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# Normal Distribution Calculator

Normal Distribution Calculator:

I want to find:

Cumulative probability: P(X < x):

Mean:

Standard deviation:

Mean:

Standard deviation:

Skip input, if you don't need it

 $P\left(X\le$ $\right)=?$ $P\left(X\ge$ $\right)=?$ $P\left(X\le ? \right)=$ $P\left(X\ge ? \right)=$ $\mathrm{P \left(}$ $\le X\ge$ $\right)\mathrm{= ?}$ $\mathrm{P \left( -?}\le X\ge \mathrm{? \right) =}$

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This normal distribution calculator assists you to do to and fro calculations among cumulative probability and standard random variable. Also, you could now determine the area under the bell curve by subjecting this standard normal curve calculator.

So tie your seat belts to go on a ride of normal distribution concept in more depth with us.

Let’s Continue!

## What Is Normal Distribution?

In the light of statistical analysis:

“When data provided gets very close to the central point with no bias to any side of it, then this kind of distribution is known as the normal distribution”

### Important Considerations:

You need to go through some important facts about the normal distribution which are listed below:

• The mean value represents the middle term that is present in between the data set of the value being provided.
• The number that repeats itself the most is termed the mode of the data set
• The symmetric distribution of the dataset about the mean position indicates that half of the dispersion would be lower than the mean value and the rest would lie higher than that of the mean value.

Moreover, if you wish to determine these values, then you can also use our another mean median mode range calculator to get the desired results

### Relation of Normal Distribution and Standard Deviation:

There lies a deep relation among both of these terms as they are related to the data distribution. Now what you need here is to memorise the following key points in your mind:

• If you analyse 1 single standard deviation from the mean position, you will get to know that almost 68% of the data lies within the said range
• Coming to the 2 deviations now. Here you will find about 95% of the data within the range that you could also analyse with the assistance of this best standard normal distribution calculator.
• At last, we have 3 positions away from the mean value where about 99.7 of the data dispersion will be found

The whole statistics are represented by pictorial diagram as under:

### Standard Normal Distribution:

It’s a most generic form of the data distribution from which the normal distribution is itself dragged out.

#### Actual Definition:

“A special type of distribution of data in which the mean value becomes 0 and standard deviation becomes 1 is known as the standard deviation.”

Another name used for the phenomenon is z distribution that is calculated by z score. For a standard normal distribution, the overall area under a bell curve would be equal to 1. Also, you must convert the value of variable x into a z score.

### Effect of Standard Normal Distribution on Bell Curve:

The standard distribution contracts or expands the curve of a normal distribution. Below we have a table along with its pictorial representation that display the effect that we are actually discussing.

 Curve Position or shape (relative to standard normal distribution) A (M = 0, SD = 1) Standard normal distribution B (M = 0, SD = 0.5) Squeezed, because SD < 1 C (M = 0, SD = 2) Stretched, because SD > 1 D (M = 1, SD = 1) Shifted right, because M > 0 E (M = –1, SD = 1) Shifted left, because M < 0

You can also analyse these behaviours with the help of this online normal calculator in a blink of moments.

### Normal Distribution Formulas:

Various formulas are used to calculate the normal distributions which include:

#### Probability density function (PDF):

$$f\left(x\right) = \frac{1}{𝛔\sqrt{2\pi}}e^{\frac{1}{2}\left(\frac{x-µ}{𝛔}\right)^{2}}$$

#### Standard normal distribution function:

$$f\left(x\right) = \frac{1}{\sqrt{2\pi}}e^{\frac{1}{2}x^{2}}$$

#### Cumulative distribution function (CDF):

$$F\left(x;µ,𝛔\right) = Pr\left(X≤x\right)$$

$$F\left(x;µ,𝛔\right) = \frac{1}{𝛔\sqrt{2\pi}}\int_{-\inf}^{x}\exp\left(\frac{-\left(t-µ\right)^{2}}{2𝛔^{2}}\right)$$

#### Inverse distribution function (quantile function, IDF):

$$F^{1} \left(p\right) = µ+𝛔ɸ^{1}\left(p\right)$$

$$F^{1} \left(p\right) = µ+𝛔\sqrt{2} erf^{-1} \left(2p-1\right), p∈\left(0, 1\right)$$

All of these formulas are also used by this best normal distribution calculator to determine probabilities of events that are either upper or lower of the mean.

### Normal Distribution Table:

The following table is the main source of calculating the z score (Standard Normal Distribution) and helps you to calculate the probability of a random variable either higher or below the mean value. Let’s have a look at it!

 z 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0 0 0.00399 0.00798 0.01197 0.01595 0.01994 0.02392 0.0279 0.03188 0.03586 0.1 0.03983 0.0438 0.04776 0.05172 0.05567 0.05962 0.06356 0.06749 0.07142 0.07535 0.2 0.07926 0.08317 0.08706 0.09095 0.09483 0.09871 0.10257 0.10642 0.11026 0.11409 0.3 0.11791 0.12172 0.12552 0.1293 0.13307 0.13683 0.14058 0.14431 0.14803 0.15173 0.4 0.15542 0.1591 0.16276 0.1664 0.17003 0.17364 0.17724 0.18082 0.18439 0.18793 0.5 0.19146 0.19497 0.19847 0.20194 0.2054 0.20884 0.21226 0.21566 0.21904 0.2224 0.6 0.22575 0.22907 0.23237 0.23565 0.23891 0.24215 0.24537 0.24857 0.25175 0.2549 0.7 0.25804 0.26115 0.26424 0.2673 0.27035 0.27337 0.27637 0.27935 0.2823 0.28524 0.8 0.28814 0.29103 0.29389 0.29673 0.29955 0.30234 0.30511 0.30785 0.31057 0.31327 0.9 0.31594 0.31859 0.32121 0.32381 0.32639 0.32894 0.33147 0.33398 0.33646 0.33891 1 0.34134 0.34375 0.34614 0.34849 0.35083 0.35314 0.35543 0.35769 0.35993 0.36214 1.1 0.36433 0.3665 0.36864 0.37076 0.37286 0.37493 0.37698 0.379 0.381 0.38298 1.2 0.38493 0.38686 0.38877 0.39065 0.39251 0.39435 0.39617 0.39796 0.39973 0.40147 1.3 0.4032 0.4049 0.40658 0.40824 0.40988 0.41149 0.41308 0.41466 0.41621 0.41774 1.4 0.41924 0.42073 0.4222 0.42364 0.42507 0.42647 0.42785 0.42922 0.43056 0.43189 1.5 0.43319 0.43448 0.43574 0.43699 0.43822 0.43943 0.44062 0.44179 0.44295 0.44408 1.6 0.4452 0.4463 0.44738 0.44845 0.4495 0.45053 0.45154 0.45254 0.45352 0.45449 1.7 0.45543 0.45637 0.45728 0.45818 0.45907 0.45994 0.4608 0.46164 0.46246 0.46327 1.8 0.46407 0.46485 0.46562 0.46638 0.46712 0.46784 0.46856 0.46926 0.46995 0.47062 1.9 0.47128 0.47193 0.47257 0.4732 0.47381 0.47441 0.475 0.47558 0.47615 0.4767 2 0.47725 0.47778 0.47831 0.47882 0.47932 0.47982 0.4803 0.48077 0.48124 0.48169 2.1 0.48214 0.48257 0.483 0.48341 0.48382 0.48422 0.48461 0.485 0.48537 0.48574 2.2 0.4861 0.48645 0.48679 0.48713 0.48745 0.48778 0.48809 0.4884 0.4887 0.48899 2.3 0.48928 0.48956 0.48983 0.4901 0.49036 0.49061 0.49086 0.49111 0.49134 0.49158 2.4 0.4918 0.49202 0.49224 0.49245 0.49266 0.49286 0.49305 0.49324 0.49343 0.49361 2.5 0.49379 0.49396 0.49413 0.4943 0.49446 0.49461 0.49477 0.49492 0.49506 0.4952 2.6 0.49534 0.49547 0.4956 0.49573 0.49585 0.49598 0.49609 0.49621 0.49632 0.49643 2.7 0.49653 0.49664 0.49674 0.49683 0.49693 0.49702 0.49711 0.4972 0.49728 0.49736 2.8 0.49744 0.49752 0.4976 0.49767 0.49774 0.49781 0.49788 0.49795 0.49801 0.49807 2.9 0.49813 0.49819 0.49825 0.49831 0.49836 0.49841 0.49846 0.49851 0.49856 0.49861 3 0.49865 0.49869 0.49874 0.49878 0.49882 0.49886 0.49889 0.49893 0.49896 0.499 3.1 0.49903 0.49906 0.4991 0.49913 0.49916 0.49918 0.49921 0.49924 0.49926 0.49929 3.2 0.49931 0.49934 0.49936 0.49938 0.4994 0.49942 0.49944 0.49946 0.49948 0.4995 3.3 0.49952 0.49953 0.49955 0.49957 0.49958 0.4996 0.49961 0.49962 0.49964 0.49965 3.4 0.49966 0.49968 0.49969 0.4997 0.49971 0.49972 0.49973 0.49974 0.49975 0.49976 3.5 0.49977 0.49978 0.49978 0.49979 0.4998 0.49981 0.49981 0.49982 0.49983 0.49983 3.6 0.49984 0.49985 0.49985 0.49986 0.49986 0.49987 0.49987 0.49988 0.49988 0.49989 3.7 0.49989 0.4999 0.4999 0.4999 0.49991 0.49991 0.49992 0.49992 0.49992 0.49992 3.8 0.49993 0.49993 0.49993 0.49994 0.49994 0.49994 0.49994 0.49995 0.49995 0.49995 3.9 0.49995 0.49995 0.49996 0.49996 0.49996 0.49996 0.49996 0.49996 0.49997 0.49997 4 0.49997 0.49997 0.49997 0.49997 0.49997 0.49997 0.49998 0.49998 0.49998 0.49998

This standard normal table calculator also makes use of these z score values to determine the probabilities of the normal distributions.

### How Normal Distribution Calculator Works?

This normal model calculator takes a couple of clicks to calculate the probability of the standard normal distribution. Want to know how? Let’s continue to discuss!

Input:

• From the top drop-down list, select whether you want to calculate results with “Basic” or “Advanced” method
• After you make a selection, you need to select whether you wish to determine the “Normal Random Variable” or “Cumulative Probability”
• After that, go for entering all the parameter values in their designated fields
• At last, tap the calculate button

Output:

The free normal distribution calculator determines the following results:

• Normal Random variable
• Cumulative probability
• Detailed calculations

## FAQ’s:

### What is standard normal variable?

A normally distributed random variable with a mean of 0 and a standard deviation of 1 is known as a standard normal random variable. The letter Z will always be used to represent it.

### Is z-score same as standard deviation?

The Z-score shows how far a value deviates from the standard deviation. The Z-score, also known as the standard score, is the amount of standard deviations a data point deviates from the mean. The standard deviation is a measure of how much variability there is in a given data collection.

### What is the purpose of normal distribution?

For numerous reasons, we transform normal distributions to the ordinary normal distribution: The chance of an observation in a population falling above or below a certain value is calculated.

### How is normal distribution used in real life?

Following are the real world examples of the normal distribution:

• Calculating height
• Dice rolling
• Coin tossing
• Stock market exchange
• Economical income distribution
• Estimating size of shoe

### Why normal distribution is important in quantitative techniques?

The normal distribution is a reasonable model for a random variable in general when: the variable has a strong tendency to take a central value; Deviations from this core value, both positive and negative, are equally likely. As the deviations increase higher, the frequency of deviations decreases rapidly.

## Conclusion:

The normal distribution defines how the values of a variable are distributed. Because it accurately captures the distribution of values for many natural occurrences, it is the most important probability distribution in statistics. And when it comes to calculating the most precise probability values for normal distributions, this best normal distribution calculator is the one that stands out.

## References:

From the source of wikipedia: Normal distribution, Alternative parameterizations, Cumulative distribution functions, Quantile function, Properties, Symmetries and derivatives,

From the source of khan academy: Qualitative sense of normal distributions, Empirical rule

From the source of lumen learning: Z-Scores, The Empirical Rule