There are different ways of distributing data. We can spread out the data more towards the left or right. However, there is a more symmetric distribution of data where the data tends to be around a central value. It does not have a bias left or right. Such symmetrical data distributions are Normal Distributions. On plotting the values, the graph of its probability density looks like a bell; thus, it gets its name ‘bell curve.’

Here’s what we’ll cover in the article below:

- What is Normal Distribution?
- Normal Distribution Curve
- Normal Distribution Formula
- Standard Normal Distribution Table
- Normal Distribution Standard Deviation

## What is Normal Distribution?

Normal Distributions are also called the Gaussian Distributions. A normal distribution is the most significant continuous probability distribution. Early mathematicians and statisticians noticed the same shape for various distributions—so they named it the normal distribution, i.e., normally occurring distribution.

**Definition: **The Normal Distribution can be defined by the probability density function for a continuous random variable in a system. If f(x) is the probability density function, X is the random variable; then it defines a function that is integrated between the range or interval (x to x + dx). Thus, the probability of random variable X is given by considering the values between x and x+dx.

Mathematically,

f(x) ≥ 0 ∀ x ϵ (−∞,+∞)

And -∞∫+∞ f(x) = 1

### Some Properties of Normal Distributions

The properties of Normal Distributions are as follows:

- Mean = Median = Mode
- The total area under the Gaussian distribution curve equals 1.
- The normal distributions curve is unimodal (has one peak)
- The curve approaches the x-axis but does not touch it (see figure below)
- It has symmetry about the center.
- 50% of values are less than the mean, and 50% are greater than the mean
- A normal distribution curve is a bell-shaped curve

### Normal Distribution Formula

The probability density function of a normal distribution in a variable X with mean μ, and variance σ^{2} is a statistical distribution. The formula for probability density function is as follows:

#### Normal Distribution Curve

The random variables that follow the normal distribution are those whose values can find any unknown value within a given range. For instance, finding the weight of the school’s students. The distribution can take any value, but there will be a limited range like 45- 65 kg.

On the contrary, the normal distribution does not have a range limit. The range can even extend to –∞ to + ∞. These random variables are termed Continuous Variables. The Normal Distribution provides the probability of the value in a particular range for a given experiment.

The normal distribution curve is also called the bell curve because the graph of its probability density is similar to the shape of a bell. It is symmetrical on both sides of the mean. This curve shows that trials will usually give a result near the average. However, they can occasionally deviate by large amounts.

## Standard Normal Distribution (Z)

The mean helps determine the line of symmetry of a graph, and the standard deviation helps determine the data spread out. When the standard deviation value is small, it implies that the data is close to each other. Thus, the graph is narrower. When the standard deviation is large, the data is more dispersed. Thus, the graph becomes wider. Therefore, standard deviation effectively subdivides the area under the normal curve.

**Note:** The standard score is the number of standard deviations from the mean. It is also called “sigma” or “z-score.”

**Definition: **A normal distribution with a zero mean-value and standard deviation of 1 is a standard normal distribution. The standard normal distribution is represented by Z. For a standard normal distribution,

- 68% of the data falls within 1 standard deviation
- 95% of the data lie within 2 standard deviations of the mean
- 99.7% of the data lie within 3 standard deviations of the mean

**Standard Normal Distribution Table**

The following standard normal distribution table shows the area from 0 to Z-value.

Z | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |

0.0 | 0.0000 | 0.0040 | 0.0080 | 0.0120 | 0.0160 | 0.0199 | 0.0239 | 0.0279 | 0.0319 | 0.0359 |

0.1 | 0.0398 | 0.0438 | 0.0478 | 0.0517 | 0.0557 | 0.0596 | 0.0636 | 0.0675 | 0.0714 | 0.0753 |

0.2 | 0.0793 | 0.0832 | 0.0871 | 0.0910 | 0.0948 | 0.0987 | 0.1026 | 0.1064 | 0.1103 | 0.1141 |

0.3 | 0.1179 | 0.1217 | 0.1255 | 0.1293 | 0.1331 | 0.1368 | 0.1406 | 0.1443 | 0.1480 | 0.1517 |

0.4 | 0.1554 | 0.1591 | 0.1628 | 0.1664 | 0.1700 | 0.1736 | 0.1772 | 0.1808 | 0.1844 | 0.1879 |

0.5 | 0.1915 | 0.1950 | 0.1985 | 0.2019 | 0.2054 | 0.2088 | 0.2123 | 0.2157 | 0.2190 | 0.2224 |

0.6 | 0.2257 | 0.2291 | 0.2324 | 0.2357 | 0.2389 | 0.2422 | 0.2454 | 0.2486 | 0.2517 | 0.2549 |

0.7 | 0.2580 | 0.2611 | 0.2642 | 0.2673 | 0.2704 | 0.2734 | 0.2764 | 0.2794 | 0.2823 | 0.2852 |

0.8 | 0.2881 | 0.2910 | 0.2939 | 0.2967 | 0.2995 | 0.3023 | 0.3051 | 0.3078 | 0.3106 | 0.3133 |

0.9 | 0.3159 | 0.3186 | 0.3212 | 0.3238 | 0.3264 | 0.3289 | 0.3315 | 0.3340 | 0.3365 | 0.3389 |

1.0 | 0.3413 | 0.3438 | 0.3461 | 0.3485 | 0.3508 | 0.3531 | 0.3554 | 0.3577 | 0.3599 | 0.3621 |

1.1 | 0.3643 | 0.3665 | 0.3686 | 0.3708 | 0.3729 | 0.3749 | 0.3770 | 0.3790 | 0.3810 | 0.3830 |

1.2 | 0.3849 | 0.3869 | 0.3888 | 0.3907 | 0.3925 | 0.3944 | 0.3962 | 0.3980 | 0.3997 | 0.4015 |

1.3 | 0.4032 | 0.4049 | 0.4066 | 0.4082 | 0.4099 | 0.4115 | 0.4131 | 0.4147 | 0.4162 | 0.4177 |

1.4 | 0.4192 | 0.4207 | 0.4222 | 0.4236 | 0.4251 | 0.4265 | 0.4279 | 0.4292 | 0.4306 | 0.4319 |

1.5 | 0.4332 | 0.4345 | 0.4357 | 0.4370 | 0.4382 | 0.4394 | 0.4406 | 0.4418 | 0.4429 | 0.4441 |

1.6 | 0.4452 | 0.4463 | 0.4474 | 0.4484 | 0.4495 | 0.4505 | 0.4515 | 0.4525 | 0.4535 | 0.4545 |

1.7 | 0.4554 | 0.4564 | 0.4573 | 0.4582 | 0.4591 | 0.4599 | 0.4608 | 0.4616 | 0.4625 | 0.4633 |

1.8 | 0.4641 | 0.4649 | 0.4656 | 0.4664 | 0.4671 | 0.4678 | 0.4686 | 0.4693 | 0.4699 | 0.4706 |

1.9 | 0.4713 | 0.4719 | 0.4726 | 0.4732 | 0.4738 | 0.4744 | 0.4750 | 0.4756 | 0.4761 | 0.4767 |

2.0 | 0.4772 | 0.4778 | 0.4783 | 0.4788 | 0.4793 | 0.4798 | 0.4803 | 0.4808 | 0.4812 | 0.4817 |

2.1 | 0.4821 | 0.4826 | 0.4830 | 0.4834 | 0.4838 | 0.4842 | 0.4846 | 0.4850 | 0.4854 | 0.4857 |

2.2 | 0.4861 | 0.4864 | 0.4868 | 0.4871 | 0.4875 | 0.4878 | 0.4881 | 0.4884 | 0.4887 | 0.4890 |

2.3 | 0.4893 | 0.4896 | 0.4898 | 0.4901 | 0.4904 | 0.4906 | 0.4909 | 0.4911 | 0.4913 | 0.4916 |

2.4 | 0.4918 | 0.4920 | 0.4922 | 0.4925 | 0.4927 | 0.4929 | 0.4931 | 0.4932 | 0.4934 | 0.4936 |

2.5 | 0.4938 | 0.4940 | 0.4941 | 0.4943 | 0.4945 | 0.4946 | 0.4948 | 0.4949 | 0.4951 | 0.4952 |

2.6 | 0.4953 | 0.4955 | 0.4956 | 0.4957 | 0.4959 | 0.4960 | 0.4961 | 0.4962 | 0.4963 | 0.4964 |

2.7 | 0.4965 | 0.4966 | 0.4967 | 0.4968 | 0.4969 | 0.4970 | 0.4971 | 0.4972 | 0.4973 | 0.4974 |

2.8 | 0.4974 | 0.4975 | 0.4976 | 0.4977 | 0.4977 | 0.4978 | 0.4979 | 0.4979 | 0.4980 | 0.4981 |

2.9 | 0.4981 | 0.4982 | 0.4982 | 0.4983 | 0.4984 | 0.4984 | 0.4985 | 0.4985 | 0.4986 | 0.4986 |

3.0 | 0.4987 | 0.4987 | 0.4987 | 0.4988 | 0.4988 | 0.4989 | 0.4989 | 0.4989 | 0.4990 | 0.4990 |

**Example based on Normal Distributions**

**Question 1: Given the value of the random variable is 4, the standard deviation is 4, and the value of mean is 5. Calculate the probability density function of the normal distribution.**

**Solution:**

Variable x = 4; Mean = 5; Standard deviation = 4

Normal distribution formula:

On substituting values in the probability density of normal distribution, we get

f(4,5,4) = 0.0967 is the probability density function.