Homework 01 -- Data Display in Python

CS 237 Homework 01 -- Data Display in Python

In this first homework, you will become familiar with various methods of displaying the results of probability
experiments graphically in Python using Jupyter notebooks and Matplotlib. This will be a fundamental way of
understand the results of experiments throughout the course. We will cover:
Basic introduction to Python and Jupyter Notebook.
Basic introduction to Matplotlib and graphing
Basic introduction to Monte Carlo (probability) simulation.
Please follow the HW Submission Instructions
(http://www.cs.bu.edu/fac/snyder/cs237/HWSubmissionInstructions.html); any homeworks not following these
requirements may be penalized.
Anaconda Distribution of Python
Download from https://www.continuum.io/downloads (https://www.continuum.io/downloads)
Make sure it's Python 3!
Be sure to pick the correct Operating System (i.e., Windows, MAC OS, Linux)
Starting Jupyter
Open a Terminal Window
For MAC OS, open your Applications folder, then open the Utilities folder. Open the Terminal
application. (Or, type "terminal" into Spotlight.) You may want to add this to your dock.
For Windows, simply search for CMD and run the result
In the terminal run
This will bring you to the home page.
Click new and then click python3 to create a new Ipython3 notebook.
Alternatively, use the Anaconda Navigator that will be installed in your Applications folder.
jupyter notebook
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Downloading the Assignments
From the class web page download HW01 as an IPYNB file (NOT a .txt file):
As soon as you download it, Run All the cells:
Normally, you select one or more cells and then start execution using Cell - Run Cells, but it is easier to
use the keyboard shortcut: Click inside the cell to select it, then type Control-Return.
Use the Kernel menu to kill a runaway piece of code, and if weird stuff is happening and you want to reset
everything, select Kernel - Restart and Run All:
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In [1]: # Here are some imports which will be used in the code in the rest of
the lab
# Imports used for the code in CS 237
import numpy as np # arrays and functions which operat
e on array
import matplotlib.pyplot as plt # normal plotting
import seaborn as sns # Fancy plotting
import pandas as pd # Data input and manipulation
from numpy.random import seed, randint, random
from collections import Counter
%matplotlib inline
Plotting Points
The scatter(...) function is used to plot points from a list of x values and the associated y values.
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In [2]: # To plot the points (1,2), (2,3), (3,6), (4,8) we would list the x v
alues and the corresponding y values:
X = [1,2,3,4]
Y = [2,3,6,8]
print("\nThis is the list of points:",list(zip(X,Y)))
print("They must be input to the function as separate lists:")
print("\tX =",X)
print("\tY =",Y,"\n")
plt.scatter(X,Y)
plt.title('Graphing Points with scatter(X,Y)')
plt.xlabel("The X Values")
plt.ylabel("The Y Values")
plt.show()
Plotting Lines and Curves
If you call plot(...) instead of scatter(...) you will display a curve created by connecting the points
with straight lines. Essentially you can only plot straight lines between points, but if the points are close together,
you will not notice, and it will look like a smooth curve.
This is the list of points: [(1, 2), (2, 3), (3, 6), (4, 8)]
They must be input to the function as separate lists:
X = [1, 2, 3, 4]
Y = [2, 3, 6, 8]
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In [3]: # To plot a curve through the points (1,2), (2,3), (3,6), (4,8) we wo
uld use:
plt.plot([1,2,3,4], [2,3,6,8])
plt.title('A Sequence of Straight Lines')
plt.xlabel("The X Values")
plt.ylabel("The Y Values")
plt.show()
X = np.linspace(1,20,100) # returns a list of 100 equallyspaced values in the range [1..20]
Y = [np.sin(x) for x in X]
plt.plot(X,Y)
plt.title('A Smooth-Looking Curve')
plt.xlabel("The X Values")
plt.ylabel("The Y Values")
plt.show()
If you want to do both, you can simply call both functions before you call show().
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In [4]: plt.scatter([1,2,3,4], [2,3,6,8])
plt.plot([1,2,3,4], [2,3,6,8])
plt.title('A Curve Through Some Points, Showing the Points')
plt.xlabel("The X Values")
plt.ylabel("The Y Values")
plt.show()
If you want to draw a single line from to you can plot and
Here we have added a zero line to our sin curve:
(x1, y1) (x2, y2) [x1, x2] [y1, y2].
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In [5]: X = np.linspace(1,20,100) # returns a list of 100 equallyspaced values in the range [1..20]
Y = [np.sin(x) for x in X]
plt.plot(X,Y)
plt.plot([0,20],[0,0])
plt.title('A Smooth-Looking Curve')
plt.xlabel("The X Values")
plt.ylabel("The Y Values")
plt.show()
For further details on drawing plots, particularly on color and format, see the
Appendix at the end of this document
Problem Zero: Using the Numpy Random Library
We have imported a number of functions from the Numpy Random library, which you can read about here
(https://docs.scipy.org/doc/numpy/user/quickstart.html).
In [6]: # Run this cell several times and and see what happens
random()
Out[6]: 0.25437701508006305
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In [7]: # Run this cell several times and and see what happens
# List comprehensions are your best friend, learn how to use them!!!
size = 10
rands = [ random() for k in range(size) ]
rands
In [8]: # Run this cell several times and and see what happens
# This produces a Numpy 1D vector, which is
# another way to store a sequence.
size = 5
random(size)
In [9]: # In most cases, arrays are interchangeable with lists, but if you ne
ed a Python
# list, just do this:
list(random(size))
In order to make grading easier, we will "seed" the random number generation so that it always produces the
same pseudo-random sequence. However, you should generally try running your code several times without the
seed, trying it on various random sequences.
Just be sure to include the seed before running your program to submit, so that the graders can see your correct
results.
In [10]: # Run this cell several times, and see what happens
# How is it different from the result in cell [210]?
seed(0)
random(5)
Out[7]: [0.8459421725471246,
0.8956782540437821,
0.587705032211821,
0.35378549433952533,
0.07456167058026497,
0.2864312207404033,
0.5830974374308513,
0.35410819800605386,
0.7460296596860136,
0.7850564361492659]
Out[8]: array([0.74799137, 0.72214408, 0.10288086, 0.01369631, 0.85348445])
Out[9]: [0.7669604344687095,
0.13193918476106503,
0.4256066530035917,
0.06568377219402044,
0.013844805563202]
Out[10]: array([0.5488135 , 0.71518937, 0.60276338, 0.54488318, 0.4236548 ])
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Problem 1 (Plotting Points)
Part A
One of our standard examples in lecture will be a "spinner" which can be set in motion to randomly chose a real
number in the half-open interval [0..1):
We will simulate this using the Python function random() from the Numpy.random library we imported above.
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In [11]: # Use numpy's random() function to build a list of x_vals with th
e following properties:
# - x_vals list should have length num_trials
# - Each value in x_vals should be a random float in the ran
ge [0..1)
# Now plot the points (xi,0) along the x axis in the diagram belo
w
def random_line_plot(num_trials):
x_vals = [ [ random() for k in range(num_trials) ] ]
# <--- Your code here
y_vals = [ 0 ] * num_trials # <--- Your
code here
plt.figure(figsize=(12, 4))
plt.title('Graph of Uniform Distribution of '+str(num_trials)+' p
oints in 1D.',fontsize=12)
plt.grid(color='0.95')
plt.ylim(-0.2, 0.2)
plt.xlim(-0.0,1.0)
plt.plot([0,1.0],[0,0])
plt.scatter(x_vals, y_vals, marker="x",color="r")
plt.show()
# TO DO: Complete the lines of code indicated above. Then comment ou
t the seed(0)
# statement and run this cell several times to see the distribution o
f points;
# put the seed command back in before you submit!
seed(0)
random_line_plot(50)
Part B
Now we will simulate the experiment of throwing a dart at a unit square, which will produce a scatter plot of
random points in a 2D grid.
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In [12]: # TO DO: Complete the following function stub which will produce a s
catter plot of
# random points in a unit square.
def random_plane_plot(num_trials):
"""Uses numpy's random function to build a list of x and y values
with the following properties:
- x_vals should have len num_trials
- y_vals should have len num_trials
- Each value in x and y should be between 0 and 1
HINT: use the same code to build x_vals and y_vals
"""
x_vals = [ random() for k in range(num_trials) ] # <---
Your code here
y_vals = [ random() for k in range(num_trials) ] # <---
Your code here
plt.figure(num=None, figsize=(10, 10))
plt.title('Graph of Uniform Distribution of '+str(num_trials)+' p
oints in 2D.',fontsize=12)
plt.grid(color='0.95')
plt.ylim(0, 1)
plt.xlim(0,1)
plt.scatter(x_vals, y_vals,marker="x",color="r")
plt.show()

# Comment out the next line and run this cell several times to see th
e distribution of points;
# put the seed command back in before you submit!
seed(0)
random_plane_plot(500)
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[End of Problem One]
Bar Charts
If we do the exact same thing as we did with a simple plot, but use the function bar(...) we get a bar chart:
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In [13]: # To plot the points (1,2), (2,3), (3,6), (4,8) we would list the x v
alues and the corresponding y values:
plt.figure(num=None, figsize=(8, 6))
plt.bar([1,2,3,4], [2,3,6,8])
plt.title('A Bar Chart')
plt.xlabel("The X Values")
plt.ylabel("The Y Values")
plt.show()
If the Y axis is probabilities (in the range 0 .. 1), we get a distribution of the probabilities among the outcomes of
an experiment:
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In [14]: # Show the distribution of probabilities for a coin flip:
x = [0,1]
y = [0.5, 0.5]
labels = ['Heads', 'Tails']
plt.figure(num=None, figsize=(8, 6))
plt.xticks(x, labels)
plt.bar(x,y)
plt.title('Probability Distribution for Coin Flips')
plt.ylabel("Probability")
plt.xlabel("Outcomes")
plt.show()
With a few tweaks, you can create an attractive bar chart for arbitrary probability distributions (we will cover the
topic of distributions in a few weeks).
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In [15]: # Show the distribution of probabilities for flipping two dice
# We will consider this example in lecture
x = [k for k in range(2,13)]
y = [1/36,2/36,3/36,4/36,5/36,6/36,5/36,4/36,3/36,2/36,1/36]
plt.figure(num=None, figsize=(8, 6))
plt.bar(x,y, width=1.0,edgecolor='black') # <--- Note how we set t
he width and edge color
plt.title('Probability Distribution for Tossing Two Dice')
plt.ylabel("Probability")
plt.xlabel("Outcomes")
plt.show()
Histograms
If you give a list of values to hist(...) it will create a histogram counting how many of each value occur;
this list can be unordered;
You will get a cleaner display if you specify where the edges of the bins are, and make sure the edges of the
bins are visible, as shown in this example:
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In [16]: plt.figure(num=None, figsize=(8, 6))
plt.hist([1,2,4,2,6,2,4,5,6,4,6,3,4,3],bins=[0.5,1.5,2.5,3.5,4.5,5.6,
6.5],edgecolor='black')
plt.title('Sample Histogram')
plt.xlabel("Outcomes")
plt.ylabel("Frequency")
plt.show()
Problem 2
Read and understand the function role_die(...) below, which simulates the experiment of rolling a fair, sixsided die num_trials times. Note carefully:
The sample space is .
The experiment is equi-probable, i.e., the probability of any particular outcome is .
If we record the outcome for a large number of experiments, we would expect the number of outcomes to be
"evenly distributed." In other words, for a large number of trials, we would expect the probability of each
outcome to be
TO DO: For this first problem, simply provide the Python code which would display a histogram of the results of
the experiment for 100,000 trials with appropriate labels. You should use the Numpy function randint(...) ,
as shown in the next cell.
{1, 2, 3, 4, 5, 6}
1
6
k ∈ {1, 2, 3, 4, 5, 6}
≈
number of times we observed a value k
num_trials
1
6
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In [17]: # Demo of randint(lo,hi), which generates a random integer from the s
equence
# [lo,lo+1,...,hi-1].
# In other words the upper bound is exclusive, to be consistent with
the indices in lists,
# the Python range function, and so on.
randint(0,4)
In [18]: # You can also ask it for an array, since it is a Numpy function:
randint(0,4,10)
In [19]: seed(0)
def roll_die(num_trials = 100000): # <-- note the use of the
default parameter value
trials_and_results = randint(1,7,num_trials) # this creates a 1D
array of length num_trials of random integers 1..6
return trials_and_results
example_trials = roll_die() # <-- now we don't have
to specify unless different
Out[17]: 1
Out[18]: array([1, 2, 0, 2, 1, 0, 0, 0, 3, 2])
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In [20]: # Your code here
plt.figure(num=None, figsize=(8, 6))
plt.hist(example_trials,bins=[0.5,1.5,2.5,3.5,4.5,5.6,6.5],edgecolor=
'black')
plt.title('Sample Histogram')
plt.xlabel("Outcomes")
plt.ylabel("Frequency")
plt.show()
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Problem 3
Now we will display the same results showing the distribution of probabilities, instead of an explicit histogram;
since the experiment is equi-probable, if we record the outcome for a large number of experiments, we would
expect the number of outcomes to be "evenly distributed." In other words, for a large number of trials, we would
expect the probability of each outcome to be
To calculate the probabilities, you will need to count the number of occurrences of each of the outcomes,
you may find the function Counter(...) useful for this (Google "Numpy Counter" to find out how this
works);
Once you have the frequency of each outcome, divide by the total number of trials to get the probability for
each.
TO DO: Complete the function stub below which takes the list returned by roll_die(...) , or any other
experiment returning numerical results, and produces a frequency distribution; this should have the same shape
as the histogram, but the Y axis will be probabilities instead of the frequency. Again, create appropriate labels.
Demonstrate your function, again, on the list example_trials produced in Problem 1.
k ∈ {1, 2, 3, 4, 5, 6}
≈
number of times we observed a value k
num_trials
1
6
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In [21]: # Solution
seed(0)
def show_distribution(outcomes, title='Probability Distribution'):
total = len(outcomes)
count = Counter(outcomes)
probs = [count[i]/total for i in range(min(outcomes), max(outcome
s)+1)]
plt.figure(num=None, figsize=(8, 6))
plt.bar([i for i in range(min(outcomes), max(outcomes)+1)],probs,
width=1.0,edgecolor='black') # <--- Note how we set the width and
edge color
plt.title(title)
plt.ylabel("Probability")
plt.xlabel("Outcomes")
plt.show()

show_distribution(example_trials,title='Probability Distribution for
Single Die Toss')
Motivation for Monte Carlo Simulation
For the case of a fair die, the distribution is very easily computed by hand. But in general, it may be difficult to
write down an analytical solution for the distribution produced by a random experiment. This is where simulation
comes into play: instead of mathematically computing the distribution explicitly, you can use this method of
repeating experiments, and recording outcomes to understand the probabilistic rules governing some real world
event. When you can come up with an analytical result, this is a nice way of confirming its correctness!
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Problem 4
You will now do the same thing you did in the previous problems, but with a new experiment: instead of rolling
one die and recording the value, you will simulate rolling dies and recording their sum. For example, if
and the first die shows up as a 3, and the second die shows up as a 1, the sum (and the value we record) would
be 4.
TO DO: Complete the two function stubs below and then demonstrate by providing code which would print out
the probability distribution for rolling 2 dice 100,000 times.
Hint: Not required but think about how you might do this in one line using Numpy and list comprehensions
n n = 2
In [22]: # Solution
seed(0)

def roll_and_add_dice(num_dice, num_trials = 10**6):
return([sum([randint(1,7) for i in range(num_dice)]) for k in ran
ge(num_trials)])
show_distribution(roll_and_add_dice(2),title='Probability Distributio
n for Sum of Two Dice')
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Problem 5 (Monte Carlo Calculation of )
This final problem is also a Monte Carlo simulation, but this time in the continuous domain: we will calculate the
value of using a variation of Problem 1B.
We will leave this one up to you as it the exact details, but you must use the following fact: a circle inscribed in a
unit square (i.e., with side of length and area 1.0):
has as radius of 0.5 and an area of
Therefore, if you generate num_trials random points in the unit square, as in Problem 1B, and count how
many land inside the circle, you can calculate an approximation of .
For this problem, you must
(A) Draw the diagram of the unit square with inscribed circle and 500 random points, and
calculate the value of .
(B) Without drawing the diagram, calculate the value of you would get from trials.
(C) After completing (B), try to get a more accurate value for by increasing the number of trials.
Your results will depend on your machine, but for comparison, with my new Macbook Pro, I ran it
with trials while I got a cup of coffee, and it had the answer correct to 3 decimal places when
I came back. Sometimes I run big experiments overnight! The key here is to try increasingly large
numbers and see how the time increases.
</blockquote
Hint: Start by copying your code from Problem 1B. You might find Mr. Pythagoras's formula
useful.
π
π
π ∗ (0.5 ) = .
2 π
4
π
π
π 10
5
π
10
8
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In [23]: # (A)
def monte_carlo(num_trials=500):
incircle = 0
x_vals = [ random() for k in range(num_trials) ]
y_vals = [ random() for k in range(num_trials) ]
plt.figure(num=None, figsize=(10, 10), dpi=89)
plt.plot([0,1],[0,0],color='orange') # Line connecting (0,0) to
(1,0)
plt.plot([0,0],[0,1],color='orange') # Line connecting (0,0) to
(0,1)
plt.plot([0,1],[1,1],color='orange') # Line connecting (0,1) to
(1,1)
plt.plot([1,1],[0,1],color='orange') # Line connecting (1,0) to
(1,1)
theta = np.linspace(0, 2*np.pi, 100)
r = np.sqrt(0.25)
x1 = r*np.cos(theta)+0.5
x2 = r*np.sin(theta)+0.5
for (x,y) in zip(x_vals, y_vals):
if(((x-0.5)**2+(y-0.5)**2) <= 0.25):
incircle += 1
pi_val = 4*incircle/num_trials
plt.text(0.8,1.01, "The value of pi is "+str(pi_val)+".")
plt.plot(x1,x2, color='orange')
plt.title('Graph of Uniform Distribution of '+str(num_trials)+' p
oints in 2D.',fontsize=12)
plt.grid(color='0.95')
plt.scatter(x_vals, y_vals,marker="x",color="r")
plt.show()

monte_carlo()
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In [24]: # (B)
def monte_carlo_without_diagram(num_trials=10**3):
incircle=0
x_vals = [ random() for k in range(num_trials) ]
y_vals = [ random() for k in range(num_trials) ]
for (x,y) in zip(x_vals, y_vals):
if(((x-0.5)**2+(y-0.5)**2) <= 0.25):
incircle += 1
pi_val = 4*incircle/num_trials
print("Value of pi is "+str(pi_val)+".")
monte_carlo_without_diagram(num_trials=10**5)
In [25]: # (C)
monte_carlo_without_diagram(num_trials=10**8)
Value of pi is 3.1462.
Value of pi is 3.14150952.
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Appendix: Customizing Your Plots
One thing you have probably noticed is that when you write "bare-bones" code such as we have above, certain
defaults are used for the size and layout of the figure and the style of the drawing. One of the most noticable is
that when you draw multiple lines, Matplotlib will change the color each time you call the same function (notice
that this doesn't happen when calling a different function, e.g., plot followed by scatter).
Using Colors
Matplotlib cycles through a sequence of 10 colors, which is fine if that is what you want. For my taste, they are
pretty ugly, and in the next section we will show you how to use the colors you want.
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In [26]: print("\nThe 10 Matplotlib color sequence, starting at 12 o\'clock an
d going clockwise:")
plt.figure(figsize=(10,10))
for k in np.arange(0,2*np.pi,np.pi/20): # arange is l
ike range, except it allows you to use floats
plt.plot([0,np.sin(k)],[0,np.cos(k)],lw=4)
plt.title('Line Colors',fontsize=14)
plt.xlim([-1.2,1.2])
plt.ylim([-1.2,1.2])
plt.show()
Here is an example where we simply change the colors of the plot using the appropriate parameter; see a
complete list of colors here: https://matplotlib.org/2.0.2/api/colors_api.html
(https://matplotlib.org/2.0.2/api/colors_api.html)
The 10 Matplotlib color sequence, starting at 12 o'clock and going cl
ockwise:
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In [27]: # EXAMPLE: Plotting a square with lines of different colors
plt.figure(num=None, figsize=(8, 8), dpi=89)
plt.plot([0,1],[0,0],color='red') # Line connecting (0,0) to (1,0)
plt.plot([0,0],[0,1],color='green') # Line connecting (0,0) to (0,1)
plt.plot([0,1],[1,1],color='orange') # Line connecting (0,1) to (1,1)
plt.plot([1,1],[0,1],color='black') # Line connecting (1,0) to (1,1)
Out[27]: [<matplotlib.lines.Line2D at 0x7efc35fd51d0]
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Changing the Style of Plots
Here is an example showing how to
change the size of the whole figure
change the color of lines or points
change the style of lines or points
To see a complete list of lines styles see: https://matplotlib.org/2.0.2/api/lines_api.html
(https://matplotlib.org/2.0.2/api/lines_api.html)
To see a complete list of colors see: https://matplotlib.org/2.0.2/api/colors_api.html
(https://matplotlib.org/2.0.2/api/colors_api.html)
To see a complete list of marker (point) styles see: https://matplotlib.org/2.0.2/api/markers_api.html#modulematplotlib.markers (https://matplotlib.org/2.0.2/api/markers_api.html#module-matplotlib.markers)
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In [28]: # EXAMPLE: Plotting a square via lines
plt.figure(figsize=(12, 12)) # the size is (horizontal, v
ertical)
plt.title("Line Colors and Styles",fontsize=16)
plt.plot([0,10],[0,0], color='red') # Line connecting (0,0) to (1,0)
plt.plot([0,0],[0,10], color='green') # Line connecting (0,0) to (0,
1)
plt.plot([0,10],[10,10],color='orange') # Line connecting (0,1) to
(1,1)
plt.plot([10,10],[0,10],color='black') # Line connecting (1,0) to (1,
1)
plt.plot([1,9],[9,9], linewidth=5) # give a linewidth in points, d
efault is 1.0
plt.plot([1,9],[8,8], linewidth=5,color = '#eaafff') # for custom
color give the RGB value in hex
plt.plot([1,9],[7,7], linewidth=5,color='0.75') # for grey give the p
ercentage of white in quotes
plt.plot([1,9],[6,6], lw=4,linestyle='--') # Linestyles
plt.plot([1,9],[5,5], lw=3,linestyle='-.') # Linestyles
plt.plot([1,9],[4,4], lw=2,linestyle=':') # Linestyles
plt.scatter(range(1,10),[3.5]*9,marker='.') # various markers, if yo
u don't specify the colors it will cycle
plt.scatter(range(1,10),[3]*9,marker='o') # through a bunch of col
ors, starting with blue, orange, green, etc.
plt.scatter(range(1,10),[2.5]*9,marker='x')
plt.scatter(range(1,10),[2]*9,marker='s')
plt.scatter(range(1,10),[1.5]*9,marker='^')
plt.scatter(range(1,10),[1]*9,marker='D')
print()
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Et Cetera
Then you can start getting obsessive, adding gridlines, changing the background color, adding legends, text, and
so on.
Another nice feature of matplotlib is that you can insert simple Latex commands into titles and text.....
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In [29]: # Plotting a smooth curve for the function x^2
x = [i for i in range(11)]
y = [i**2 for i in x]
plt.figure(figsize=(8, 8))
plt.title('Graph of $x = x^2$')
plt.xlabel("X")
plt.ylabel("Y")
plt.grid()
plt.plot(x,y)
plt.show()
plt.figure(figsize=(8, 8))
plt.title('Graph of $y = x^2$')
plt.grid(color='w') # grid of white lines -- don't use
points with this, they look funny
plt.gca().set_facecolor('0.95') # background of light grey
plt.plot(x,y,color='b')
plt.xlabel("X")
plt.ylabel("Y")
plt.show()
plt.figure(figsize=(8, 8))
plt.title('Graph of $y = x^2$')
plt.grid(color='r',alpha=0.1) # alpha sets the transparency, 0
= invisible and 1 = normal
plt.plot(x,y,color='r',lw=0.5,label='Curve')
plt.scatter(x,y,color='r',marker='o',label='Points')
plt.legend()
plt.xlabel("X")
plt.ylabel("Y")
plt.show()
plt.figure(figsize=(8, 8))
plt.title('Graph of $y = x^2$',fontsize=16)
plt.xlabel("X",fontsize=14)
plt.ylabel("Y",rotation=0,fontsize=14)
plt.grid(color='0.95')
plt.text(0,90,"The title has been enlarged from default 12 points to
16 points.")
plt.text(0,80,"Notice that the $y$ axis label is rotated to be uprigh
t, \nand the x and y labels are also bigger, at 14 points.") # lowe
r left corner of text string is at point (0,60)
plt.text(0,60,"When drawing points and lines together it looks \nbett
er if you make the lines thinner.")
plt.text(0,40,"Honestly I think it is also better to just use\nthe de
fault marker (circles) when you draw \nlines, these triangles are kin
da fussy\nand they don't seem to be centered on \nthe data point.")
plt.plot(x,y,color='b',lw=0.5)
plt.scatter(x,y,color='b',marker='^')
plt.show()
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In [ ]: