Standard Normal Distribution Formula

Formula to Calculate Standard Normal Distribution

Standard Normal Distribution is a type of probability distribution that is symmetric about the average or the mean, depicting that the data near the average or the mean are occurring more frequently when compared to the data which is far from the average or the mean. A score on the standard normal distribution can be termed as the “Z-score”.

Standard Normal Distribution Formula is represented as below-

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For eg:
Source: Standard Normal Distribution Formula (wallstreetmojo.com)

Where,

• X is a normal random variable
• µ is the average or the mean
• σ is the standard deviation

Then we need to derive probability from the above table.

Explanation

The standard normal distribution in order words referred to as the Z-distribution has the following properties:

1. It has an average or says the mean of zero.
2. It has a standard deviation, which is equal to 1.

Using the standard normal table, we can find out the areas under the density curve. Z-score is sore on the standard normal distribution and should be interpreted as the number of standard deviations where the data point is below or above the average or the mean.

A negative Z-Score shall indicate a score that is below the mean or the average, while A positive Z-Score shall indicate that the data point is above the mean or the average.

The standard normal distribution follows the 68-95-99.70 Rule, which is also called as the Empirical RuleEmpirical RuleEmpirical Rule in Statistics states that almost all (95%) of the observations in a normal distribution lie within 3 Standard Deviations from the Mean.read more, and as per that Sixty eight percent of the given data or the values shall fall within 1 standard deviation of the average or the mean, while ninety-five percent shall fall within 2 standard deviations, and finally, the ninety-nine decimal seven percent of the value or the data shall fall within 3 standard deviations of the average or of the mean.

Examples

Example #1

Consider the mean given to you like 850, standard deviation as 100. You are required to calculate Standard Normal Distribution for a score above 940.

Solution:

Use the following data for the calculation of standard normal distribution.

So, the calculation of z score can be done as follows-

Z – score = ( X – µ ) / σ

= (940 – 850) / 100

Z Score will be –

Z Score = 0.90

Now using the above table of the standard normal distribution, we have a value for 0.90 as 0.8159, and we need to calculate the score above that which is P(Z >0.90).

We need the right path to the table. Hence, the probability would be 1 – 0.8159, which is equal to 0.1841.

Thus, only 18.41% of the scores lie above 940.

Example #2

Sunita takes private tuition classes for mathematics subjects, and currently, she has around 100 students enrolled under her. After the 1^st test she took for her students, she got the following average numbers, scored by them, and have ranked them percentile-wise. 

Solution:

First, we plot what we are targeting, which is the left side of the cure. P(Z<75).

Use the following data for the calculation of standard normal distribution.

For that, we need to calculate the mean and the standard deviation first.

The calculation of meanCalculation Of MeanMean 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 can be done as follows-

Mean = (98 + 40 + 55 + 77 + 76 + 80 + 85 + 82 + 65 + 77) / 10

Mean = 73.50

The calculation of standard deviation can be done as follows-

Standard deviation = √[∑(x – x) / (n-1)]

Standard deviation = 16.38

So, the calculation of z score can be done as follows-

Z – score= ( X – µ ) / σ

= (75 – 73.50) / 16.38

Z Score will be –

Z Score = 0.09

Now using the above table of a standard normal distribution, we have value for 0.09 as 0.5359 and that is the value for P (Z <0.09).

Hence 53.59% of the students scored below 75.

Example #3

Vista limited is an electronic equipment showroom. It wants to analyze its consumer behavior. It has around 10,000 customers around the city. On average, the customer spends 25,000 when it comes to its shop. However, the spending varies significantly as customers spend from 22,000 to 30,000 and the average of this variance around 10,000 customers that management of vista limited has come up with is around 500.

The management of Vista limited has approached you, and they are interested to know what proportion of their customers spend more than 26,000? Assume that customer’s spending figures are normally distributed.

Solution:

First, we plot what we are targeting, which is the left side of the cure. P(Z>26000).

Use the following data for the calculation of standard normal distribution.

The calculation of z scoreCalculation Of Z ScoreThe Z-score of raw data refers to the score generated by measuring how many standard deviations above or below the population mean the data, which helps test the hypothesis under consideration. In other words, it is the distance of a data point from the population mean that is expressed as a multiple of the standard deviation.read more can be done as follows-

Z – score= ( X – µ ) / σ

=(26000 – 25000) / 500

Z Score will be-

Z Score = 2

The calculation of standard normal distribution can be done as follows-

Standard normal distribution will be-

Now using the above table of the standard normal distribution, we have a value for 2.00, which is 0.9772, and now we need to calculate for P(Z >2).

We need the right path to the table. Hence, the probability would be 1 – 0.9772, which is equal to 0.0228.

Hence 2.28% of the consumers spend above 26000.

Relevance and Use

To make an informed and a proper decision, one needs to convert all of the scores to a similar scale. One needs to standardize those scores, converting all of them to the standard normal distribution using the Z score method, with a single standard deviation and a single average or the mean. Majorly this is used in the field of statistics and also in the field of finance that too by traders.

Many statistical theories have attempted to model the prices of the asset (in fields of finance) under the main assumption that they shall follow this kind of normal distribution. Price distributions mostly tend to have fatter tails and, hence have 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, which is greater than 3 in real-life scenarios. Such assets have been observed to have the price movements which are greater than 3 standard deviations beyond the average or the mean and more often than the expected assumption in a normal distributionNormal DistributionNormal 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 a key role in assets return calculation and in risk management strategy.read more.

Recommended Articles

This has been a guide to Standard Normal Distribution Formula. Here we learn how to calculate standard normal distribution (Z-score) with practical examples and a downloadable excel template. You can learn more about excel modeling from the following articles –

• Calculate Poisson Distribution
• Normal Distribution Formula
• Calculation of Binomial Distribution
• Formula of Sampling Distribution