Lab 8

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Western University *

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1000A

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Statistics

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Jan 9, 2024

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11/2/22, 5:35 PM Lab 8 localhost:8888/nbconvert/html/Documents/Python_examples/Lab_8/Lab 8.ipynb?download=false 1/4 Lab 8 In this lab we discuss simple random sampling, systematic sampling, and stratified random sampling. Simple Random Sampling random.sample: https://www.w3schools.com/python/ref_random_sample.asp (https://www.w3schools.com/python/ref_random_sample.asp) In [1]: import numpy as np import pandas as pd import random In [2]: # Random sampling using random.sample() # random.sample returns unique random elements from a sequence or set. This is sampling without replacement. random . seed( 52 ) ## setting the seed so that we all get the same sample names = [ "Roger" , "Jack" , "John" , "Jason" , "Laura" , "Mariya" , "Martina" , "Lauren" ] sampled_list1 = random . sample(names, 3 ) print (sampled_list1) In [3]: # if we change the seed another sample is obtained random . seed( 17 ) sampled_list2 = random . sample(names, 3 ) print (sampled_list2) In [4]: # an alternative way to sample is to assign a number to each name (or ID) in the list and then sample the numbers # generating 8 consecutive numbers corresponding to the positions of each name in the list: sequence_numbers = list ( range ( 8 )) print (sequence_numbers) # Python starts with zero random . seed( 17 ) # same seed as before to obtain the same sample of names as in the code cell above sample_list3 = random . sample(sequence_numbers, 3 ) # randomly sampling 3 number positions print (sample_list3) In [5]: [names[i] for i in sample_list3 ] # getting the names corresponding to the sampled number positions In [6]: # Getting a sample array from a multidimensional array array = np . array([[ 2 , 5 , 7 ], [ 5 , 11 , 16 ], [ 6 , 13 , 19 ], [ 7 , 15 , 22 ], [ 8 , 17 , 25 ]]) print ( "2D array \n " , array) In [7]: random . seed( 48 ) random_rows = random . sample( range ( 5 ), 2 ) # randomly selecting two row indices, without replacement print (random_rows) In [8]: array[random_rows, :] Systematic Sampling Systematic sampling is a type of sampling where we obtain a sample by going through a list of the population at fixed intervals from a randomly chosen starting point. ['Laura', 'Roger', 'Mariya'] ['Martina', 'Lauren', 'John'] [0, 1, 2, 3, 4, 5, 6, 7] [6, 7, 2] Out[5]: ['Martina', 'Lauren', 'John'] 2D array [[ 2 5 7] [ 5 11 16] [ 6 13 19] [ 7 15 22] [ 8 17 25]] [4, 2] Out[8]: array([[ 8, 17, 25], [ 6, 13, 19]])

11/2/22, 5:35 PM Lab 8 localhost:8888/nbconvert/html/Documents/Python_examples/Lab_8/Lab 8.ipynb?download=false 2/4 np.arange: https://numpy.org/doc/stable/reference/generated/numpy.arange.html (https://numpy.org/doc/stable/reference/generated/numpy.arange.html) In [9]: # Let's assume we are interested in sampling from a population of 15 students with the following ID list: df_students = pd . DataFrame({ 'ID' :np . arange( 1 , 16 ) . tolist()}) df_students In [10]: # Defining the function for systematic sampling def systematic_sampling (df, starting_index, step): indices = np . arange(starting_index, len (df), step = step) systematic_sample = df . iloc[indices] return systematic_sample In [11]: # Obtaining a systematic sample of size 5 # Because 15/3=5, choose one of the first 3 IDs on the list at random and then every 3rd ID after that. random . seed( 68 ) random_start = random . randint( 0 , 2 ) print (random_start) # another way # random.seed(68) # random_start = random.sample(range(3),1) # print(random_start) In [12]: systematic_sample = systematic_sampling(df = df_students, starting_index = random_start, step =3 ) systematic_sample # recall that Python starts at position 0, so position 2 corresponds to ID = 3 Stratified Random Sampling Another type of sampling is stratified random sampling, in which a population is split into groups and a certain number of members from each group are randomly selected to be included in the sample. Stratified Random Sampling Using Counts Out[9]: ID 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 2 Out[12]: ID 2 3 5 6 8 9 11 12 14 15

11/2/22, 5:35 PM Lab 8 localhost:8888/nbconvert/html/Documents/Python_examples/Lab_8/Lab 8.ipynb?download=false 3/4 In [13]: # Suppose we have the following dataframe containing the ID of 8 students from 2 different undergrad programs. # This is our population list. df = pd . DataFrame({ 'ID' :np . arange( 1 , 9 ) . tolist(), 'program' :[ 'Stats' ] *4 + [ 'Math' ] *4 }) # 4 students in Stats, 4 students in Math df In [14]: # random sample of 2 Stats students df_Stats = df[df[ 'program' ] == 'Stats' ] df_Stats random_rows = random . sample( range ( 4 ), 2 ) #randomly selecting 2 students from the 4 Stats students in the populatio n print (df_Stats . iloc[random_rows]) In [15]: # random sample of 2 Math students df_Math = df[df[ 'program' ] == 'Math' ] df_Math random_rows = random . sample( range ( 4 ), 2 ) #randomly selecting 2 students from the 4 Math students in the population print (df_Math . iloc[random_rows]) In [16]: # Alternative way using one line of code and additional Python functions DataFrame.groupby: https://pandas.pydata.org/docs/reference/api/pandas.DataFrame.groupby.html (https://pandas.pydata.org/docs/reference/api/pandas.DataFrame.groupby.html) DataFrame.apply: https://pandas.pydata.org/docs/reference/api/pandas.DataFrame.apply.html (https://pandas.pydata.org/docs/reference/api/pandas.DataFrame.apply.html) DataFrame.sample: https://pandas.pydata.org/docs/reference/api/pandas.DataFrame.sample.html (https://pandas.pydata.org/docs/reference/api/pandas.DataFrame.sample.html) In [17]: # Stratified random sampling by randomly selecting 2 students from each program to be included in the sample df . groupby( 'program' , group_keys = False ) . apply( lambda x:x . sample( 2 )) Stratified Random Sampling Using Proportions Out[13]: ID program 0 1 Stats 1 2 Stats 2 3 Stats 3 4 Stats 4 5 Math 5 6 Math 6 7 Math 7 8 Math ID program 3 4 Stats 2 3 Stats ID program 4 5 Math 6 7 Math Out[17]: ID program 4 5 Math 6 7 Math 0 1 Stats 1 2 Stats

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11/2/22, 5:35 PM Lab 8 localhost:8888/nbconvert/html/Documents/Python_examples/Lab_8/Lab 8.ipynb?download=false 4/4 In [18]: # Now suppose we have the following population dataframe with 25% Stats and 75% Math students. df = pd . DataFrame({ 'ID' :np . arange( 1 , 9 ) . tolist(), 'program' :[ 'Stats' ] *2 + [ 'Math' ] *6 }) # 2 students in Stats, 6 students in Math df np.rint: https://numpy.org/doc/stable/reference/generated/numpy.rint.html (https://numpy.org/doc/stable/reference/generated/numpy.rint.html) In [19]: # Stratified random sampling such that the proportion of students in each program sample # matches the proportion of students from each program in the population dataframe N = 4 # sample size # So, the sample must contain 1 random student from Stats and 3 from Math to maintain the population proportions In [20]: # random sample of Stats students df_Stats = df[df[ 'program' ] == 'Stats' ] df_Stats random_rows = random . sample( range ( 2 ), 1 ) #sampling 1 student from the 2 Stats students in the population print (df_Stats . iloc[random_rows]) In [21]: # random sample of Math students df_Math = df[df[ 'program' ] == 'Math' ] df_Math random_rows = random . sample( range ( 6 ), 3 ) #sampling 3 students from the 6 Math students in the population print (df_Math . iloc[random_rows]) Alternative way: np.rint: https://numpy.org/doc/stable/reference/generated/numpy.rint.html (https://numpy.org/doc/stable/reference/generated/numpy.rint.html) In [22]: df . groupby( 'program' , group_keys = False ) . apply( lambda x:x . sample( int (np . rint(N * len (x) / len (df))))) In [ ]: Out[18]: ID program 0 1 Stats 1 2 Stats 2 3 Math 3 4 Math 4 5 Math 5 6 Math 6 7 Math 7 8 Math ID program 0 1 Stats ID program 5 6 Math 7 8 Math 4 5 Math Out[22]: ID program 4 5 Math 2 3 Math 3 4 Math 0 1 Stats