Handling Highly Skewed Data
WHY DO WE CARE SO MUCH ABOUT NORMALITY?
Most of the parametric machine learning models like LDA, Linear Regression and any more assume that the data is normally distributed. If this assumption fails the model fails to give accurate predictions.
Everyone Learn statistics but they fail to apply But We are here to make it easy.
WHAT IS NORMAL DISTRIBUTION?
A probability distribution with the mean 0 and standard deviation of 1 is known as standard normal distribution or Gaussian distribution. A normal distribution is symmetric about the mean and follows a bell shaped curve. And almost 99.7% of the values lie within 3 standard deviation. The mean, median and mode of a normal distribution are equal.
Handling Skewness
The skewness of a distribution is defined as the lack of symmetry. In a symmetrical distribution, the Mean, Median and Mode are equal. The normal distribution has a skewness of 0.
Skewness tells us about the distribution of our data.
Skewness is of two types:
Positive skewness: When the tail on the right side of the distribution is longer or fatter, we say the data is positively skewed.
For a positive skewness mean > median > mode.
Negative skewness: When the tail on the left side of the distribution is longer or fatter, we say that the distribution is negatively skewed.
For a negative skewness mean < median < mode.
Let’s have clear picture of skewness
Skewness of a distribution is defined as the lack of symmetry. In a symmetrical distribution, the Mean, Median and Mode are equal to each other. The normal distribution has a skewness of 0. Skewness tells us about where most of the values are concentrated on an ascending scale.
Now, the question is when we can say our data is moderately skewed or heavily skewed?
The thumb rule is:
If the skewness is between 0.5 to +0.5 then we can say data is fairly symmetrical. If the skewness is between 1 to 0.5 or 0.5 to 1 then data is moderately skewed.
And if the skewness is less than 1 and greater than +1 then our data is heavily skewed.
Types of Skewness
Positive skewness: In simple words, if the skewness is greater than 0 then the distribution is positively skewed. The tail on the right side of the distribution will be longer or flatter. If the data is positively skewed than most of values will be concentrated below the average value of the data.
Negative skewness: If the skewness is less than 0 then the distribution is negatively skewed. For negatively skewed data, most of the values will be concentrated above the average value and tail on the left side of the distribution will be longer of flatter.
What does skewness tells us?
To understand this better consider a example.
Consider house prices ranging from 100k to 1,000,000 with the average being 500,000.
If the peak of the distribution is in left side that means our data is positively skewed and most of the houses are being sold at the price less than the average.
If the peak of the distribution is in right side that means our data is negatively skewed and most of the houses are being sold at the price greater than the average.
Skewness of a data indicates the direction and relative magnitude of a distribution’s deviation from the normal distribution. Skewness considers the extremes of the dataset rather than concentrating only on the average. Investors need to look at the extremes while judging the return from market as they are less likely to depend on the average value to work out.
Many models assume normal distribution but in reality data points may not be perfectly symmetric. If the data are skewed, then this kind of model will always underestimate the skewness risk. The more the data is skewed the less accurate the model will be.
Here,
skew of raw data is positive and greater than 1, right tail of the data is skewed. skew of raw data is negative and less than 1, left tail of the data is skewed.
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All dataset used can be downloaded from this link:
https://drive.google.com/file/d/1mMbTuA7NsGGK5qxJC1gxjP4IfY9vTEP_/view?usp=sharing
About Dataset
This dataset tells the details about the cars. The car Dataset contains the following attributes:
index: Unnamed: 0(index values)
price: The sale price of the vehicle in the ad brand: The brand of car
model: model of the vehicle
year: The vehicle registration year
title_status: This feature included binary classification, which are clean title vehicles and salvage insurance mileage: miles traveled by vehicle
color: Color of the vehicle
vinThe: vehicle identification number is a collection of 17 characters (digits and capital letters)
lot: A lot number is an identification number assigned to a particular quantity or lot of material from a single manufacturer. For cars, a lot number is combined with a serial number to form the Vehicle Identification Number.
state: The location in which the car is being available for purchase country: The location in which the car is being available for purchase condition: Time
In  [3]:  import pandas as pd import numpy as np 
In  [4]:  #reading the dataset data = pd.read_csv(r’cars_datasets.csv’) 
In  [5]:  #look at dataset data.head() 
Out[5]:
Unnamed:
0
price brand model year title_status mileage color vin lot state coun
 0 6300 toyota cruiser 2008 clean vehicle 0 black jtezu11f88k007763 159348797 new
jersey
usa
 1 2899 ford se 2011 clean vehicle 0 silver 2fmdk3gc4bbb02217 166951262 tennessee usa
 2 5350 dodge mpv 2018 clean vehicle 0 silver 3c4pdcgg5jt346413 167655728 georgia usa
 3 25000 ford door 2014 clean vehicle 0 blue 1ftfw1et4efc23745 167753855 virginia usa
 4 27700 chevrolet 1500 2018 clean vehicle 0 red 3gcpcrec2jg473991 167763266 florida usa
In [6]:
#Removing the unnamed: 0 column
data = data.drop([‘Unnamed: 0’], axis = 1)
In [7]:
data.head()
Out[7]:
price brand model year title_status mileage color vin lot state country conditio
 6300 toyota cruiser 2008 clean vehicle 0 black jtezu11f88k007763 159348797 new
jersey
usa 10 days left
 2899 ford se 2011 clean vehicle 0 silver 2fmdk3gc4bbb02217 166951262 tennessee usa 6 days lef
 5350 dodge mpv 2018 clean vehicle 0 silver 3c4pdcgg5jt346413 167655728 georgia usa 2 days lef
 25000 ford door 2014 clean vehicle 0 blue 1ftfw1et4efc23745 167753855 virginia usa 22 hours
left
 27700 chevrolet 1500 2018 clean vehicle 0 red 3gcpcrec2jg473991 167763266 florida usa 22 hours
left
In [8]:
#Exploring the dataset information
data.info()
Out[8]:
<class ‘pandas.core.frame.DataFrame’> RangeIndex: 2499 entries, 0 to 2498 Data columns (total 12 columns):
price  2499  nonnull  int64 
brand  2499  nonnull  object 
model  2499  nonnull  object 
year  2499  nonnull  int64 
title_status  2499  nonnull  object 
mileage  2499  nonnull  float64 
color  2499  nonnull  object 
vin  2499  nonnull  object 
lot  2499  nonnull  int64 
state  2499  nonnull  object 
country  2499  nonnull  object 
condition  2499  nonnull  object 
dtypes: float64(1), int64(3), object(8) memory usage: 234.4+ KB
In [9]:
#checking dimension of dataset
data.shape
Out[9]:
(2499, 12)
In [10]:
#importing library for visualizing dataset and plotting the histogram for price attributes
import seaborn as sns
data[‘price’].hist(grid = False)
Out[10]:
<matplotlib.axes._subplots.AxesSubplot at 0x1dee79edb38>
In [11]:
# Checking the skewness of Price column of dataset
data[‘price’].skew()
Out[11]:
0.9227307836499805
In [12]:
#density plot
sns.distplot(data[‘price’], hist = True)
Out[12]:
<matplotlib.axes._subplots.AxesSubplot at 0x1dee8a0a048>
Note: As we can see the skewed values lies between the 0.5 to 1 range. so, data is moderately skewed and right skewed(but it’s fine to train the model with it). Let’s explore another attribute.
In [13]:
# checking the skewness of mileage column of dataset
data[‘mileage’].skew()
Out[13]:
7.0793210165347915
In [14]:
sns.distplot(data[‘mileage’], hist = True)
Out[14]:
<matplotlib.axes._subplots.AxesSubplot at 0x1dee8b2fa20>
Note: As we can see the skewed values lies between 1 and greater than +1 then our data is heavily skewed. so, data is heavily skewed. data[‘mileage’] is rightskewed by looking at the graph and skewed values.
How to handle these skewed data?
Transformation
In data analysis transformation is the replacement of a variable by a function of that variable: for example, replacing a variable x by the square root of x or the logarithm of x. In a stronger sense, a transformation is a replacement that changes the shape of distribution or relationship.
Steps to do the transformation
 Draw a graph(histogram and density plot) of the data to see how far patterns in data match the simplest ideal patterns.
 check the range of the data. Because Transformations will have little effect if the range checks the skewness by statistical methods(decide right and left skewness).
 apply the methods (explained in detail below) to handle the skewness based on the skewed
Reasons for using transformations
There are many reasons for transformation.
 Convenience
 Reducing skewness
 Equal spreads
 Linear relationships
 Additive relationships
 Convenience: A transformed scale may be as natural as the original scale and more convenient for a specific purpose. for example percentage rather than the original
 Reducing skewness: A transformation may be used to reduce A distribution that is symmetric or nearly so is often easier to handle and interpret than a skewed distribution.
 To handle the right skewness, we use:
 logarithms (best for it)
 roots[square root and cube root] (good)
 reciprocals (weak)
 To handle left skewness, we use:
 squares
 cubes
 higher
 Equal spreads: A transformation may be used to produce approximately equal spreads, despite marked variations in level, which again makes data easier to handle and interpret. Each data set or subset having about the same spread or variability is a condition called homoscedasticity and it’s opposite is called heteroscedasticity
 Linear relationships: When looking at relationships between variables, it is often far easier to think about patterns that are approximately linear than about patterns that are highly curved. This is vitally important when using linear regression, which amounts to fitting such patterns to
 Additive relationships: Relationships are often easier to analyses when additive rather than (say)
So,
y = a + bx
In which two terms a and bx are added is easier to deal with, than
y = ax^b
In which two terms a and x^b are multiplied. Additivity is a vital issue in analysis of variance (in Stata, anova, oneway, etc.).
To Handle Right Skewedness
log Transformation
The log transformation is widely used in research to deal with skewed data. It is the best method to handle the rightskewed data.
Why log?
The normal distribution is widely used in basic research studies to model continuous outcomes. Unfortunately, the symmetric bellshaped distribution often does not adequately describe the observed data from research projects. Quite often data arising in real studies are so skewed that standard statistical analyses of these data yield invalid results.
Many methods have been developed to test the normality assumption of observed data. When the distribution of the continuous data is nonnormal, transformations of data are applied to make the data as “normal” as possible and, thus, increase the validity of the associated statistical analyses.
Popular use of the log transformation is to reduce the variability of data, especially in data sets that include outlying observations. Again, contrary to this popular belief, log transformation can often increase – not reduce
– the variability of data whether or not there are outliers.
Why not?
Using transformations in general and log transformation in particular can be quite problematic. If such an approach is used, the researcher must be mindful about its limitations, particularly when interpreting the relevance of the analysis of transformed data for the hypothesis of interest about the original data.
In [15]:
#performing the log transformation using numpy
log_mileage = np.log(data[‘mileage’]) log_mileage.head(15)
Out[15]:
0 12.521310
1 12.157680
2 10.586332
3 11.068917
4 8.802973
5 10.726807
6 11.912037
7 10.065819
8 9.145375
9 11.057503
10 11.588552
11 10.587846
12 10.039285
13 11.839708
14 11.520467
Name: mileage, dtype: float64
In [16]:
#checking the skewness after the logtransformation
log_mileage.skew()
Out[16]:
nan
It’s giving us nan because there are some values as the zero. In log transformation, it deals with only the positive and negative numbers, not with zero. The log is the range in between ( infinity to infinity) but greater or less than zero. For better understanding you can check the log graph below:
Note: In the graph, the line at zero deviates towards the positive infinity. so, If you are getting zeros inside the data, refer to root Transformation.
2. Root Transformation
2.1. Square root Transformation
The square root means x to x^(1/2) = sqrt(x), is a transformation with a moderate effect on distribution shape. it is weaker than the logarithm and the cube root.
It is also used for reducing right skewness, and also has the advantage that it can be applied to zero values. Note that the square root of an area has the units of a length. It is commonly applied to counted data, especially if the values are mostly rather small.
In [17]:
#calculating the square root for data[‘mileage’] column
sqrt_mileage = np.sqrt(data[‘mileage’])
sqrt_mileage.head(15)
Out[17]:
0 523.561840
1 436.522623
2 198.972360
3 253.270606
4 81.572054
5 213.450228
6 386.069942
7 153.378617
8 96.803926
9 251.829307
10 328.414372
11 199.123078
12 151.357193
13 372.357355
14 317.422431
Name: mileage, dtype: float64
In [18]:
#calculation skewness after calculating the square root & we can observe the change in the value of skewness
sqrt_mileage.skew()
Out[18]:
1.6676282633339148
In [19]:
#visualising by density plot
sns.distplot(sqrt_mileage, hist = True)
Out[19]:
<matplotlib.axes._subplots.AxesSubplot at 0x1dee8a5d710>
Note: In the previous case, we got the nan because of zero, but the square root transformation has reduced the skewed values from 7.07 to 1.66. Which is nearer to zero compared to 7.07.
2.2. Cube root Transformation
The cube root means x to x^(1/3). This is a fairly strong transformation with a substantial effect on distribution shape,
It is weaker than the logarithm but stronger than the square root transformation.
It is also used for reducing right skewness and has the advantage that it can be applied to zero and negative values. Note that the cube root of a volume has the units of a length. It is commonly applied to rainfall data.
In [20]:
#calculating the cube root for the column
data[‘mileage’] column
cube_root_mileage = np.cbrt(data[‘mileage’])
cube_root_mileage.head(15)
Out[20]:

In [21]:
#calculation skewness after calculating the cube root
cube_root_mileage.skew()
Out[22]:
0.6866069687334178
In [23]:
#visualising by density plot
sns.distplot(cube_root_mileage, hist = True)
Out[23]:
<matplotlib.axes._subplots.AxesSubplot at 0x1dee8cce358>
Note: In logarithm transformation, we got the nan because of zero, and in the square root transformation it has reduced the skewed values from 7.07 to 1.66. but now in cube root transformation, the skewed values reduced to
0.68. and it is very much near to zero compare to 1.66 and 7.07.
3. Reciprocals Transformation
The reciprocal, x to 1/x, with its sibling the negative reciprocal, x to 1/x, is a very strong transformation with a drastic effect on distribution shape.
It cannot be applied to zero values. Although it can be applied to negative values, it is not useful unless all values are positive.
For Example: we might want to multiply or divide the results of taking the reciprocal by some constant, such as 100 or 1000, to get numbers that are easy to manage, but that it has no effect on skewness or linearity.
In [24]:
#calculating the reciprocal for the column data[‘mileage’] column
recipr_mileage = np.reciprocal(data[‘mileage’])
recipr_mileage.head(15)
c:\users\Hemant\appdata\local\programs\python\python37\lib\sitepackages\pandas\core\series.py:679: RuntimeWarning: divide by z ero encountered in reciprocal
result = getattr(ufunc, method)(*inputs, **kwargs)
Out[24]:
0  0.000004  
1  0.000005  
2  0.000025  
3  0.000016  
4  0.000150  
5  0.000022  
6  0.000007  
7  0.000043  
8  0.000107  
9  0.000016  
10  0.000009  
11  0.000025  
12  0.000044  
13  0.000007  
14  0.000010  
Name:  mileage,  dtype: float64 
In [25]:
recipr_mileage.skew()
Out[25]:
nan
Note: It’s giving output as nan because there are some values as the zero. In reciprocal transformation, it’s good to deal with negative numbers, not with zero.
To Handle Left skewness
In [26]:
#let’s create small dataset & have alook on left skewness
Data = [[‘sameer’,10],[‘pankaj’,20],[‘sam’,30],[‘Hemant’,48],[‘vivek’,62],[‘ram’,87],[‘suman’, 93], [‘anup’,85],[‘mohit’,60],[‘sandeep’,75],[‘ajeet’,84],
[‘yash’,90], [‘sam’, 99], [‘deepak’, 92],[‘vikas’, 99],[‘rajiv’,95],[‘sivam’, 94],[‘akash’, 89],[‘vinod’,90],[‘Kundan’,87],[‘Abhay’,99]]
#converting in dataframe
df = pd.DataFrame(Data, columns=[‘Name’,’Marks’] )
df.head(10)
Out[26]:

In [27]:
#ploting the Density & histogram plot
import seaborn as sns
sns.distplot(df[‘Marks’], hist = True)
Out[27]:
<matplotlib.axes._subplots.AxesSubplot at 0x1dee8deb630>
In [28]:
#checking the skewness
df[‘Marks’].skew()
Out[28]:
1.4076657771292151
Note: If the skewness is less than 1 and greater than +1 then our data is heavily skewed. Our data is leftskewed here; the skewed value is less than 1. Let’s try to make it symmetric.
squares Transformation
The square, x to x^2, has a moderate effect on distribution shape and it could be used to reduce left skewness.
Squaring usually makes sense only if the variable concerned is zero or positive, given that (x)^2 and x^2 are identical.
In [29]:
#calculating the square for the column df[‘Marks’]
column Square_marks = np.square(df[‘Marks’])
Square_marks.head(10)
Out[29]:

Name: Marks, dtype: int64
In [30]:
#checking the skewness
Square_marks.skew()
Out[30]:
0.9341854225868288
In [31]:
#plotting the density and histogram plot
sns.distplot(Square_marks, hist=True)
Out[31]:
<matplotlib.axes._subplots.AxesSubplot at 0x1dee8e72f98>
Note: After applying the square Transformation, we are getting the skewed value as 0.93. If the skewed value lies in between 1 to 0.5 then data is moderately skewed. Let’s try some other transformation.
Cubes Transformation
The cube, x to x³, has a better effect on distribution shape than squaring and it could be used to reduce left skewness.
In [32]:
#calculating the Cubes for the column df[‘Marks’]
column cube_marks = np.power(df[‘Marks’], 3)
cube_marks.head(10)
Out[32]:
0  1000 
1  8000 
2  27000 
3  110592 
4  238328 
5  658503 
6  804357 
7  614125 
8  216000 
9  421875 
Name: Marks, dtype: int64
In [33]:
#calculating the skewness
cube_marks.skew()
Out[33]:
0.6133662709032679
In [34]:
#plotting the density and histogram plot
sns.distplot(cube_marks, hist= True)
Out[34]:
<matplotlib.axes._subplots.AxesSubplot at 0x1dee8ee2240>
Note: After applying the cube transformation, the skewed value is 0.6 , and If the skewed value lies in between 1 to 0.5 then data is moderately skewed. Let’s try some other transformation.
higher powers
When simple transformation like square and cubes doesn’t reduce the skewness in the data distribution, we can use higher powers to transform to data. It is only useful in left skewness.
In [35]:
#calculating the Higher power(power = 4) for the column df[‘Marks’] column
higher_power_4 = np.power(df[‘Marks’], 4)
higher_power_4.head(15)
Out[35]:
0 10000
1 160000
2 810000
3 5308416
4 14776336
5 57289761
6 74805201
7 52200625
8 12960000
9 31640625
10 49787136
11 65610000
12 96059601
13 71639296
14 96059601
Name: Marks, dtype: int64
In [36]:
#calculating the skewness
higher_power_4.skew()
Out[36]:
0.3563776896040546
In [37]:
#plotting the density and histogram
sns.distplot(higher_power_4, hist = True)
Out[37]:
<matplotlib.axes._subplots.AxesSubplot at 0x1dee8f03668>
Note: After applying the higher power (power = 4) the skewness is changed from 1.4 to 0.3. If the skewness is between 0.5 to +0.5 then we can say data is fairly symmetrical. So, finally, we have got the best result and we got the skew value as 0.3.
In case if we would not have got this skew value still after applying these many powers, we can increase the power to get a better result.
You can check out the below for better understanding:
In [38]:
#applying the higher power(power = 5) and calculating the skewness
higher_power_5 = np.power(df[‘Marks’], 5)
higher_power_5.skew()
Out[38]:
0.12781688683710232
In [39]:
#plotting the density and histogram
sns.distplot(higher_power_5, hist = True)
Out[39]:
<matplotlib.axes._subplots.AxesSubplot at 0x1dee8fb60b8>
In [40]:
#applying the higher power(power = 5) and calculating the skewness
higher_power_6 = np.power(df[‘Marks’], 6) higher_power_6.skew()
Out[40]:
0.08406219634567366
In [41]:
#plotting the density and histogram
sns.distplot(higher_power_6, hist= True)
Out[41]:
<matplotlib.axes._subplots.AxesSubplot at 0x1dee8fb63c8>
Note: Finally, we have got the skewed value as 0.08 (almost 0), and we can see the data is not symmetrically distributed.
Let’s look into some more examples,
In [42]:
import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
from scipy.stats import skew, skewtest, norm
from scipy.stats import kurtosis
In [43]:
data1=pd.read_csv(‘house1.csv’)
In [44]:
data1.head(2)
Out[44]:
Id  MSSubClass  MSZoning  LotFrontage  LotArea  Street  Alley  LotShape  LandContour  Utilities  …  PoolArea 
0 1  60  RL  65.0  8450  Pave  NaN  Reg  Lvl  AllPub  …  0 
1 2  20  RL  80.0  9600  Pave  NaN  Reg  Lvl  AllPub  …  0 
2 rows × 81 columns
Let’s check the distribution of the “SalePrice”
In [45]:
data1[‘SalePrice’].describe()
Out[45]:

Name: SalePrice, dtype: float64
Here we can see that Mean (180921) is greater than the median(163000) and the maximum is 3.5 times 75%. (The distribution is positively skewed).
We can say that most of the house prices are below the average.
Let’s plot and check
In [46]:
#Plot and check the distribution
plt.figure(figsize=(12,6))
sns.distplot(data1[‘SalePrice’],fit=norm, color =”r”)
plt.show()
Out[46]:
The histogram confirms that our dataset is positively skewed.
Now let’s check the measure of skewness and kurtosis
In [47]:
print(“Skew of raw data: %f” % data1[‘SalePrice’].skew()) #check skewness
print(“Kurtosis of raw data: %f” % kurtosis(data1[‘SalePrice’],fisher = False)) #check kurtosis
Out[47]:
Skew of raw data: 1.882876 Kurtosis of raw data: 9.509812
Let’s work with highly kurtosis data.
Note::In this notebook we have focused on skewed dataset & soon how to handle highly skewed dataset will be released soon
Here, the skew of raw data is positive and greater than 1, and kurtosis is greater than 3, right tail of the data is skewed. So, our data in this case is positively skewed and leptokurtic.
Note If we are keeping ‘fisher=True’, then kurtosis of normal distribution will be 0. Similarly, kurtosis >0 will be leptokurtic and kurtosis < 0 will be Platykurtic
Log Transformation
A logarithm is defined only for positive values so we can’t apply log transformation on 0 and negative numbers.
Logarithmic transformation is a convenient means of transforming a highly skewed variable into a more normalized dataset. When modelling variables with nonlinear relationships, the chances of producing errors may also be skewed negatively. Using the logarithm of one or more variables improves the fit of the model by transforming the distribution of the features to a more normallyshaped bell curve.
In [48]:
#log transformation
data1[‘Log_SalePrice’] = np.log(data1[‘SalePrice’])
#check distribution,skewness and kurtosis
sns.distplot(data1[‘Log_SalePrice’], fit=norm,color =”r”)
print(“Skew after Log Transformation: %f” % data1[‘Log_SalePrice’].skew())
print(“Kurtosis after Log Transformation: %f” % kurtosis(data1[‘Log_SalePrice’],fisher = False)) data1[‘Log_SalePrice’].describe()
Skew after Log Transformation: 0.121335 Kurtosis after Log Transformation: 3.802656
Out[48]:
count  1460.000000 
mean  12.024051 
std  0.399452 
min  10.460242 
25%  11.775097 
50%  12.001505 
75%  12.273731 
max  13.534473 
Name: Log_SalePrice, dtype: float64
Now if you look the distribution it is close to normal distribution. We have also reduced the skewness and the kurtosis.
Let’s apply a linear regression model and check how well model performs before and after we apply log transformation.
In [49]:
import statsmodels.api as sm
from statsmodels.formula.api import ols
f = ‘SalePrice~GrLivArea’
model = ols(formula=f, data=data1).fit()
fig = plt.figure(figsize =(15,8))
fig = sm.graphics.plot_regress_exog(model, ‘GrLivArea’, fig=fig)
We can notice that it has a cone shape where the data points essentially scatter off as we increase in GrLivArea.
Apply Log on GrLivArea
In [50]:
#log transformation on GrLivArea
data1[‘Log_GrLivArea’] = np.log(data1[‘GrLivArea’])
Apply Linear Regression on Log transformed SalePrice and GrLivArea
In [51]:
import statsmodels.api as sm
from statsmodels.formula.api import ols
f = ‘Log_SalePrice~Log_GrLivArea’
model = ols(formula=f, data=data1).fit()
fig = plt.figure(figsize =(15,8))
fig = sm.graphics.plot_regress_exog(model, ‘Log_GrLivArea’, fig=fig)
We can now see the relationship as a per cent change. By applying the logarithm to the variables, there is a much more distinguished and or adjusted linear regression line through the base of the data points, resulting in a better prediction model.
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