# Can anyone explain me StandardScaler?

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### Question :

Can anyone explain me StandardScaler?

I am unable to understand the page of the `StandardScaler` in the documentation of `sklearn`.

Can anyone explain this to me in simple terms?

The idea behind `StandardScaler` is that it will transform your data such that its distribution will have a mean value 0 and standard deviation of 1.
In case of multivariate data, this is done feature-wise (in other words independently for each column of the data).
Given the distribution of the data, each value in the dataset will have the mean value subtracted, and then divided by the standard deviation of the whole dataset (or feature in the multivariate case).

Intro: I assume that you have a matrix `X` where each row/line is a sample/observation and each column is a variable/feature (this is the expected input for any `sklearn` ML function by the way — `X.shape` should be `[number_of_samples, number_of_features]`).

Core of method: The main idea is to normalize/standardize i.e. `? = 0` and `? = 1` your features/variables/columns of `X`, individually, before applying any machine learning model.

`StandardScaler()` will normalize the features i.e. each column of X, INDIVIDUALLY, so that each column/feature/variable will have `? = 0` and `? = 1`.

I am quoting “each value in the dataset will have the sample mean value subtracted” — This is neither true nor correct.

Example:

``````from sklearn.preprocessing import StandardScaler
import numpy as np

# 4 samples/observations and 2 variables/features
data = np.array([[0, 0], [1, 0], [0, 1], [1, 1]])
scaler = StandardScaler()
scaled_data = scaler.fit_transform(data)

print(data)
[[0, 0],
[1, 0],
[0, 1],
[1, 1]])

print(scaled_data)
[[-1. -1.]
[ 1. -1.]
[-1.  1.]
[ 1.  1.]]
``````

Verify that the mean of each feature (column) is 0:

``````scaled_data.mean(axis = 0)
array([0., 0.])
``````

Verify that the std of each feature (column) is 1:

``````scaled_data.std(axis = 0)
array([1., 1.])
``````

The maths: UPDATE 08/2020: Concerning the input parameters `with_mean` and `with_std` to `False`/`True`, I have provided an answer here: StandardScaler difference between “with_std=False or True” and “with_mean=False or True”

How to calculate it: StandardScaler performs the task of Standardization. Usually a dataset contains variables that are different in scale. For e.g. an Employee dataset will contain AGE column with values on scale 20-70 and SALARY column with values on scale 10000-80000.
As these two columns are different in scale, they are Standardized to have common scale while building machine learning model.

This is useful when you want to compare data that correspond to different units. In that case, you want to remove the units. To do that in a consistent way of all the data, you transform the data in a way that the variance is unitary and that the mean of the series is 0.

Following is a simple working example to explain how standarization calculation works. The theory part is already well explained in other answers.

``````>>>import numpy as np
>>>data = [[6, 2], [4, 2], [6, 4], [8, 2]]
>>>a = np.array(data)

>>>np.std(a, axis=0)
array([1.41421356, 0.8660254 ])

>>>np.mean(a, axis=0)
array([6. , 2.5])

>>>from sklearn.preprocessing import StandardScaler
>>>scaler = StandardScaler()
>>>scaler.fit(data)
>>>print(scaler.mean_)

#Xchanged = (X??)/?  WHERE ? is Standard Deviation and ? is mean
>>>z=scaler.transform(data)
>>>z
``````

Calculation

As you can see in the output, mean is [6. , 2.5] and std deviation is [1.41421356, 0.8660254 ]

Data is (0,1) position is 2
Standardization = (2 – 2.5)/0.8660254 = -0.57735027

Data in (1,0) position is 4
Standardization = (4-6)/1.41421356 = -1.414

Result After Standardization Check Mean and Std Deviation After Standardization Note: -2.77555756e-17 is very close to 0.

References

The answers above are great, but I needed a simple example to alleviate some concerns that I have had in the past. I wanted to make sure it was indeed treating each column separately. I am now reassured and can’t find what example had caused me concern. All columns ARE scaled separately as described by those above.

### CODE

``````import pandas as pd
import scipy.stats as ss
from sklearn.preprocessing import StandardScaler

data= [[1, 1, 1, 1, 1],[2, 5, 10, 50, 100],[3, 10, 20, 150, 200],[4, 15, 40, 200, 300]]

df = pd.DataFrame(data, columns=['N0', 'N1', 'N2', 'N3', 'N4']).astype('float64')

sc_X = StandardScaler()
df = sc_X.fit_transform(df)

num_cols = len(df[0,:])
for i in range(num_cols):
col = df[:,i]
col_stats = ss.describe(col)
print(col_stats)
``````

## OUTPUT

``````DescribeResult(nobs=4, minmax=(-1.3416407864998738, 1.3416407864998738), mean=0.0, variance=1.3333333333333333, skewness=0.0, kurtosis=-1.3599999999999999)
DescribeResult(nobs=4, minmax=(-1.2828087129930659, 1.3778315806221817), mean=-5.551115123125783e-17, variance=1.3333333333333337, skewness=0.11003776770595125, kurtosis=-1.394993095506219)
DescribeResult(nobs=4, minmax=(-1.155344148338584, 1.53471088361394), mean=0.0, variance=1.3333333333333333, skewness=0.48089217736510326, kurtosis=-1.1471008824318165)
DescribeResult(nobs=4, minmax=(-1.2604572012883055, 1.2668071116222517), mean=-5.551115123125783e-17, variance=1.3333333333333333, skewness=0.0056842140599118185, kurtosis=-1.6438177182479734)
DescribeResult(nobs=4, minmax=(-1.338945389819976, 1.3434309690153527), mean=5.551115123125783e-17, variance=1.3333333333333333, skewness=0.005374558840039456, kurtosis=-1.3619131970819205)
``````

# NOTE:

The scipy.stats module is correctly reporting the “sample” variance, which uses (n – 1) in the denominator. The “population” variance would use n in the denominator for the calculation of variance. To understand better, please see the code below that uses scaled data from the first column of the data set above:

## Code

``````import scipy.stats as ss

sc_Data = [[-1.34164079], [-0.4472136], [0.4472136], [1.34164079]]
col_stats = ss.describe([-1.34164079, -0.4472136, 0.4472136, 1.34164079])
print(col_stats)
print()

mean_by_hand = 0
for row in sc_Data:
for element in row:
mean_by_hand += element
mean_by_hand /= 4

variance_by_hand = 0
for row in sc_Data:
for element in row:
variance_by_hand += (mean_by_hand - element)**2
sample_variance_by_hand = variance_by_hand / 3
sample_std_dev_by_hand = sample_variance_by_hand ** 0.5

pop_variance_by_hand = variance_by_hand / 4
pop_std_dev_by_hand = pop_variance_by_hand ** 0.5

print("Sample of Population Calcs:")
print(mean_by_hand, sample_variance_by_hand, sample_std_dev_by_hand, 'n')
print("Population Calcs:")
print(mean_by_hand, pop_variance_by_hand, pop_std_dev_by_hand)
``````

## Output

``````DescribeResult(nobs=4, minmax=(-1.34164079, 1.34164079), mean=0.0, variance=1.3333333422778562, skewness=0.0, kurtosis=-1.36000000429325)

Sample of Population Calcs:
0.0 1.3333333422778562 1.1547005422523435

Population Calcs:
0.0 1.000000006708392 1.000000003354196
``````

After applying `StandardScaler()`, each column in X will have mean of 0 and standard deviation of 1.