Prerequisite : Python statistics | variance()
pvariance() function helps to calculate the variance of an entire, rather than that of a sample. The only difference between variance() and pvariance() is that while using variance(), only the sample mean is taken into consideration, while during pvariance(), the mean of entire population is taken into consideration.
Population variance is just similar to sample variance, it tells how data points in a specific population are spread out. It is the average of the distance from the data-points to the mean of the data-set, squared. The population variance is a parameter of the population and is not dependent on research methods or sampling practices.
Syntax : pvariance( [data], mu)
Parameters :
[data] : An iterable with real valued numbers.
mu (optional): Takes actual mean of data-set/ population as value.
Returnype : Returns the actual population variance of the values passed as parameter.
Exceptions :
StatisticsError is raised for data-set less than 2-values passed as parameter.
Impossible values when the value provided as mu doesn’t match actual mean of the data-set.
Code #1 :
Python3
# Python code to demonstrate the# use of pvariance()# importing statistics moduleimport statistics# creating a random population listpopulation = (1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.9, 2.2, 2.3, 2.4, 2.6, 2.9, 3.0, 3.4, 3.3, 3.2)# Prints the population varianceprint("Population variance is %s" %(statistics.pvariance(population))) |
Output :
Population variance is 0.6658984375
Code #2 : Demonstrates pvariance() on a different range of population trees.
Python3
# Python code to demonstrate pvariance()# on various range of population sets# importing statistics modulefrom statistics import pvariance# importing fractions module as Ffrom fractions import Fraction as F# Population tree for a set of positive integerspop1 = (1, 2, 3, 5, 4, 6, 1, 2, 2, 3, 1, 3, 7, 8, 9, 1, 1, 1, 2, 6, 7, 8, 9, )# Creating a population tree for# a set of negative integerspop2 = (-36, -35, -34, -32, -30, -31, -33, -33, -33, -38, -36, -35, -34, -38, -40, -31, -32)# Creating a population tree for# a set of fractional numberspop3 = (F(1, 3), F(2, 4), F(2, 3), F(3, 2), F(2, 5), F(2, 2), F(1, 1), F(1, 4), F(1, 2), F(2, 1))# Creating a population tree for# a set of decimal valuespop4 = (3.45, 3.2, 2.5, 4.6, 5.66, 6.43, 4.32, 4.23, 6.65, 7.87, 9.87, 1.23, 1.00, 1.45, 10.12, 12.22, 19.88)# Print the population variance for# the created population treesprint("Population variance of set 1 is % s" %(pvariance(pop1))) print("Population variance of set 2 is % s" %(pvariance(pop2))) print("Population variance of set 3 is % s" %(pvariance(pop3))) print("Population variance of set 4 is % s" %(pvariance(pop4))) |
Output :
Population variance of set 1 is 7.913043478260869 Population variance of set 2 is 7.204152249134948 Population variance of set 3 is 103889/360000 Population variance of set 4 is 21.767923875432526
Code #3 : Demonstrates the use of mu parameter.
Python3
# Python code to demonstrate the use# of 'mu' parameter on pvariance()# importing statistics moduleimport statistics# Apparently, the Python interpreter doesn't# even check whether the value entered for mu# is the actual mean of data-set or not.# Thus providing incorrect value would# lead to impossible answers# Creating a population tree of the# age of kids in a localitytree = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 12, 12, 12, 13, 1, 2, 12, 2, 2, 2, 3, 4, 5, 5, 5, 5, 6, 6, 6)# Finding the mean of population treem = statistics.mean(tree)# Using the mu parameter# while using pvariance()print("Population Variance is % s" %(statistics.pvariance(tree, mu = m))) |
Output :
Population Variance is 14.30385015608741
Code #4 : Demonstrate the difference between pvariance() and variance()
Python3
# Python code to demonstrate the # difference between pvariance() # and variance()# importing statistics moduleimport statistics# Population tree and extract# a sample from ittree = (1.1, 1.22, .23, .55, .67, 2.33, 2.81, 1.54, 1.2, 0.2, 0.1, 1.22, 1.61)# Sample extract from population treesample = (1.22, .23, .55, .67, 2.33, 2.81, 1.54, 1.2, 0.2)# Print sample variance and as # well as population varianceprint ("Variance of whole population is %s" %(statistics.pvariance(tree))) print ("Variance of sample from population is %s " % (statistics.variance(sample)))# Print the difference in both population # variance and sample varianceprint("\n")print("Difference in Population variance" "and Sample variance is % s" %(abs(statistics.pvariance(tree) - statistics.variance(sample)))) |
Output :
Variance of the whole population is 0.6127751479289941 Variance of the sample from population is 0.8286277777777779 Difference in Population variance and Sample variance is 0.21585262984878373
Note : We can see from the above sample example that Population Variance and Sample Variance doesn’t differ by a huge value.
Code #5 : Demonstrates StatisticsError
Python3
# Python code to demonstrate StatisticsError# importing statistics moduleimport statistics# creating an empty population setpop = ()# will raise StatisticsErrorprint(statistics.pvariance(pop)) |
Output :
Traceback (most recent call last):
File "/home/fa112e1405f09970eeddd48214318a3c.py", line 10, in
print(statistics.pvariance(pop))
File "/usr/lib/python3.5/statistics.py", line 603, in pvariance
raise StatisticsError('pvariance requires at least one data point')
statistics.StatisticsError: pvariance requires at least one data point
Applications :
The applications of Population Variance is much similar to Sample Variance, although the range of population variance is much larger than sample variance. Population variance is only to be used when the variance of an entire population is to be calculated, otherwise for calculating the variance of a sample, variance() is preferred. Population Variance is a very important tool in Statistics and handling huge amounts of data. Like, when the omniscient mean is unknown (sample mean) then variance is used as biased estimator.
