5.1. Statistics¶
5.1.1. Mean¶
Compute the arithmetic mean along the specified axis.
The arithmetic mean is the sum of the elements along the axis divided by the number of elements.
The average is taken over the flattened array by default, otherwise over the specified axis.
import numpy as np
a = np.array([1, 2, 3])
np.mean(a)
# 2.0
np.mean(a, axis=0)
# array([2., 3.])
np.mean(a, axis=1)
# Traceback (most recent call last):
# IndexError: tuple index out of range
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6]])
np.mean(a)
# 3.5
np.mean(a, axis=0)
# array([2.5, 3.5, 4.5])
np.mean(a, axis=1)
# array([2., 5.])
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
np.mean(a)
# 5.0
np.mean(a, axis=0)
# array([4., 5., 6.])
np.mean(a, axis=1)
# array([2., 5., 8.])
5.1.2. Average¶
Compute the weighted average along the specified axis.
import numpy as np
a = np.array([1, 2, 3])
np.average(a)
# 2.0
np.average(a, axis=0)
# 2.0
np.average(a, axis=1)
# Traceback (most recent call last):
# IndexError: tuple index out of range
np.average(a, weights=[1, 1, 2])
# 2.25
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6]])
np.average(a)
# 3.5
np.average(a, axis=0)
# array([2.5, 3.5, 4.5])
np.average(a, axis=1)
# array([2., 5.])
np.average(a, weights=[[1, 0, 2],
[2, 0, 1]])
# 3.5
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
np.average(a)
# 5.0
np.average(a, axis=0)
# array([4., 5., 6.])
np.average(a, axis=1)
# array([2., 5., 8.])
np.average(a, weights=[[1, 0, 2],
[2, 0, 1],
[1./4, 1./2, 1./3]])
# 4.2
5.1.3. Median¶
Compute the median along the specified axis
import numpy as np
a = np.array([1, 2, 3])
np.median(a)
# 2.0
np.median(a, axis=0)
# 2.0
np.median(a, axis=1)
# Traceback (most recent call last):
# numpy.AxisError: axis 1 is out of bounds for array of dimension 1
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6]])
np.median(a)
# 3.5
np.median(a, axis=0)
# array([2.5, 3.5, 4.5])
np.median(a, axis=1)
# array([2., 5.])
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
np.median(a)
# 5.0
np.median(a, axis=0)
# array([4., 5., 6.])
np.median(a, axis=1)
# array([2., 5., 8.])
import numpy as np
a = np.array([1, 2, 3, 4])
np.median(a)
# 2.5
5.1.4. Variance¶
Compute the variance along the specified axis.
Variance of the array elements is a measure of the spread of a distribution.
The variance is the average of the squared deviations from the mean, i.e.,
var = mean(abs(x - x.mean())**2)The variance is computed for the flattened array by default, otherwise over the specified axis.
import numpy as np
a = np.array([1, 2, 3])
np.var(a)
# 0.6666666666666666
np.var(a, axis=0)
# 0.6666666666666666
np.var(a, axis=1)
# Traceback (most recent call last):
# IndexError: tuple index out of range
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6]])
np.var(a)
# 2.9166666666666665
np.var(a, axis=0)
# array([2.25, 2.25, 2.25])
np.var(a, axis=1)
# array([0.66666667, 0.66666667])
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
np.var(a)
# 6.666666666666667
np.var(a, axis=0)
# array([6., 6., 6.])
np.var(a, axis=1)
# array([0.66666667, 0.66666667, 0.66666667])
5.1.5. Standard Deviation¶
Compute the standard deviation along the specified axis.
Standard deviation is a measure of the spread of a distribution, of the array elements.
The standard deviation is the square root of the average of the squared deviations from the mean, i.e.,
std = sqrt(mean(abs(x - x.mean())**2))The standard deviation is computed for the flattened array by default, otherwise over the specified axis.
import numpy as np
a = np.array([1, 2, 3])
np.std(a)
# 0.816496580927726
np.std(a, axis=0)
# 0.816496580927726
np.std(a, axis=1)
# Traceback (most recent call last):
# IndexError: tuple index out of range
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6]])
np.std(a)
# 1.707825127659933
np.std(a, axis=0)
# array([1.5, 1.5, 1.5])
np.std(a, axis=1)
# array([0.81649658, 0.81649658])
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
np.std(a)
# 2.581988897471611
np.std(a, axis=0)
# array([2.44948974, 2.44948974, 2.44948974])
np.std(a, axis=1)
# array([0.81649658, 0.81649658, 0.81649658])
5.1.6. Covariance¶
Estimate a covariance matrix, given data and weights
Covariance indicates the level to which two variables vary together
ddof- Delta Degrees of Freedom
import numpy as np
a = np.array([1, 2, 3])
np.cov(a)
# array(1.)
np.cov(a, ddof=0)
# array(0.66666667)
np.cov(a, ddof=1)
# array(1.)
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6]])
np.cov(a)
# array([[1., 1.],
# [1., 1.]])
np.cov(a, ddof=0)
# array([[0.66666667, 0.66666667],
# [0.66666667, 0.66666667]])
np.cov(a, ddof=1)
# array([[1., 1.],
# [1., 1.]])
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
np.cov(a)
# array([[1., 1., 1.],
# [1., 1., 1.],
# [1., 1., 1.]])
np.cov(a, ddof=0)
# array([[0.66666667, 0.66666667, 0.66666667],
# [0.66666667, 0.66666667, 0.66666667],
# [0.66666667, 0.66666667, 0.66666667]])
np.cov(a, ddof=1)
# array([[1., 1., 1.],
# [1., 1., 1.],
# [1., 1., 1.]])
5.1.7. Correlation coefficient¶
measure of the linear correlation between two variables X and Y
Pearson correlation coefficient (PCC)
Pearson product-moment correlation coefficient (PPMCC)
bivariate correlation
Figure 5.2. Examples of scatter diagrams with different values of correlation coefficient (ρ) [NumpyWik19b]¶
import numpy as np
a = np.array([1, 2, 3])
np.corrcoef(a)
# 1.0
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6]])
np.corrcoef(a)
# array([[1., 1.],
# [1., 1.]])
import numpy as np
a = np.array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
np.corrcoef(a)
# array([[1., 1., 1.],
# [1., 1., 1.],
# [1., 1., 1.]])
import numpy as np
a = np.array([[1, 2, 1],
[5, 4, 3]])
np.corrcoef(a)
# array([[1., 0.],
# [0., 1.]])
import numpy as np
a = np.array([[3, 1, 3],
[5, 5, 3]])
np.corrcoef(a)
# array([[ 1. , -0.5],
# [-0.5, 1. ]])
import numpy as np
a = np.array([[5, 2, 1],
[2, 4, 5]])
np.corrcoef(a)
# array([[ 1. , -0.99587059],
# [-0.99587059, 1. ]])