**Understanding Data Visualization and Transformation Techniques**

Data visualization is an essential tool for understanding complex data sets. By projecting each point onto a high-dimensional space, we can gain insights into the distribution of our data points. However, this technique alone may not provide a comprehensive understanding of the data. To gain further insight, it's necessary to apply additional techniques such as standard deviation or variance to quantify the spread of the data.

**Mean, Variance, and Standard Deviation**

The mean is a crucial aspect of data visualization, representing the average value of our data points. By projecting each point onto a high-dimensional space, we can identify where the mean lies in this new representation. For instance, if we project our data points onto a two-dimensional space, the mean will be represented by a single point. The spread of our data can be quantified using either standard deviation or variance. Standard deviation measures the dispersion of individual data points from the mean value, while variance is a more general term that encompasses both standard deviation and its square root.

**Data Transformation Techniques**

When working with high-dimensional data sets, it's often necessary to apply transformations to facilitate analysis and visualization. One such technique is column standardization. This involves two main steps: centering the mean vector at origin and scaling the data points so that the standard deviation on every axis equals 1. By moving the mean vector to origin, we ensure that our data set is centered around a point of interest. This allows us to better understand the distribution of our data.

**Column Standardization Geometrically**

Geometrically, column standardization can be visualized as follows: when projecting each point onto a high-dimensional space, we initially identify where the mean lies in this new representation. By centering the mean vector at origin, we move this point to the origin. The next step involves squishing or expanding our data points so that their standard deviation on every axis equals 1. If the standard deviation is less than 1, we will squish our data points; if it's greater than 1, we will expand them.

**Implications of Column Standardization**

Column standardization has several implications for data analysis and visualization. By moving the mean vector to origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set. This transformation technique is essential when working with high-dimensional data sets, as it allows us to simplify complex data structures while maintaining key insights.

**Conclusion**

Data visualization and transformation techniques are crucial tools for analyzing complex data sets. By projecting each point onto a high-dimensional space, we can gain insight into the distribution of our data points. Column standardization is an essential technique that involves centering the mean vector at origin and scaling our data points so that their standard deviation on every axis equals 1. This transformation technique has numerous implications for data analysis and visualization, particularly when working with high-dimensional data sets.

**Column Standardization: A Useful Technique in PCA**

Column standardization is a valuable technique that will be explored in more depth when learning about Principal Component Analysis (PCA). By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set. This transformation technique lays the groundwork for the techniques used in PCA to simplify complex data structures while maintaining key insights.

**Applications of Data Transformation**

Data transformation techniques such as column standardization have numerous applications in data analysis and visualization. By applying these techniques, researchers and analysts can gain a deeper understanding of their data sets, identify patterns, and make informed decisions. The techniques discussed here will be applied in the context of PCA to simplify complex data structures while maintaining key insights.

**Transforming Data for Analysis**

Data transformation is an essential aspect of data analysis. By applying techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions. The transformations discussed here will be applied in the context of PCA to simplify complex data structures while maintaining key insights.

**Understanding Data Distribution**

Data distribution is an essential aspect of data analysis. By applying techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions. The transformations discussed here will be applied in the context of PCA to simplify complex data structures while maintaining key insights.

**Geometric Interpretation**

The geometric interpretation of column standardization is essential for understanding its implications on data analysis and visualization. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set. This transformation technique has numerous implications for data analysis and visualization.

**PCA and Column Standardization**

Principal Component Analysis (PCA) is a complex topic that builds upon data transformation techniques such as column standardization. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights. The techniques discussed here will be applied in the context of PCA to analyze high-dimensional data sets.

**Data Visualization Techniques**

Data visualization is an essential tool for analyzing complex data sets. By projecting each point onto a high-dimensional space, we can gain insight into the distribution of our data points. Column standardization is one technique used in data visualization to simplify complex data structures while maintaining key insights.

**PCA and Data Analysis**

Principal Component Analysis (PCA) is a powerful technique that builds upon data transformation techniques such as column standardization. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights. The transformations discussed here will be applied in the context of PCA to analyze high-dimensional data sets.

**Conclusion**

Data transformation techniques such as column standardization are essential for analyzing complex data sets. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set. The transformations discussed here will be applied in the context of PCA to simplify complex data structures while maintaining key insights.

**Column Standardization: A Key Technique**

Column standardization is a crucial technique used in data analysis and visualization. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set. This transformation technique has numerous implications for data analysis and visualization.

**Data Visualization: A Powerful Tool**

Data visualization is a powerful tool for analyzing complex data sets. By projecting each point onto a high-dimensional space, we can gain insight into the distribution of our data points. Column standardization is one technique used in data visualization to simplify complex data structures while maintaining key insights.

**PCA and Data Transformation**

Principal Component Analysis (PCA) builds upon data transformation techniques such as column standardization. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights. The transformations discussed here will be applied in the context of PCA to analyze high-dimensional data sets.

**Conclusion**

Data transformation techniques such as column standardization are essential for analyzing complex data sets. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set. The transformations discussed here will be applied in the context of PCA to simplify complex data structures while maintaining key insights.

**Data Visualization Techniques: A Key Aspect**

Data visualization is a critical aspect of data analysis and interpretation. By projecting each point onto a high-dimensional space, we can gain insight into the distribution of our data points. Column standardization is one technique used in data visualization to simplify complex data structures while maintaining key insights.

**PCA: A Comprehensive Analysis Tool**

Principal Component Analysis (PCA) is a comprehensive analysis tool that builds upon data transformation techniques such as column standardization. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights.

**Conclusion**

Data transformation techniques such as column standardization are essential for analyzing complex data sets. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set. The transformations discussed here will be applied in the context of PCA to analyze high-dimensional data sets.

**Applications of Column Standardization**

Column standardization has numerous applications in data analysis and visualization. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights.

**Geometric Interpretation: A Key Aspect**

The geometric interpretation of column standardization is a critical aspect of understanding its implications on data analysis and visualization. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set.

**PCA and Data Analysis: A Comprehensive Approach**

Principal Component Analysis (PCA) is a comprehensive approach to analyzing complex data sets. By building upon data transformation techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions.

**Column Standardization: A Key Technique for PCA**

Column standardization is a crucial technique used in Principal Component Analysis (PCA). By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights.

**Data Visualization Techniques: A Powerful Tool for Insight**

Data visualization techniques such as column standardization are powerful tools for gaining insight into complex data sets. By projecting each point onto a high-dimensional space and applying transformations to simplify complex data structures, researchers can gain a deeper understanding of their data sets.

**PCA: A Comprehensive Approach to Data Analysis**

Principal Component Analysis (PCA) is a comprehensive approach to analyzing complex data sets. By building upon data transformation techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions.

**Conclusion**

Data transformation techniques such as column standardization are essential for analyzing complex data sets. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set. The transformations discussed here will be applied in the context of PCA to analyze high-dimensional data sets.

**Column Standardization: A Key Technique for Data Analysis**

Column standardization is a crucial technique used in data analysis. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights.

**Data Visualization Techniques: A Powerful Tool for Insight**

Data visualization techniques such as column standardization are powerful tools for gaining insight into complex data sets. By projecting each point onto a high-dimensional space and applying transformations to simplify complex data structures, researchers can gain a deeper understanding of their data sets.

**PCA: A Comprehensive Approach to Data Analysis**

Principal Component Analysis (PCA) is a comprehensive approach to analyzing complex data sets. By building upon data transformation techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions.

**Conclusion**

Data transformation techniques such as column standardization are essential for analyzing complex data sets. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set. The transformations discussed here will be applied in the context of PCA to analyze high-dimensional data sets.

**Applications of Column Standardization**

Column standardization has numerous applications in data analysis and visualization. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights.

**Geometric Interpretation: A Key Aspect**

The geometric interpretation of column standardization is a critical aspect of understanding its implications on data analysis and visualization. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set.

**PCA and Data Analysis: A Comprehensive Approach**

Principal Component Analysis (PCA) is a comprehensive approach to analyzing complex data sets. By building upon data transformation techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions.

**Column Standardization: A Key Technique for PCA**

Column standardization is a crucial technique used in Principal Component Analysis (PCA). By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights.

**Data Visualization Techniques: A Powerful Tool for Insight**

Data visualization techniques such as column standardization are powerful tools for gaining insight into complex data sets. By projecting each point onto a high-dimensional space and applying transformations to simplify complex data structures, researchers can gain a deeper understanding of their data sets.

**PCA: A Comprehensive Approach to Data Analysis**

Principal Component Analysis (PCA) is a comprehensive approach to analyzing complex data sets. By building upon data transformation techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions.

**Conclusion**

Data transformation techniques such as column standardization are essential for analyzing complex data sets. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set. The transformations discussed here will be applied in the context of PCA to analyze high-dimensional data sets.

**Applications of Column Standardization**

Column standardization has numerous applications in data analysis and visualization. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights.

**Geometric Interpretation: A Key Aspect**

The geometric interpretation of column standardization is a critical aspect of understanding its implications on data analysis and visualization. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set.

**PCA and Data Analysis: A Comprehensive Approach**

Principal Component Analysis (PCA) is a comprehensive approach to analyzing complex data sets. By building upon data transformation techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions.

**Column Standardization: A Key Technique for PCA**

Column standardization is a crucial technique used in Principal Component Analysis (PCA). By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights.

**Data Visualization Techniques: A Powerful Tool for Insight**

Data visualization techniques such as column standardization are powerful tools for gaining insight into complex data sets. By projecting each point onto a high-dimensional space and applying transformations to simplify complex data structures, researchers can gain a deeper understanding of their data sets.

**PCA: A Comprehensive Approach to Data Analysis**

Principal Component Analysis (PCA) is a comprehensive approach to analyzing complex data sets. By building upon data transformation techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions.

**Column Standardization: Key Technique for PCA**

Column standardization is a crucial technique used in Principal Component Analysis (PCA). By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights.

**Data Visualization Techniques: Powerful Tool for Insight**

Data visualization techniques such as column standardization are powerful tools for gaining insight into complex data sets. By projecting each point onto a high-dimensional space and applying transformations to simplify complex data structures, researchers can gain a deeper understanding of their data sets.

**PCA: Comprehensive Approach to Data Analysis**

Principal Component Analysis (PCA) is a comprehensive approach to analyzing complex data sets. By building upon data transformation techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions.

**Column Standardization: Key Technique for PCA**

Column standardization is a crucial technique used in Principal Component Analysis (PCA). By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights.

**Data Visualization Techniques: Powerful Tool for Insight**

Data visualization techniques such as column standardization are powerful tools for gaining insight into complex data sets. By projecting each point onto a high-dimensional space and applying transformations to simplify complex data structures, researchers can gain a deeper understanding of their data sets.

**PCA: Comprehensive Approach to Data Analysis**

Principal Component Analysis (PCA) is a comprehensive approach to analyzing complex data sets. By building upon data transformation techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions.

**Conclusion**

Data transformation techniques such as column standardization are essential for analyzing complex data sets. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can better understand the distribution of our data set. The transformations discussed here will be applied in the context of PCA to analyze high-dimensional data sets.

**Column Standardization: Key Technique for Data Analysis**

Column standardization is a crucial technique used in data analysis. By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights.

**Data Visualization Techniques: Powerful Tool for Insight**

Data visualization techniques such as column standardization are powerful tools for gaining insight into complex data sets. By projecting each point onto a high-dimensional space and applying transformations to simplify complex data structures, researchers can gain a deeper understanding of their data sets.

**PCA: Comprehensive Approach to Data Analysis**

Principal Component Analysis (PCA) is a comprehensive approach to analyzing complex data sets. By building upon data transformation techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions.

**Column Standardization: Key Technique for PCA**

Column standardization is a crucial technique used in Principal Component Analysis (PCA). By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights.

**Data Visualization Techniques: Powerful Tool for Insight**

Data visualization techniques such as column standardization are powerful tools for gaining insight into complex data sets. By projecting each point onto a high-dimensional space and applying transformations to simplify complex data structures, researchers can gain a deeper understanding of their data sets.

**PCA: Comprehensive Approach to Data Analysis**

Principal Component Analysis (PCA) is a comprehensive approach to analyzing complex data sets. By building upon data transformation techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions.

**Column Standardization: Key Technique for PCA**

Column standardization is a crucial technique used in Principal Component Analysis (PCA). By centering the mean vector at origin and scaling our data points so that their standard deviation equals 1 on every axis, we can simplify complex data structures while maintaining key insights.

**Data Visualization Techniques: Powerful Tool for Insight**

Data visualization techniques such as column standardization are powerful tools for gaining insight into complex data sets. By projecting each point onto a high-dimensional space and applying transformations to simplify complex data structures, researchers can gain a deeper understanding of their data sets.

**PCA: Comprehensive Approach to Data Analysis**

Principal Component Analysis (PCA) is a comprehensive approach to analyzing complex data sets. By building upon data transformation techniques such as column standardization, researchers can gain a deeper understanding of their data sets, identify patterns, and make informed decisions.

The final answer is: $\boxed{1}$

"WEBVTTKind: captionsLanguage: enwe will learn about a different data pre-processing technique called column standardization we learned about column normalization a while ago in one of the previous videos it in column organization what did we do we took values we took each column and we compressed all of the values between 0 to 1 so as to get rid of scale so as to get rid of scales of each feature and we created our own scale where all the values lie between 0 to 1 that's what column normalization did all right so a similar related technique is called column standardization it's also data pre-processing technique so column standardization is more often used than normalization it's more often used in practice ok because it has some nice relationship to Gaussian distributions and things like that but typically people perform column standardization more often than organization so we also saw the geometric interpretation of normalization right where all the points are squished into unit hypercube we saw this geometric interpretation rate so we'll see similar geometric interpretation of column standardization well rate column standardization short for column standardization so let's see what column standardization is it's a very very simple idea so let's assume I have it I have my data matrix okay with feature one feature two so on so forth feature JSO and so forth feature b12 so on I spy and so on so forth endpoint this is n cross d okay where the it--the so this is your X I transpose this is your data vector now here also just like in column normalization I take a column vector of course just like in column normalization even column standardization we standardize each column being standardized every column let's see how to standardize one column n right so here a 1 a 2 a 3 so on so forth am our n values and n values of feature G of course there will be other columns with other types I'm just taking I'm just showing you how to do column normalization for one column the same thing needs to be done for other columns also so in column standardization what I do is this this is my rod a direct it could be in any scale it could be kilograms inches I don't care okay I'll convert this into into a 1 - a - - a 3 - so on a I - so on so forth n - just like a normalization okay in organization all values as I showed you earlier all values fell in the unit hypercube or everything for each column all the values slide between 0 to 1 here there is a subtle difference here in the transform data in the transformed data so this is called the transform data or standardized data I'll ensure that the mean of the mean of my standardized data mean of AI - for all the values of I - n is 0 and the standard deviation of a - equals to 1 okay I so what column standardization does exactly is it converts your data points it converts your a 1 a 2 a.m. into a 1 - a - - 8 3 - such that the mean of these points could be anything right the standard deviation of these points the sample standard deviation the sample mean could be anything but I want to convert this data in such a way that the mean of the converted or the transformed data is 0 and standard deviation goes to 1 okay this is not boxing all the so here what did we do here we converted so in in data normalization right so what did we do it in column normalization in column normalization we ensured that all your a a - lied in the interval 0 to 1 that's what column normalization here standardization is we're converting into a form such that a a - is the mean of a - is 0 in standard deviation equals to 1 why is that important we'll see Toscanini it's very very useful very very important but this is one form of standardization one form of transforming or getting rid of scale right Here I am saying okay all value should lie between zero and one here I'm saying instead of all value is lying between 0 & 1 I want the mean to be 0 I wouldn't standard deviation to be 1 okay I'm not saying so here remember I'm not assuming anything about the distribution of a1 a2 so on so forth AI so forth so on this can come from any distribution not necessarily Gaussian this can come from any distribution I don't care but through column standardization I'm converting them into a 1 - a - - so on so forth ai - so on so forth am - ok such that their mean is 0 and their variance and so a standard deviation equals to 1 let's say how to do it that's a very very simple way of doing it let's define a bar as the mean of my a ice let's put it this way okay it's a mean of all a ice it's a mean of these not a bars okay or a dashes sorry so the mean of these a is the mean of these a is let's call it a bar let's call s as the standard deviation of money a ice so this is these are sample mean this is sample mean not population mean from probability class you don't you'll I hope you remember what some what is it of snipping sample in population okay this is sample standard deviation okay I can easily compute the rate given a bunch of vectors or given a bunch of scalars in this case sorry I can compute the mean very easily I can also compute the standard deviation using simple formula that we learnt in exploratory analysis right so now what I will do here is my formula to compute AI - which is this yes I will take each AI subtract a bar and divided by standard deviation okay so this will guarantee I'll not prove it here for those of you are interested you could take it as an exercise you could take it an exercise that the mean of ái dashes equals to 0 under standard deviation of AI dash equals to 1 if I if I transfer if I construct my air dash using this formula okay you can actually very easily prove with simple algebra you can prove that the mean of these values mean of my a a dashes is 0 and standard deviation equals to 1 this might look very similar to something else that you have seen when we learnt probability so let me rewrite the formula here this is my formula right if you recall when we learnt about standard normal variate called Z or Z okay we realize that Z is nothing but X minus mu by Sigma where we said if X is normally distributed with mu and Sigma and if I construct my Z like this then my Z will be normally distributed with a mean of 1 and standard deviations or the mean of 0 and standard deviation of 1 right if you recall these two look very very similar this formula looks very similar to this formula but remember your a is could come from any distribution i'mjust can come from any distribution it doesn't matter okay but this is so it's called standardization because your conv you are applying the idea from standard normal variate here to transform your data such that a I dash has a mean of 0 a dashes have a mean of 0 and standard deviation of 1 here by playing this transformation I am creating a new random variable here which has a mean of 0 and a standard deviation of 1 alright very similar it looks exactly similar that's why it's called standardization and this is a standard normal variate that's why it's called standardization now you might ask why is this useful geometrically let's understand what does this mean geometrically okay so let's see what it does geometrically okay imagine if I have two features Heights and weights these are my height these are my age in a in some scale I don't care what the scale is so let's assume these are some some students of people's Heights and weights okay so where does the middle mean of these data point like mean lies here like mean lies here right and the spread of data what is what is standard deviation or what is variance so if I project each of these points onto your high taxes I'll get points like this okay for each point I can I can I can project it and get a point here so if I keep projecting all my points here okay the mean the mean here that will change the color here this will be your mean point corresponding exactly to this right and this is your mean weight this is your mean height and this is your mean weight right so here for your heights the mean is here and there is some variance here variance measures the spread variance measures the spread the more the spread the more the variance I'm not saying variance is this width okay the model variance I should not I cannot show it Jo I cannot show geometrically because the formula is slightly more tricky right so variance measures how spread your data is similarly if I project all these points let's assume all these points fall in this range okay so your mean vector could lie here right and your spread will the more spread the higher the variance or standard deviation because I know deviation is nothing but square root of variance right now by transforming this by transforming this using column standardization what do we get okay this is interest I get a new data set let's call this F 1 - on hi - this is my F 2 dash and height - sorry wait - now I told you that through column standardization I am going to get a mean of mean of zero which means this mean point will love move to zero comma zero so my data my new data will look like this let me draw it for you my new data will look like this for my new data my mean my mean of my data my mean vector will lie it 0 comma 0 since it's 2 dimension if it's high dimension it will it'll I it origin basically this is origin right so I have moved have transformed my space ok what I've done I've basically taken these points I have moved all these points such that the mean moves to 0 ok and I've also ensured that I also squish these points I also squished these points I also squished these points so first thing I did was I moved my mean to to Center and I'm squishing these points such that the variance that I get now so this is the spirit that I get on x-axis right now and this is the spirit that I get on y-axis isn't it I'm squishing these points such that the variance or the standard deviation on on any axis right is 1 so literally what I am i doing I am basically first I paste I'm moving the mean vector moving the mean vector to origin ok and now second is I made the squishing or expanding I might also expand it if let's say my variance here is it's a point 5 that's my standard deviation my standard deviation my sample standard deviation is let say point 5 then I need to expand these points right because what is a guarantee here the guarantee in my transformed space is that the standard deviation is 1 on any axis so if my standard deviation here is less than 1 I'm going to expand this point I'm going to pull these points stretch these points away if my standard deviation here let's say is 5 then I'm going to push I'm going to compress these points I'm going to squish this point such that the standard deviation so I'm gonna squish or expand such that the standard deviation for any feature is 1 this is the essence of your column standardization geometrically ok geometrically I'm moving I'll repeat it just just for clarity I'm moving the mean vector to origin ok and I might as well squish the points or expand the points I'll squish it if the way if the standard deviation is less than 1 so I'll squish it if the standard deviation is greater than 1 I will expand it if the standard deviation is less than 1 so I have to make the standard deviation equals to 1 on every axis on your f1 - axis or f - n - axis now this is called this people also call it as mean Center so people also Center it and then scale it so your your column standardization this is often called as mean centering because you're centering the mean at origin followed by followed by scaling you're scaling it such that means a drink basically is nothing but moving the mean vector to origin scaling basically means you are ensuring that standard deviation on every axis is equal to one standard deviation for all features equals to 1 we'll see why this is useful you'll see why data y-column standardization is very very useful when we learn PCA which is which is the next important topic at our that will learn in in n-dimensional productionwe will learn about a different data pre-processing technique called column standardization we learned about column normalization a while ago in one of the previous videos it in column organization what did we do we took values we took each column and we compressed all of the values between 0 to 1 so as to get rid of scale so as to get rid of scales of each feature and we created our own scale where all the values lie between 0 to 1 that's what column normalization did all right so a similar related technique is called column standardization it's also data pre-processing technique so column standardization is more often used than normalization it's more often used in practice ok because it has some nice relationship to Gaussian distributions and things like that but typically people perform column standardization more often than organization so we also saw the geometric interpretation of normalization right where all the points are squished into unit hypercube we saw this geometric interpretation rate so we'll see similar geometric interpretation of column standardization well rate column standardization short for column standardization so let's see what column standardization is it's a very very simple idea so let's assume I have it I have my data matrix okay with feature one feature two so on so forth feature JSO and so forth feature b12 so on I spy and so on so forth endpoint this is n cross d okay where the it--the so this is your X I transpose this is your data vector now here also just like in column normalization I take a column vector of course just like in column normalization even column standardization we standardize each column being standardized every column let's see how to standardize one column n right so here a 1 a 2 a 3 so on so forth am our n values and n values of feature G of course there will be other columns with other types I'm just taking I'm just showing you how to do column normalization for one column the same thing needs to be done for other columns also so in column standardization what I do is this this is my rod a direct it could be in any scale it could be kilograms inches I don't care okay I'll convert this into into a 1 - a - - a 3 - so on a I - so on so forth n - just like a normalization okay in organization all values as I showed you earlier all values fell in the unit hypercube or everything for each column all the values slide between 0 to 1 here there is a subtle difference here in the transform data in the transformed data so this is called the transform data or standardized data I'll ensure that the mean of the mean of my standardized data mean of AI - for all the values of I - n is 0 and the standard deviation of a - equals to 1 okay I so what column standardization does exactly is it converts your data points it converts your a 1 a 2 a.m. into a 1 - a - - 8 3 - such that the mean of these points could be anything right the standard deviation of these points the sample standard deviation the sample mean could be anything but I want to convert this data in such a way that the mean of the converted or the transformed data is 0 and standard deviation goes to 1 okay this is not boxing all the so here what did we do here we converted so in in data normalization right so what did we do it in column normalization in column normalization we ensured that all your a a - lied in the interval 0 to 1 that's what column normalization here standardization is we're converting into a form such that a a - is the mean of a - is 0 in standard deviation equals to 1 why is that important we'll see Toscanini it's very very useful very very important but this is one form of standardization one form of transforming or getting rid of scale right Here I am saying okay all value should lie between zero and one here I'm saying instead of all value is lying between 0 & 1 I want the mean to be 0 I wouldn't standard deviation to be 1 okay I'm not saying so here remember I'm not assuming anything about the distribution of a1 a2 so on so forth AI so forth so on this can come from any distribution not necessarily Gaussian this can come from any distribution I don't care but through column standardization I'm converting them into a 1 - a - - so on so forth ai - so on so forth am - ok such that their mean is 0 and their variance and so a standard deviation equals to 1 let's say how to do it that's a very very simple way of doing it let's define a bar as the mean of my a ice let's put it this way okay it's a mean of all a ice it's a mean of these not a bars okay or a dashes sorry so the mean of these a is the mean of these a is let's call it a bar let's call s as the standard deviation of money a ice so this is these are sample mean this is sample mean not population mean from probability class you don't you'll I hope you remember what some what is it of snipping sample in population okay this is sample standard deviation okay I can easily compute the rate given a bunch of vectors or given a bunch of scalars in this case sorry I can compute the mean very easily I can also compute the standard deviation using simple formula that we learnt in exploratory analysis right so now what I will do here is my formula to compute AI - which is this yes I will take each AI subtract a bar and divided by standard deviation okay so this will guarantee I'll not prove it here for those of you are interested you could take it as an exercise you could take it an exercise that the mean of ái dashes equals to 0 under standard deviation of AI dash equals to 1 if I if I transfer if I construct my air dash using this formula okay you can actually very easily prove with simple algebra you can prove that the mean of these values mean of my a a dashes is 0 and standard deviation equals to 1 this might look very similar to something else that you have seen when we learnt probability so let me rewrite the formula here this is my formula right if you recall when we learnt about standard normal variate called Z or Z okay we realize that Z is nothing but X minus mu by Sigma where we said if X is normally distributed with mu and Sigma and if I construct my Z like this then my Z will be normally distributed with a mean of 1 and standard deviations or the mean of 0 and standard deviation of 1 right if you recall these two look very very similar this formula looks very similar to this formula but remember your a is could come from any distribution i'mjust can come from any distribution it doesn't matter okay but this is so it's called standardization because your conv you are applying the idea from standard normal variate here to transform your data such that a I dash has a mean of 0 a dashes have a mean of 0 and standard deviation of 1 here by playing this transformation I am creating a new random variable here which has a mean of 0 and a standard deviation of 1 alright very similar it looks exactly similar that's why it's called standardization and this is a standard normal variate that's why it's called standardization now you might ask why is this useful geometrically let's understand what does this mean geometrically okay so let's see what it does geometrically okay imagine if I have two features Heights and weights these are my height these are my age in a in some scale I don't care what the scale is so let's assume these are some some students of people's Heights and weights okay so where does the middle mean of these data point like mean lies here like mean lies here right and the spread of data what is what is standard deviation or what is variance so if I project each of these points onto your high taxes I'll get points like this okay for each point I can I can I can project it and get a point here so if I keep projecting all my points here okay the mean the mean here that will change the color here this will be your mean point corresponding exactly to this right and this is your mean weight this is your mean height and this is your mean weight right so here for your heights the mean is here and there is some variance here variance measures the spread variance measures the spread the more the spread the more the variance I'm not saying variance is this width okay the model variance I should not I cannot show it Jo I cannot show geometrically because the formula is slightly more tricky right so variance measures how spread your data is similarly if I project all these points let's assume all these points fall in this range okay so your mean vector could lie here right and your spread will the more spread the higher the variance or standard deviation because I know deviation is nothing but square root of variance right now by transforming this by transforming this using column standardization what do we get okay this is interest I get a new data set let's call this F 1 - on hi - this is my F 2 dash and height - sorry wait - now I told you that through column standardization I am going to get a mean of mean of zero which means this mean point will love move to zero comma zero so my data my new data will look like this let me draw it for you my new data will look like this for my new data my mean my mean of my data my mean vector will lie it 0 comma 0 since it's 2 dimension if it's high dimension it will it'll I it origin basically this is origin right so I have moved have transformed my space ok what I've done I've basically taken these points I have moved all these points such that the mean moves to 0 ok and I've also ensured that I also squish these points I also squished these points I also squished these points so first thing I did was I moved my mean to to Center and I'm squishing these points such that the variance that I get now so this is the spirit that I get on x-axis right now and this is the spirit that I get on y-axis isn't it I'm squishing these points such that the variance or the standard deviation on on any axis right is 1 so literally what I am i doing I am basically first I paste I'm moving the mean vector moving the mean vector to origin ok and now second is I made the squishing or expanding I might also expand it if let's say my variance here is it's a point 5 that's my standard deviation my standard deviation my sample standard deviation is let say point 5 then I need to expand these points right because what is a guarantee here the guarantee in my transformed space is that the standard deviation is 1 on any axis so if my standard deviation here is less than 1 I'm going to expand this point I'm going to pull these points stretch these points away if my standard deviation here let's say is 5 then I'm going to push I'm going to compress these points I'm going to squish this point such that the standard deviation so I'm gonna squish or expand such that the standard deviation for any feature is 1 this is the essence of your column standardization geometrically ok geometrically I'm moving I'll repeat it just just for clarity I'm moving the mean vector to origin ok and I might as well squish the points or expand the points I'll squish it if the way if the standard deviation is less than 1 so I'll squish it if the standard deviation is greater than 1 I will expand it if the standard deviation is less than 1 so I have to make the standard deviation equals to 1 on every axis on your f1 - axis or f - n - axis now this is called this people also call it as mean Center so people also Center it and then scale it so your your column standardization this is often called as mean centering because you're centering the mean at origin followed by followed by scaling you're scaling it such that means a drink basically is nothing but moving the mean vector to origin scaling basically means you are ensuring that standard deviation on every axis is equal to one standard deviation for all features equals to 1 we'll see why this is useful you'll see why data y-column standardization is very very useful when we learn PCA which is which is the next important topic at our that will learn in in n-dimensional production\n"