## What is Normal Distribution in Statistics?

Normal Distribution is a bell-shaped frequency distribution curve which helps describe all the possible values a random variable can take within a given range with most of the distribution area is in the middle and few are in the tails, at the extremes. This distribution has two key parameters: the mean (µ) and the standard deviation (σ) which plays key role in assets return calculation and in risk management strategy.

### How to Interpret Normal Distribution

The above figure shows that the statistical normal distribution is a bell-shaped curve. The range of possible outcomes of this distribution is the whole real numbers lying between -∞ to +∞. The tails of the bell curve extend on both sides of the chart (+/-) without limits.

- Approximately 68% of all observation fall within +/- one standard deviation(σ)
- Approximately 95% of all observation fall within +/- two standard deviations (σ)
- Approximately 99% of all observation fall within +/- three standard deviations (σ)

It has a skewnessSkewnessSkewness is the deviation or degree of asymmetry shown by a bell curve or the normal distribution within a given data set. If the curve shifts to the right, it is considered positive skewness, while a curve shifted to the left represents negative skewness.read more of zero (symmetry of a distribution). If the distribution of data is asymmetric, then the distribution is uneven if the data set has skewness greater than zero or positive skewness. Then, the right tail of the distribution is more prolonged than the left, and for negative skewness (less than zero) left tail will be longer than the right tail.

It has a kurtosis of 3 (measures peakedness of a distribution), which indicates distribution is neither too peaked nor too thin tails. If the kurtosisKurtosisKurtosis in statistics is used to describe the distribution of the data set and depicts to what extent the data set points of a particular distribution differ from the data of a normal distribution. It determines whether the data is heavy-tailed or light-tailed.read more is more than three than distribution is more peaked with fatter tails, and if the kurtosis is less than three, then it has thin tails, and the peak point is lower than the normal distribution.

### Characteristics

- They represent a family of distribution where mean & deviation determine the shape of the distribution.
- The mean, median, and mode of this distribution are all equal.
- Half of the values are to the left of the center and the other half to the right.
- The total value under the standard curve will always be one.
- Most likely, distribution is at the center, and fewer values lie in the tail end.

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For eg:

Source: Normal Distribution (wallstreetmojo.com)

### Transformation (Z)

The Probability density function(PDF) of a random variable (X) following distribution is given by:

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For eg:

Source: Normal Distribution (wallstreetmojo.com)

**where -∞ < x < ∞ ; -∞ < µ < ∞ ; σ > 0**

Where,

- F(x) = Normal probability Function
- x = Random variable
- µ = Mean of distribution
- σ = Standard deviationStandard DeviationStandard deviation (SD) is a popular statistical tool represented by the Greek letter 'σ' to measure the variation or dispersion of a set of data values relative to its mean (average), thus interpreting the data's reliability.read more of the distribution
- π = 3.14159
- e = 2.71828

#### Transformation Formula

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For eg:

Source: Normal Distribution (wallstreetmojo.com)

Where,

*X*= Random variable

### Examples of Normal Distribution in Statistics

Let’s discuss the following examples.

#### Example #1

Suppose a company has 10000 employees and multiple salaries structure as per the job role in which employee works. The salaries are generally distributed with the population meanPopulation MeanThe population mean is the mean or average of all values in the given population and is calculated by the sum of all values in population denoted by the summation of X divided by the number of values in population which is denoted by N.read more of µ = $60,000, and the population standard deviation σ = $15000. What will be the probability that randomly selected employee has a salary less than $45000 annually.

**Solution**

As shown in the above figure, to answer this question, we need to find out the area under the normal curve from 45 to the left side tail. Also, we need to use Z- table value to get the right answer.

Firstly, we need to convert the given mean and standard deviation into a standard normal distributionStandard Normal DistributionThe standard normal distribution is a symmetric probability distribution about the average or the mean, depicting that the data near the average or the mean are occurring more frequently than the data far from the average or the norm. Thus, the score is termed “Z-score”.read more with mean (µ)= 0 and standard deviation (σ) =1 using the transformation formula.

After the conversion, we need to look up the Z- table to find out the corresponding value, which will give us the correct answer.

Given,

- Mean (µ) = $60,000
- Standard deviation (σ) = $15000
- Random Variable (x) = $45000

Transformation (z) = (45000 – 60000 / 15000)

** Transformation (z) = -1**

Now the value that is equivalent to -1 in Z-table is 0.1587, which represents the area under the curve from 45 to the way to left. It indicated that when we randomly select an employee, the probability of making less than $45000 a year is 15.87%.

#### Example #2

Now keeping the same scenario as above, find out the probability that randomly selected employee earns more than $80,000 a year using the normal distribution.

**Solution**

So in this question, we need to find out the shaded area from 80 to right tail using the same formula.

Given,

- Mean (µ) = $60,000
- Standard deviation (σ) = $15000
- Random Variable (X) = $80,000

Transformation (z) = (80000 – 60000 /15000)

**Transformation (z) = 1.33**

As per the Z-table, the equivalent value of 1.33 is 0.9082 or 90.82%, which shows that the probability of randomly selecting employees earning less than $80,000 annually is 90.82%.

But as per the question, we need to determine the probability of the random employees earning more than $80,000 a year, so we need to subtract value from 100.

- Random Variable (X) = 100% – 90.82%
**Random Variable (X) =****9.18%**

So the probability that employees earn more than $80,000 per year is 9.18%.

### Uses

- The stock market technical chart is often a bell curveBell CurveBell Curve graph portrays a normal distribution which is a type of continuous probability. It gets its name from the shape of the graph which resembles to a bell. read more, allowing analysts and investors to make statistical inferences about expected return and risk of stocks.
- It is used in the real-world, like to determine the most probable best time taken by pizza companies to deliver pizza and many more real applications.
- Used in comparing heights of a given population set in which most people will have an average size with very few people having above average or below average height.
- They are used in determining the average academic performance of students, which helps to compare the rank of students.

### Conclusion

Normal distribution finds applications in data science and data analytics. Advanced technologies like Artificial Intelligence and machine learning used along with this distribution can give better data quality, which will help individuals and companies in effective decision making.

### Recommended Articles

This has been a guide to what is Normal Distribution in Statistics and its definition. Here we discuss examples of normal distribution along with its characteristics and uses. You can learn more about financing from the following articles –

- NORMDIST Excel
- Log Normal Distribution
- Formula of Normal Distribution
- Create Normal Distribution Graph in Excel

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