of the series.

Bell curve is used to describe a normal probability distribution (can also be referred as Gaussian distribution) in a graphical depiction and whose shape which is the bell-shaped curve is created by the underlying standard deviations from the median.

The formula for Bell Curve as per below:

Where,

• μ is mean
• σ is a standard deviation
• π is 3.14159
• e is 2.71828

Explanation

• The mean is denoted by μ which denotes the center or the mid-point of the distribution.
• The horizontal symmetry about the vertical line which is x = μ as there is square in the exponent.
• The standard deviation is denoted by σ and is related to the spread of the distribution. As σ increases, the normal distribution will spread out more. Specifically, the distribution’s peak is not as high, and the distribution’s tail shall become thicker.
• π is constant pi and has an infinite which is not repeating decimal expansion.
• e represents another constant and is also transcendental and irrational like pi.
•  There is a non-positive sign in the exponent, and rest of the terms are squared in the exponent.  Which means exponent will always be negative. And because of that, the function is an increasing function for all x < mean μ. The opposite is true when all x > mean μ.
• Another horizontal asymptote corresponds to the horizontal line y which equals 0 which would mean that the graph of the function will never touch the x-axis and will have a zero.
• The square root term will normalize the formula which means that when one integrates the function for searching the area under the curve where the whole area will be under the curve and it is one and that corresponds to 100%.
• This formula is related to a normal distribution and is used for calculating probabilities.

Examples

Example# 1

Consider the mean given to you like 950, standard deviation as 200. You are required to calculate y for x = 850 using the bell curve equation.

Solution:

Use the following data for the calculation of bell-shaped curve

First, we are given all the values i.e. mean as 950, standard deviation as 200 and x as 850, we just need to plug in the figures in the formula and try to calculate the y.

Calculation of bell curve (y) can be done as follows –

The formula for Bell Shaped Curve as per below:

y = 1/(200√2*3.14159)^e^{-(850 – 950)/2*(200^2)}

y will be –

y = 0.0041

After doing the above math (check excel template) we have the value of y as 0.0041.

Example# 2

Sunita is a runner and is preparing for the upcoming Olympics and she wants to determine that the race she is going to run has perfect timing calculation as a split delay can cause her the gold in Olympics. Her brother is a statistician and he noted that the mean timing of her sister is 10.33 seconds whereas the standard deviation of her timing is 0.57 seconds which is quite risky as such split delay can cause her to win gold in the Olympics. Using the bell shaped curve equation, what is the probability of Sunita completing the race in 10.22 seconds?

Solution:

Use the following data for the calculation of bell curve

First, we are given all the values i.e. mean as 10.33 seconds, standard deviation as 0.57 seconds and x as 10.22, we just need to plug in the figures in the formula and try to calculate the y.

Calculation of bell curve (y) can be done as follows –

The formula for Bell Curve as per below:

y = 1/(0.57√2*3.14159)^e^{-(850 – 950)/2*(200^2)}

y will be –

y = 0.7045

After doing the above math (check excel template) we have the value of y as 0.7045.

Example# 3

Hari-baktii limited is an audit firm. It has recently received statutory audit of ABC bank and they have noted that in last few audits they have picked up incorrect sample which was giving misrepresentation of the population for example in case of receivable the sample they picked up was depicting that receivable was genuine but later it was discovered that receivable population had many dummy entries.

So now they are trying to analyze what is the probability of picking up the bad sample which would generalize the population as correct even though the sample was not a correct representation of that population. They have an article assistant who is good at statistics and recently he has learned about the bell curve equation.

So, he decides to use that formula to find the probability of picking up at least 7 incorrect samples. He went into the history of the firm and found that the average incorrect sample they collect from a population is between 5 to 10 and the standard deviation is 2.

Solution:

Use the following data for the calculation of bell curve

First, we need to take the average of the 2 numbers given i.e. for mean as (5+10)/2 which is 7.50, standard deviation as 2 and x as 7, we just need to plug in the figures in the formula and try to calculate the y.

Calculation of bell curve (y) can be done as follows –

The formula for Bell Curve as per below:

y = 1/(2√2*3.14159)^e^{-(7 – 7.5)/2*(2^2)}

y will be –

y = 0.2096

After doing the above math (check excel template) we have the value of y as 0.2096

So, there is a 21% chance that this time also they could take 7 incorrect samples in the audit.

Relevance and Uses

This function will be used to describe the events which are physical i.e. the number of events is humongous. In simple words, one may not be able to predict what the outcome of the item will perform if there are a whole ton of observations, but one shall be able to predict what those shall do a whole. Take an example, suppose one has a gas jar at a constant temperature, the normal distribution or the bell curve will enable that person to figure out the probability of one particle which shall move at a certain velocity.

The financial analyst will often use the normal probability distribution or say the bell curve while analyzing the returns of overall market sensitivity or of security.

E.g. stocks which display a bell curve are usually the blue-chip ones and those shall have the lower volatility and often more behavioral patterns which shall be predictable and hence they make use of the normal probability distribution or bell curve of a stock’s previous returns to make assumptions about the expected returns.

Recommended Articles

This has been a guide to Bell Curve Formula. Here we learn how to calculate bell-shaped curve (y) using its formula along with practical examples and downloadable excel template. You can learn more about financial analysis from the following articles –

• Laffer Curve Limitations
• Formula of Standard Normal Distribution
• Formula of Binomial Distribution
• Excel Lognormal Distribution
• Excel Weibull Distribution