Bell Curve Formula Examples What Is Bell Shaped Graph

What is the Bell Curve?

The formula for the bell curve is as per below:

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Where,

μ is meanσ is a standard deviationπ is 3.14159e is 2.71828

Explanation

The meanMeanMean refers to the mathematical average calculated for two or more values. There are primarily two ways: arithmetic mean, where all the numbers are added and divided by their weight, and in geometric mean, we multiply the numbers together, take the Nth root and subtract it with one.read more denotes μ, which denotes the distribution’s center or midpoint.The horizontal symmetry about the vertical line is x = μ as there is a square in the exponent.The standard deviation denotes 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, not repeating decimal expansion.E represents another constant and is also transcendental and irrational, like pi.The exponent has a non-positive sign, and the rest of the terms are squared. So it means the exponent will always be negative. And because of that, the function increases 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, meaning that the function’s graph will never touch the x-axis and will have a zero.The square root in the excelSquare Root In The ExcelThe Square Root function is an arithmetic function built into Excel that is used to determine the square root of a given number. To use this function, type the term =SQRT and hit the tab key, which will bring up the SQRT function. Moreover, this function accepts a single argument.read more 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, it is one, corresponding to 100%.This formula is related to a normal distribution used for calculating probabilities.

Examples

Example# 1

Consider the mean given to you, like 950, the standard deviation of 200. Then, it would help if you calculated y for x = 850 using the bell curve equation.

Solution:

Use the following data for the calculation.

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

The formula for bell-shaped curve as per below:

y = 1/(200√23.14159)^e-(850 – 950)/2(200^2)

y will be –

y = 0.0041

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

Example# 2

Sunita is a runner preparing for the upcoming Olympics and wants to determine that the race she will run has perfect timing calculations 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 a split delay can cause her to win gold in the Olympics. So, using the bell-shaped curve equation, what is the probability of Sunita completing the race in 10.22 seconds?

Solution:

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

y = 1/(0.57√23.14159)^e-(850 – 950)/2(200^2)

y = 0.7045

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

Example# 3

Hari-baktii Ltd. is an audit firm. It has recently received a statutory audit from ABC bank. They have noted that in the last few audits, they picked up an incorrect sample that misrepresented the population. For example, the sample they picked up depicted that they were genuine in the case of receivables. Still, later they discovered that the receivable population had many dummy entries.

So now, they are trying to analyze 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 firm’s history and found that the average incorrect sample they collect from a population is between 5 to 10, and the standard deviation is 2.

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. Then, we just need to plug in the figures in the formula and calculate the y.

y = 1/(2√23.14159)^e-(7 – 7.5)/2(2^2)

y = 0.2096

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

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

Relevance and Uses

One will use this function to describe the physical events, i.e., the number of events is humongous. In simple words, one may not be able to predict the outcome of the item if there are a ton of observations, but one shall be able to predict what those shall do as a whole. For example, suppose one has a gas jar at a constant temperature. Then, the normal distribution or the bell curve will enable that person to figure out the probability of one particle moving at a specific velocity.

The financial analyst will often use the normal probability distributionProbability DistributionProbability distribution could be defined as the table or equations showing respective probabilities of different possible outcomes of a defined event or scenario. In simple words, its calculation shows the possible outcome of an event with the relative possibility of occurrence or non-occurrence as required.read more or the bell curve while analyzing the returns of overall market sensitivity or security.

E.g., stocks that 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 use 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.

What is the Bell Curve?#

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Relevance and Uses#

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