How To Draw Normal Distribution Curve With Mean And Standard Deviation

Introduction to the Normal Distribution

Have you lot heard of the bell curve? It tends to be amongst the most discussed water-cooler topics among people around the globe. For a long fourth dimension, a bell curve dictated the professional person assessment of an employee and was a beloved or dreaded topic, depending on who to spoke to!

Take a look at this image:

Source: empxtrack.com

What do you lot remember the shape of the curve signifies? Equally a data scientist (or an aspiring one), yous should be able to answer that question at the drop of a lid. The idea behind a bell curve, among many other applications, is that of a normal distribution.

The normal distribution is a core concept in statistics, the backbone of data scientific discipline. While performing exploratory data analysis, we first explore the data and aim to find its probability distribution, right? And estimate what – the most mutual probability distribution is Normal Distribution.

Check out these 3 very common examples of the normal distribution:

As you tin clearly see, the Birth weight, the IQ Score, and stock price return oftentimes form a bell-shaped curve. Similarly, there are many other social and natural datasets that follow Normal Distribution.

I more than reason why Normal Distribution becomes essential for data scientists is the Central Limit Theorem. This theorem explains the magic of mathematics and is the foundation for hypothesis testing techniques.

In this article, we will be understanding the significance and different properties of Normal Distribution and how we can use those backdrop to bank check the Normality of our information.

Table of Contents

• Backdrop of Normal Distribution
• Empirical Dominion for Normal Distribution
• What is a Standard Normal Distribution?
• Getting Familiar with Skewed Distribution
  •  Left Skewed Distribution
  • Right Skewed Distribution
• How to check the Normality of a Distribution
  • • Histogram
    • KDE Plots
    • Q_Q Plots
    • Skewness
    • Kurtosis
• Python Code to Implement and Understand Normal Distribution

Backdrop of Normal Distribution

We phone call this Bong-shaped curve a Normal Distribution. Carl Friedrich Gauss discovered it so sometimes we also phone call information technology a Gaussian Distribution as well.

We can simplify the Normal Distribution'southward Probability Density by using merely two parameters: 𝝻 Hateful and 𝛔two. This curve is symmetric around the Hateful. Also as you can run across for this distribution, the Mean, Median, and Mode are all the same.

One more important phenomena of a normal distribution is that information technology retains the normal shape throughout, unlike other probability distributions that change their backdrop after a transformation. For a Normal Distribution:

• Production of two Normal Distribution results into a Normal Distribution
• The Sum of 2 Normal Distributions is a Normal Distribution
• Convolution of 2 Normal Distribution is too a Normal Distribution
• Fourier Transformation of a Normal Distribution is also Normal

Starting to realize the power of this incredible concept?

Empirical Rule for Normal Distribution

Have you heard of the empirical rule? Information technology'southward a commonly used concept in statistics (and in a lot of operation reviews too):

According to the Empirical Rule for Normal Distribution:

• 68.27% of data lies within 1 standard deviation of the hateful
• 95.45% of data lies inside ii standard deviations of the mean
• 99.73% of data lies within 3 standard deviations of the mean

Thus, about all the information lies inside 3 standard deviations. This dominion enables us to check for Outliers and is very helpful when determining the normality of any distribution.

What is a Standard Normal Distribution?

Standard Normal Distribution is a special example of Normal Distribution when 𝜇 = 0 and 𝜎  = one. For whatsoever Normal distribution, nosotros can convert it into Standard Normal distribution using the formula:

To understand the importance of converting Normal Distribution into Standard Normal Distribution,  let'southward suppose at that place are two students: Ross and Rachel. Ross scored 65 in the exam of paleontology and Rachel scored eighty in the fashion designing exam.

Can we conclude that Rachel scored meliorate than Ross?

No, because the way people performed in paleontology may be different from the way people performed in fashion designing. The variability may non be the same here.

And then, a direct comparison by just looking at the scores will non piece of work.

Now, allow'southward say the paleontology marks follow a normal distribution with mean 60 and a standard deviation of 4. On the other hand, the fashion designing marks follow a normal distribution with mean 79 and standard deviation of 2.

We will have to calculate the z score by standardization of both these distributions:

Thus, Ross scored 1.25 standard deviations in a higher place the mean score while Rachel scored just 0.5 standard deviations to a higher place the mean score. Hence nosotros can say that Ross Performed better than Rachel.

Allow'due south Talk About the Skewed Distribution

Normal Distribution is symmetric, which ways its tails on ane side are the mirror image of the other side. But this is not the instance with most datasets. Generally, data points cluster on one side more than than the other. We call these types of distributions Skewed Distributions.

Left Skewed Distribution

When information points cluster on the right side of the distribution, and so the tail would be longer on the left side. This is the property of Left Skewed Distribution. The tail is longer in the negative management and so we likewise call information technology Negatively Skewed Distribution.

Hither, Mode > Median > Mean.

In the Normal Distribution, Mean, Median and Mode are equal merely in a negatively skewed distribution, we express the full general relationship between the central tendency measured every bit:

Mode > Median > Hateful

Right Skewed Distribution

When information points cluster on the left side of the distribution, then the tail would be longer on the right side. This is the belongings of Right Skewed Distribution. Here, the tail is longer in the positive management and so nosotros also call it Positively Skewed Distribution.

In a positively skewed distribution, we limited the general relationship between the fundamental trend measures as:

Mode < Median < Mean

How to Cheque the Normality of a Distribution

The big question! To check the normality of data, permit's accept an example where we accept the data of the marks of 1000 students for Mathematics, English, and History. You can observe the Dataset hither.

You can find the code in the later section of this article.

Let's see a few unlike means to check the normality of the distribution that we have.

Histogram

• A Histogram visualizes the distribution of data over a continuous interval
• Each bar in a histogram represents the tabulated frequency at each interval/bin
• In simple words, height represents the frequency for the respective bin (interval)

As you lot tin see hither, Mathematics follows the Normal Distribution, English follows the right-skewed distribution and History follows the left-skewed distribution.

KDE Plots

Histogram results can vary wildly if you set up different numbers of bins or simply alter the offset and end values of a bin. To overcome this, we tin can make employ of the density part.

A density plot is a smoothed, continuous version of a histogram estimated from the data. The most common form of estimation is known as kernel density estimation (KDE). In this method, a continuous curve (the kernel) is fatigued at every individual data signal and all of these curves are then added together to make a single smooth density estimation.

Q_Q Plot

Quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way.

• 2 quantile is known as the Median
• 4 quantile is known equally the Quartile
• x quantile is known as the Decile
• 100 quantile is known as the Percentile

10 quantile will divide the Normal Distribution into 10 parts each having 10 % of the information points. The Q-Q plot or quantile-quantile plot is a scatter plot created by plotting two sets of quantiles against one another.

Here, we will plot theoretical normal distribution quantiles and compare them against observed data quantiles:

For Mathematics Marks, values follow the directly line indicating that they come up from a Normal Distribution. On the other side for English Marks, larger values are larger every bit expected from a Normal Distribution and smaller values are not as small as expected from a Normal Distribution which is besides the case in a right-skewed distribution.

While for History Marks, larger values are not every bit large as expected from a Normal Distribution and smaller values are smaller as expected from a Normal Distribution which happens to be the instance in a left-skewed distribution.

The Concept of Skewness

Skewness is also another measure out to check for normality which tells us the amount and management of the skewed data points. Generally for the value of Skewness:

• If the value is less than -0.five, nosotros consider the distribution to be negatively skewed or left-skewed where data points cluster on the right side and the tails are longer on the left side of the distribution
• Whereas if the value is greater than 0.v, we consider the distribution to be positively skewed or correct-skewed where information points cluster on the left side and the tails are longer on the right side of the distribution
• And finally, if the value is betwixt -0.5 and 0.v, we consider the distribution to be approximately symmetric

What is Kurtosis?

Another numerical mensurate to check for Normality is Kurtosis. Kurtosis gives the information regarding tailedness which basically indicates the data distribution forth the tails.

For the symmetric type of distribution, the Kurtosis value will be close to Zero. We telephone call such types of distributions every bit Mesokurtic distribution. Its tails are similar to Gaussian Distribution.

If in that location are extreme values nowadays in the information, then it means that more information points will lie forth with the tails. In such cases, the value of One thousand will be greater than zero. Here, Tail volition exist fatter and will have longer distribution. Nosotros call such types of distributions as Leptokurtic Distribution. As nosotros can clearly see here, the tails are fatter and denser as compared to Gaussian Distribution:

If there is a depression presence of extreme values compared to Normal Distribution, so lesser data points will lie along the tail. In such cases, the Kurtosis value will be less than nothing. We telephone call such types of distributions as Platykurtic Distribution. It volition take a thinner tail and a shorter distribution in comparison to Normal distribution.

Python Code to Understand Normal Distribution

Here's the full Python code to implement and empathise how a normal distribution works.

import numpy equally np import pandas as pd import seaborn equally sns import matplotlib.pyplot as plt import statsmodels.api every bit sm

df = pd.read_csv('Marks.csv')

def UVA_numeric(data):     var_group = data.columns     size = len(var_group)     plt.effigy(figsize = (7*size,3), dpi = 400)      #looping for each variable     for j,i in enumerate(var_group):                   # calculating descriptives of variable         mini = data[i].min()         maxi = data[i].max()         ran = data[i].max()-information[i].min()         hateful = data[i].mean()         median = data[i].median()         st_dev = data[i].std()         skew = data[i].skew()         kurt = information[i].kurtosis()          # calculating points of standard deviation         points = mean-st_dev, mean+st_dev          #Plotting the variable with every information         plt.subplot(ane,size,j+ane)         sns.distplot(data[i],hist=True, kde=True)                  sns.lineplot(points, [0,0], color = 'blackness', label = "std_dev")         sns.scatterplot([mini,maxi], [0,0], color = 'orange', label = "min/max")         sns.scatterplot([mean], [0], colour = 'reddish', label = "mean")         sns.scatterplot([median], [0], color = 'bluish', label = "median")         plt.xlabel('{}'.format(i), fontsize = 20)         plt.ylabel('density')         plt.title('std_dev = {}; kurtosis = {};\nskew = {}; range = {}\nmean = {}; median = {}'.format((circular(points[0],2),circular(points[1],2)),                                                                                                        circular(kurt,2),                                                                                                        round(skew,ii),                                                                                                        (round(mini,2),circular(maxi,2),round(ran,2)),                                                                                                        round(hateful,2),                                                                                                        circular(median,two)))        

UVA_numeric(df)

End Notes

In this commodity, we followed a step by stride procedure to understand the fundamentals of Normal Distribution. We likewise understood the concepts of determining the Normality, like Histogram, KDE, Q_Q Plot, Skewness, and Kurtosis.

For more than details nearly statistical concepts, y'all can as well read these articles:

• vi Common Probability Distributions every data science professional should know
• Introduction to Central Limit Theorem

Did you find this article useful? Can you retrieve of any other distributions similar to Normal distribution like F-distributions, Chi-Square distribution or t-distribution, and their applications? Let me know in the comments section below and we tin come up upwardly with more ideas to explore them.

Source: https://www.analyticsvidhya.com/blog/2020/04/statistics-data-science-normal-distribution/

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