Formula to Calculate Standard Normal Distribution

The standard normal distribution formula is below:

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Where,

  • X is a normal random variableµ is the average or the meanσ 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:

  • First, it has an average or says a mean of zero.It has a standard deviation that is equal to 1.

Using the standard normal table, we can find the areas under the density curve. The Z-score is the score on the standard normal distribution. One should interpret it 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 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, also called 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. Per that rule, sixty-eight percent of the given data or the values shall fall within 1 standard deviation of the average or the mean. In comparison, ninety-five percent shall fall within 2 standard deviations. Finally, the ninety-nine decimal seven percent of the value of the data shall fall within 3 standard deviations of the average or the mean.

Examples

Example #1

Consider the mean given to you, like 850, with a standard deviation of 100. You are required to calculate a 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

Using the above table of the standard normal distribution, we have a value of 0.90 as 0.8159. Therefore, we must calculate the score above P(Z >0.90).

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

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

Example #2

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

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

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 as follows-

Z – score= ( X – µ ) / σ

= (75 – 73.50) / 16.38

Z Score = 0.09

Using the above table of standard normal distribution, we have a value of 0.09 as 0.5359, which is the value for P (Z <0.09).

Hence, 53.59% of the students scored below 75.

Example #3

Vista Ltd. is an electronic equipment showroom. It wants to analyze its consumer behavior and 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 as customers spend from 22,000 to 30,000. The average variance of around 10,000 customers that the management of Vista Ltd. has come up with is around 500.

The management of Vista Ltd. has approached you. They are interested to know what proportion of their customers spend more than 26,000. Assume that customers’ spending figures are normally distributed.

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

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 as follows-

=(26000 – 25000) / 500

Z-score will be-

Z-score = 2

The calculation of standard normal distribution can be as follows-

Standard normal distribution will be-

Using the above table of the standard normal distribution, we have a value of 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, equal to 0.0228.

Hence 2.28% of the consumers spend above 26,000.

Relevance and Use

To make an informed and proper decision, one needs to convert all scores to a similar scale. Then, one needs to standardize those scores, converting 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 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. However, price distributions mostly tend to have fatter tails. 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 observe price movements 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.

This article 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 DistributionCalculate Poisson DistributionPoisson distribution refers to the process of determining the probability of events repeating within a specific timeframe.read moreNormal Distribution FormulaNormal Distribution FormulaNormal distribution is a distribution that is symmetric i.e. positive values and the negative values of the distribution can be divided into equal halves and therefore, mean, median and mode will be equal. It has two tails one is known as the right tail and the other one is known as the left tail.read moreCalculation of Binomial DistributionFormulaFormulaA sampling distribution is the probability-based distribution of detailed statistics. It helps calculate means, range, standard deviation, and variance for the undertaken sample. For a sample size of more than 30, the formula is: µ͞x =µ and σ͞x =σ / √n
  • read more of Sampling Distribution