2020/2/4 Assignment6.ipynb - Colaboratory
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This notebook is licensed under CC BY-NC 4.0 by Lindsey Kuper.
Assignment 6: Variance and Standard Deviation
Notebook Format
This is a homework notebook. It consists of various types of cells:
Text cells: you should read them!
Code cells: you should run them, as they may set up the problems that you are asked to solve.
Solution cells: These are cells where you should enter a solution. You will see a marker in these
should be inserted:
    # YOUR CODE HERE
Test cells: These cells contain some tests, and are worth some points. You should run the cells
you understood the question, and whether the output of your code is produced in the correct for
When we grade your notebook, we will run many additional tests in addition to the ones you see here
credit by hard-coding the desired output!)
Instructions
To work on your notebook, we recommend using Colab. Working in Colab has many benets:
You don't have to maintain a working Python environment on your own machine; you can work f
connection.
Colab preserves the revision history, which is useful for many reasons.
Your work is automatically saved in Google Drive.
In Colab, go to "File > Save a copy in Drive..." and then you'll have your own copy to work on.
Working on Your Notebook
Before you turn your nished notebook in, it's a good idea to sure everything runs as expected. First, f
"Runtime > Factory reset runtime"). Then, run all the cells (go to "Runtime > Run all") in Colab.
Submitting Your Notebook
2020/2/4 Assignment6.ipynb - Colaboratory
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Submit your work as follows:
Download the notebook from Colab, clicking on "File > Download .ipynb".
Upload the resulting le to this Google form.
Deadline: 9pm Monday, February 10.
You can submit multiple times, and the last submission before the deadline will be used to assign you
Make sure you ll in any place that says YOUR CODE HERE or "YOUR ANSWER HERE", as well as your
collaborated with in the below cells. For collaborators, list anyone who (for example) discussed the ge
pointed you to useful Python documentation -- collaboration of that kind is encouraged. Remember th
not allowed.
NAME = ""
COLLABORATORS = ""
After you submit your notebook, your instructor at some point will retreat to a secret hideout, put on s
mysterious scripts. These will generate two things:
Your grade, which goes into a spreadsheet.
Feedback, shared to you as a PDF le on Google Drive. The PDF shows your work, your grade, th
and any comments left by the instructor and the TAs.
What Happens Next?
Make sure to run the following cell to make sure that the Python testing framework, nose , is installed
to work.)
Testing
try:
    from nose.tools import assert_equal, assert_almost_equal
    from nose.tools import assert_true, assert_false
    from nose.tools import assert_not_equal, assert_greater_equal
except:
    !pip install nose
    from nose.tools import assert_equal, assert_almost_equal
    from nose.tools import assert_true, assert_false
    from nose.tools import assert_not_equal, assert_greater_equal
Variance and Standard Deviation
2020/2/4 Assignment6.ipynb - Colaboratory
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Problem 1: Computing variance
For this problem, you will write a function variance that takes a list whose elements are numbers (
single number. (If you don't remember how to compute variance, check the lecture notebook; it's one
standard deviation.)
You should not use NumPy or any other Python libraries for this problem. For now, worry only about c
Passing an empty list to variance should result in a ZeroDivisionError exception being raised, al
raise it yourself.
def variance(data):
    """Takes a list of numbers and returns their variance."""
    # YOUR CODE HERE
    raise NotImplementedError()
### Tests for variance
 
assert_almost_equal(variance([3, 4, 5, 6]), 1.25)
assert_almost_equal(variance([3, 4, 8]), 4.666666666666667)
assert_almost_equal(variance([0.0, 0.0, 0, 0]), 0)
assert_almost_equal(variance([1.5, 2.5]), 0.25)
assert_almost_equal(variance([18, 19, 20, 21, 22]), 2.0)
assert_almost_equal(variance([0, 43, 1, 1, 55]), 575.2)
 
try:
    variance([])
except(ZeroDivisionError):
    pass
 
For this problem, you will write a function standard_deviation that takes a list whose elements are
returns their standard deviation, a single number. You may call the variance method dened above
may use sqrt from the math library, which we have already imported for you.
Passing an empty list to standard_deviation should result in a ZeroDivisionError exception be
explicitly raise it yourself.
Problem 2: Computing standard deviation
from math import sqrt
 
def standard_deviation(data):
    """Takes a list of numbers and returns their standard deviation."""
    # YOUR CODE HERE
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    raise NotImplementedError()
### Tests for standard_deviation
 
assert_almost_equal(standard_deviation([3, 4, 5, 6]), 1.1180339887499)
assert_almost_equal(standard_deviation([3, 4, 8]), 2.1602468994693)
assert_almost_equal(standard_deviation([0.0, 0.0, 0, 0]), 0)
assert_almost_equal(standard_deviation([1.5, 2.5]), 0.5)
assert_almost_equal(standard_deviation([18, 19, 20, 21, 22]), 1.4142135623731)
assert_almost_equal(standard_deviation([0, 43, 1, 1, 55]), 23.983327542274)
 
try:
    standard_deviation([])
except(ZeroDivisionError):
    pass
 
Below is the EfficientAveragerator class we saw during lecture, with one new property for you to
In EfficientAveragerator , we compute average and standard deviation in an incremental fashion,
everything when a new value arrives in our data set. We want to compute variance using the same inc
EfficientAveragerator already has everything we need to write such an incremental denition of v
Fill in the denition of the variance property in the cell below. It should take no more than 1-2 lines o
any other fancy library. (Hint: it will look a lot like the denition of std .)
Problem 3: A variance method for the EfficientAveragerator class
import numpy as np
 
class EfficientAveragerator:
    
    def __init__(self):
        self.sum_x = 0.
        self.sum_x_sq = 0.
        self.n = 0
        
    def add(self, x):
        # We compute the sum of the x, to compute their average.
        self.sum_x += x 
        # Sum of the x^2, so we can later compute the average of the x^2. 
        self.sum_x_sq += x * x
        self.n += 1
        
    @property
    def avg(self):
        return self.sum_x / self.n
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    @property
    def variance(self):
        """Returns the variance of all the data points seen so far."""
        # YOUR CODE HERE
        raise NotImplementedError()
        
    @property
    def std(self):
        average = self.avg # To avoid calling self.avg twice.
        return np.sqrt(self.sum_x_sq / self.n - average * average)
### Tests for EfficientAveragerator.variance
 
ea1 = EfficientAveragerator()
ea1.add(3)
ea1.add(4)
ea1.add(8)
assert_almost_equal(ea1.variance, 4.666666666666667)
 
ea2 = EfficientAveragerator()
for num in [0, 43, 1, 1, 55]:
    ea2.add(num)
assert_almost_equal(ea2.variance, 575.2)
 
ea3 = EfficientAveragerator()
for _ in range(2000000):
    ea3.add(1)
    ea3.add(4)
assert_almost_equal(ea3.variance, 2.25)
 
2020/2/4 Assignment6.ipynb - Colaboratory
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