Bell Curve

What is the Bell Curve?

Bell Curve is a normal probability distribution of variables that is plotted on the graph and is like a shape of a bell where highest or top point of the curve represents most probable event out of all the data of the series.

The formula for Bell Curve as per below:

Bell Curve Formula
Bell-Curve-Formula

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For eg:
Source: Bell Curve (wallstreetmojo.com)

Where,

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

Explanation

Examples

You can download this Bell Curve Formula Excel Template here – Bell Curve Formula Excel Template

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.

Bell Curve Formula Example 1

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.

The formula for Bell-Shaped Curve as per below:

Bell Curve Formula
Example 1.1

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

y will be –

Example 1.2

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.

Bell Curve Formula Example 2

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.

The formula for Bell Curve as per below:

Bell Curve Formula
Example 2.1

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

y will be –

Example 2.2

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 a statutory audit of ABC bank, and they have noted that in the last few audits, they have picked up an incorrect sample which was giving misrepresentation of the population for example, in the 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 statisticsStatisticsStatistics is the science behind identifying, collecting, organizing and summarizing, analyzing, interpreting, and finally, presenting such data, either qualitative or quantitative, which helps make better and effective decisions with relevance.read more, 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 seven 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.

Bell Curve Formula Example 3

First, we need to take the average of the two 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.

The formula for Bell Curve as per below:

Bell Curve Formula
Example 3.1

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

y will be –

Example 3.2

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 specific velocity.

The financial analyst will often use the normal probability distributionProbability DistributionProbability distribution is the calculation that shows the possible outcome of an event with the relative possibility of occurrence or non-occurrence as required. It is a mathematical function that gives results as per the possible events.read more or say the bell curve while analyzing the returns of overall market sensitivity or 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. Hence, they make use of the normal probability distribution or bell curve of a stock’s previous returns to make assumptions about the expected returnsExpected ReturnsThe Expected Return formula is determined by applying all the Investments portfolio weights with their respective returns and doing the total of results. Expected return = (p1 * r1) + (p2 * r2) + ………… + (pn * rn), where, pi = Probability of each return and ri = Rate of return with probability. read more.

Recommended Articles

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

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