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Standard Deviation
What does high variance mean? What does low variance mean? Is variance expressed in the same units as the data? What is standard deviation expressed as?
Population Variance
Variance is a measure of the spread of the data. The formula to calculate population variance is:\(
\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2
\)
A high variance means that data are spread over a large range.
A low variance means that data are clustered close together.
""" Matrices / Population Variance
"""
import numpy as np
A = np.array([
[1, 2, 3],
[4, 5, 6],
[7, 8, 9],
])
B = np.array([
[1, 2, 3],
[4, 5, 6],
[7, 8, 900], # Look Here
])
def population_variance(X):
N = X.size
avg = np.mean(X)
variance = (1/N) * np.sum((X - avg)**2)
return variance
A_variance = population_variance(A)
B_variance = population_variance(B)
assert B_variance > A_variance
print("A_variance = ", A_variance.round(2))
print("B_variance = ", B_variance.round(2))
print("np.var(A) = ", np.var(A).round(2)) # build-in
print("np.var(B) = ", np.var(B).round(2))
"""
A_variance = 6.67
B_variance = 79206.67
np.var(A) = 6.67
np.var(B) = 79206.67
"""
Standard Deviation
Standard deviation is more intuitive than variance. It is expressed in the same units as data.\(
\sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N} (x_i - \mu)^2}
\)
""" Matrices / Standard deviation
"""
import numpy as np
A = np.array([
[1, 2, 3],
[4, 5, 6],
[7, 8, 9],
])
B = np.array([
[1, 2, 3],
[4, 5, 6],
[7, 8, 900],
])
# Algorithms
def variation(M):
N = A.size
mean = np.mean(A)
variation = (1/N) * np.sum((A - mean)**2) # population variance
return variation
def standard_deviation(M):
return np.sqrt(variation(M))
return
# Native
def variation_np(M):
return np.var(M)
def standard_deviation_np(M):
return np.std(M)
assert variation(A).round(14) == variation_np(A).round(14)
assert standard_deviation(A) == standard_deviation_np(A)
print("A = \n", A)
print("B = \n", B)
print("np.var(A) variation =", np.var(A))
print("np.var(B) variation =", np.var(B))
print("np.std(A) standard deviation = ", np.std(A))
print("np.std(B) standard deviation = ", np.std(B))
"""
A =
[[1 2 3]
[4 5 6]
[7 8 9]]
B =
[[ 1 2 3]
[ 4 5 6]
[ 7 8 900]]
np.var(A) variation = 6.666666666666667
np.var(B) variation = 79206.66666666667
np.std(A) standard deviation = 2.581988897471611
np.std(B) standard deviation = 281.43678982440565
"""
Last update: 224 days ago
