Analyzing Patient Data
OverviewTeaching: 30 min
Exercises: 0 minQuestions
How can I process tabular data files in Python?Objectives
Explain what a library is and what libraries are used for.
Import a Python library and use the functions it contains.
Read tabular data from a file into a program.
Assign values to variables.
Select individual values and subsections from data.
Perform operations on arrays of data.
Plot simple graphs from data.
In this lesson we will learn how to manipulate the inflammation dataset with Python. Before we discuss how to deal with many data points, we will show how to store a single value on the computer.
You can get output from python by typing math into the console:
weight_kg = 55
A variable is a name for a value,
Python’s variables must begin with a letter and are case sensitive.
We can create a new variable by assigning a value to it using
When we are finished typing and press Shift+Enter,
the notebook runs our command.
Once a variable has a value, we can print it to the screen:
and do arithmetic with it (remember, there are 2.2 pounds per kilogram):
print('weight in pounds:', 2.2 * weight_kg)
weight in pounds: 121.0
As the example above shows, we can print several things at once by separating them with commas.
We can also change a variable’s value by assigning it a new one:
weight_kg = 57.5 print('weight in kilograms is now:', weight_kg)
weight in kilograms is now: 57.5
If we imagine the variable as a sticky note with a name written on it, assignment is like putting the sticky note on a particular value:
This means that assigning a value to one variable does not change the values of other variables. For example, let’s store the subject’s weight in pounds in a variable:
#There are 2.2 pounds per kilogram. weight_lb = 2.2 * weight_kg print('weight in kilograms:', weight_kg, 'and in pounds:', weight_lb)
weight in kilograms: 57.5 and in pounds: 126.5
and then change
weight_kg = 100.0 print('weight in kilograms is now:', weight_kg, 'and weight in pounds is still:', weight_lb)
weight in kilograms is now: 100.0 and weight in pounds is still: 126.5
weight_lb doesn’t remember where its value came from,
it isn’t automatically updated when
This is different from the way spreadsheets work.
Who’s Who in Memory
You can use the
%whoscommand at any time to see what variables you have created and what modules you have loaded into the computer’s memory. As this is an IPython command, it will only work if you are in an IPython terminal or the Jupyter Notebook.
Variable Type Data/Info -------------------------------- weight_kg float 100.0 weight_lb float 126.5
Words are useful, but what’s more useful are the sentences and stories we build with them. Similarly, while a lot of powerful, general tools are built into languages like Python, specialized tools built up from these basic units live in libraries that can be called upon when needed.
In order to load our inflammation data, we need to access (import in Python terminology) a library called NumPy. In general you should use this library if you want to do fancy things with numbers, especially if you have matrices or arrays. We can import NumPy using:
Importing a library is like getting a piece of lab equipment out of a storage locker and setting it up on the bench. Libraries provide additional functionality to the basic Python package, much like a new piece of equipment adds functionality to a lab space. Just like in the lab, importing too many libraries can sometimes complicate and slow down your programs - so we only import what we need for each program. Once we’ve imported the library, we can ask the library to read our data file for us:
array([[ 0., 0., 1., ..., 3., 0., 0.], [ 0., 1., 2., ..., 1., 0., 1.], [ 0., 1., 1., ..., 2., 1., 1.], ..., [ 0., 1., 1., ..., 1., 1., 1.], [ 0., 0., 0., ..., 0., 2., 0.], [ 0., 0., 1., ..., 1., 1., 0.]])
numpy.loadtxt(...) is a function call
that asks Python to run the function
loadtxt which belongs to the
This dotted notation is used everywhere in Python:
the thing that appears before the dot contains the thing that appears after.
As an example, John Smith is the John that belongs to the Smith family,
We could use the dot notation to write his name
loadtxt is a function that belongs to the
numpy.loadtxt has two parameters:
the name of the file we want to read
and the delimiter that separates values on a line.
These both need to be character strings (or strings for short),
so we put them in quotes.
Since we haven’t told it to do anything else with the function’s output,
the notebook displays it.
In this case,
that output is the data we just loaded.
only a few rows and columns are shown
... to omit elements when displaying big arrays).
To save space,
Python displays numbers as
1. instead of
when there’s nothing interesting after the decimal point.
Our call to
numpy.loadtxt read our file
but didn’t save the data in memory.
To do that,
we need to assign the array to a variable. Just as we can assign a single value to a variable, we can also assign an array of values
to a variable using the same syntax. Let’s re-run
numpy.loadtxt and save the returned data:
data = numpy.loadtxt(fname='inflammation-01.csv', delimiter=',')
This statement doesn’t produce any output because we’ve assigned the output to the variable
If we want to check that the data have been loaded,
we can print the variable’s value:
[[ 0. 0. 1. ..., 3. 0. 0.] [ 0. 1. 2. ..., 1. 0. 1.] [ 0. 1. 1. ..., 2. 1. 1.] ..., [ 0. 1. 1. ..., 1. 1. 1.] [ 0. 0. 0. ..., 0. 2. 0.] [ 0. 0. 1. ..., 1. 1. 0.]]
Now that the data are in memory,
we can manipulate them.
let’s ask what type of thing
data refers to:
The output tells us that
data currently refers to
an N-dimensional array, the functionality for which is provided by the NumPy library.
These data correspond to arthritis patients’ inflammation.
The rows are the individual patients, and the columns
are their daily inflammation measurements.
A Numpy array contains one or more elements of the same type. The
typefunction will only tell you that a variable is a NumPy array but won’t tell you the type of thing inside the array. We can find out the type of the data contained in the NumPy array.
This tells us that the NumPy array’s elements are floating-point numbers.
With the following command, we can see the array’s shape:
The output tells us that the
data array variable contains 60 rows and 40 columns. When we created the
data to store our arthritis data, we didn’t just create the array; we also
created information about the array, called members or
attributes. This extra information describes
the same way an adjective describes a noun.
data.shape is an attribute of
data which describes the dimensions of
We use the same dotted notation for the attributes of variables
that we use for the functions in libraries
because they have the same part-and-whole relationship.
If we want to get a single number from the array, we must provide an index in square brackets after the variable name, just as we do in math when referring to an element of a matrix. Our inflammation data has two dimensions, so we will need to use two indices to refer to one specific value:
print('first value in data:', data[0, 0])
first value in data: 0.0
print('middle value in data:', data[30, 20])
middle value in data: 13.0
data[30, 20] accesses the element at row 30, column 20. While this expression may not surprise you,
data[0, 0] might.
Programming languages like Fortran, MATLAB and R start counting at 1
because that’s what human beings have done for thousands of years.
Languages in the C family (including C++, Java, Perl, and Python) count from 0
because it represents an offset from the first value in the array (the second
value is offset by one index from the first value). This is closer to the way
that computers represent arrays (if you are interested in the historical
reasons behind counting indices from zero, you can read
Mike Hoye’s blog post).
As a result,
if we have an M×N array in Python,
its indices go from 0 to M-1 on the first axis
and 0 to N-1 on the second.
It takes a bit of getting used to,
but one way to remember the rule is that
the index is how many steps we have to take from the start to get the item we want.
In the Corner
What may also surprise you is that when Python displays an array, it shows the element with index
[0, 0]in the upper left corner rather than the lower left. This is consistent with the way mathematicians draw matrices but different from the Cartesian coordinates. The indices are (row, column) instead of (column, row) for the same reason, which can be confusing when plotting data.
An index like
[30, 20] selects a single element of an array,
but we can select whole sections as well.
we can select the first ten days (columns) of values
for the first four patients (rows) like this:
[[ 0. 0. 1. 3. 1. 2. 4. 7. 8. 3.] [ 0. 1. 2. 1. 2. 1. 3. 2. 2. 6.] [ 0. 1. 1. 3. 3. 2. 6. 2. 5. 9.] [ 0. 0. 2. 0. 4. 2. 2. 1. 6. 7.]]
“Start at index 0 and go up to, but not including, index 4.”
the up-to-but-not-including takes a bit of getting used to,
but the rule is that the difference between the upper and lower bounds is the number of values in the slice.
We don’t have to start slices at 0:
[[ 0. 0. 1. 2. 2. 4. 2. 1. 6. 4.] [ 0. 0. 2. 2. 4. 2. 2. 5. 5. 8.] [ 0. 0. 1. 2. 3. 1. 2. 3. 5. 3.] [ 0. 0. 0. 3. 1. 5. 6. 5. 5. 8.] [ 0. 1. 1. 2. 1. 3. 5. 3. 5. 8.]]
We also don’t have to include the upper and lower bound on the slice. If we don’t include the lower bound, Python uses 0 by default; if we don’t include the upper, the slice runs to the end of the axis, and if we don’t include either (i.e., if we just use ‘:’ on its own), the slice includes everything:
small = data[:3, 36:] print('small is:') print(small)
The above example selects rows 0 through 2 and columns 36 through to the end of the array.
small is: [[ 2. 3. 0. 0.] [ 1. 1. 0. 1.] [ 2. 2. 1. 1.]]
Arrays also know how to perform common mathematical operations on their values. The simplest operations with data are arithmetic: addition, subtraction, multiplication, and division. When you do such operations on arrays, the operation is done element-by-element. Thus:
doubledata = data * 2.0
will create a new array
each elements of which is twice the value of the corresponding element in
print('original:') print(data[:3, 36:]) print('doubledata:') print(doubledata[:3, 36:])
original: [[ 2. 3. 0. 0.] [ 1. 1. 0. 1.] [ 2. 2. 1. 1.]] doubledata: [[ 4. 6. 0. 0.] [ 2. 2. 0. 2.] [ 4. 4. 2. 2.]]
If, instead of taking an array and doing arithmetic with a single value (as above), you did the arithmetic operation with another array of the same shape, the operation will be done on corresponding elements of the two arrays. Thus:
tripledata = doubledata + data
will give you an array where
tripledata[0,0] will equal
and so on for all other elements of the arrays.
print('tripledata:') print(tripledata[:3, 36:])
tripledata: [[ 6. 9. 0. 0.] [ 3. 3. 0. 3.] [ 6. 6. 3. 3.]]
Often, we want to do more than add, subtract, multiply, and divide array elements.
NumPy knows how to do more complex operations, too.
If we want to find the average inflammation for all patients on all days,
we can ask NumPy to compute
data’s mean value:
Not All Functions Have Input
Generally, a function uses inputs to produce outputs. However, some functions produce outputs without needing any input. For example, checking the current time doesn’t require any input.
import time print(time.ctime())
'Sat Mar 26 13:07:33 2016'
For functions that don’t take in any arguments, we still need parentheses (
()) to tell Python to go and do something for us.
NumPy has lots of useful functions that take an array as input. Let’s use three of those functions to get some descriptive values about the dataset. We’ll also use multiple assignment, a convenient Python feature that will enable us to do this all in one line.
maxval, minval, stdval = numpy.max(data), numpy.min(data), numpy.std(data) print('maximum inflammation:', maxval) print('minimum inflammation:', minval) print('standard deviation:', stdval)
Here we’ve assigned the return value from
numpy.max(data) to the variable
maxval, the value from
minval, and so on.
maximum inflammation: 20.0 minimum inflammation: 0.0 standard deviation: 4.61383319712
Mystery Functions in IPython
How did we know what functions NumPy has and how to use them? If you are working in the IPython/Jupyter Notebook, there is an easy way to find out. If you type the name of something followed by a dot, then you can use tab completion (e.g. type
numpy.and then press tab) to see a list of all functions and attributes that you can use. After selecting one, you can also add a question mark (e.g.
numpy.cumprod?), and IPython will return an explanation of the method! This is the same as doing
When analyzing data, though, we often want to look at variations in statistical values, such as the maximum inflammation per patient or the average inflammation per day. One way to do this is to create a new temporary array of the data we want, then ask it to do the calculation:
patient_0 = data[0, :] # 0 on the first axis (rows), everything on the second (columns) print('maximum inflammation for patient 0:', patient_0.max())
maximum inflammation for patient 0: 18.0
Everything in a line of code following the ‘#’ symbol is a comment that is ignored by Python. Comments allow programmers to leave explanatory notes for other programmers or their future selves.
We don’t actually need to store the row in a variable of its own. Instead, we can combine the selection and the function call:
print('maximum inflammation for patient 2:', numpy.max(data[2, :]))
maximum inflammation for patient 2: 19.0
What if we need the maximum inflammation for each patient over all days (as in the next diagram on the left) or the average for each day (as in the diagram on the right)? As the diagram below shows, we want to perform the operation across an axis:
To support this functionality, most array functions allow us to specify the axis we want to work on. If we ask for the average across axis 0 (rows in our 2D example), we get:
[ 0. 0.45 1.11666667 1.75 2.43333333 3.15 3.8 3.88333333 5.23333333 5.51666667 5.95 5.9 8.35 7.73333333 8.36666667 9.5 9.58333333 10.63333333 11.56666667 12.35 13.25 11.96666667 11.03333333 10.16666667 10. 8.66666667 9.15 7.25 7.33333333 6.58333333 6.06666667 5.95 5.11666667 3.6 3.3 3.56666667 2.48333333 1.5 1.13333333 0.56666667]
As a quick check, we can ask this array what its shape is:
(40,) tells us we have an N×1 vector,
so this is the average inflammation per day for all patients.
If we average across axis 1 (columns in our 2D example), we get:
[ 5.45 5.425 6.1 5.9 5.55 6.225 5.975 6.65 6.625 6.525 6.775 5.8 6.225 5.75 5.225 6.3 6.55 5.7 5.85 6.55 5.775 5.825 6.175 6.1 5.8 6.425 6.05 6.025 6.175 6.55 6.175 6.35 6.725 6.125 7.075 5.725 5.925 6.15 6.075 5.75 5.975 5.725 6.3 5.9 6.75 5.925 7.225 6.15 5.95 6.275 5.7 6.1 6.825 5.975 6.725 5.7 6.25 6.4 7.05 5.9 ]
which is the average inflammation per patient across all days.
The mathematician Richard Hamming once said,
“The purpose of computing is insight, not numbers,”
and the best way to develop insight is often to visualize data.
Visualization deserves an entire lecture of its own,
but we can explore a few features of Python’s
matplotlib library here.
While there is no official plotting library,
matplotlib is the de facto standard.
we will import the
pyplot module from
and use two of its functions to create and display a heat map of our data:
import matplotlib.pyplot image = matplotlib.pyplot.imshow(data) matplotlib.pyplot.show()
Blue pixels in this heat map represent low values, while yellow pixels represent high values. As we can see, inflammation rises and falls over a 40-day period.
Some IPython Magic
If you’re using an IPython / Jupyter notebook, you’ll need to execute the following command in order for your matplotlib images to appear in the notebook when
%indicates an IPython magic function - a function that is only valid within the notebook environment. Note that you only have to execute this function once per notebook.
Let’s take a look at the average inflammation over time:
ave_inflammation = numpy.mean(data, axis=0) ave_plot = matplotlib.pyplot.plot(ave_inflammation) matplotlib.pyplot.show()
we have put the average per day across all patients in the variable
matplotlib.pyplot to create and display a line graph of those values.
The result is a roughly linear rise and fall,
which is suspicious:
we might instead expect a sharper rise and slower fall.
Let’s have a look at two other statistics:
max_plot = matplotlib.pyplot.plot(numpy.max(data, axis=0)) matplotlib.pyplot.show()
min_plot = matplotlib.pyplot.plot(numpy.min(data, axis=0)) matplotlib.pyplot.show()
The maximum value rises and falls smoothly, while the minimum seems to be a step function. Neither trend seems particularly likely, so either there’s a mistake in our calculations or something is wrong with our data. This insight would have been difficult to reach by examining the numbers themselves without visualization tools.
You can group similar plots in a single figure using subplots.
This script below uses a number of new commands. The function
creates a space into which we will place all of our plots. The parameter
tells Python how big to make this space. Each subplot is placed into the figure using
add_subplot method. The
add_subplot method takes 3 parameters. The first denotes
how many total rows of subplots there are, the second parameter refers to the
total number of subplot columns, and the final parameter denotes which subplot
your variable is referencing (left-to-right, top-to-bottom). Each subplot is stored in a
different variable (
axes3). Once a subplot is created, the axes can
be titled using the
set_xlabel() command (or
Here are our three plots side by side:
import numpy import matplotlib.pyplot data = numpy.loadtxt(fname='inflammation-01.csv', delimiter=',') fig = matplotlib.pyplot.figure(figsize=(10.0, 3.0)) axes1 = fig.add_subplot(1, 3, 1) axes2 = fig.add_subplot(1, 3, 2) axes3 = fig.add_subplot(1, 3, 3) axes1.set_ylabel('average') axes1.plot(numpy.mean(data, axis=0)) axes2.set_ylabel('max') axes2.plot(numpy.max(data, axis=0)) axes3.set_ylabel('min') axes3.plot(numpy.min(data, axis=0)) fig.tight_layout() matplotlib.pyplot.show()
The call to
loadtxt reads our data,
and the rest of the program tells the plotting library
how large we want the figure to be,
that we’re creating three subplots,
what to draw for each one,
and that we want a tight layout.
(If we leave out that call to
the graphs will actually be squeezed together more closely.)
Scientists Dislike Typing
We will always use the syntax
import numpyto import NumPy. However, in order to save typing, it is often suggested to make a shortcut like so:
import numpy as np. If you ever see Python code online using a NumPy function with
np.loadtxt(...)), it’s because they’ve used this shortcut. When working with other people, it is important to agree on a convention of how common libraries are imported.
Check Your Understanding
Draw diagrams showing what variables refer to what values after each statement in the following program:
mass = 47.5 age = 122 mass = mass * 2.0 age = age - 20
Sorting Out References
What does the following program print out?
first, second = 'Grace', 'Hopper' third, fourth = second, first print(third, fourth)
A section of an array is called a slice. We can take slices of character strings as well:
element = 'oxygen' print('first three characters:', element[0:3]) print('last three characters:', element[3:6])
first three characters: oxy last three characters: gen
What is the value of
element[:4]? What about
oxyg en oxygen
element[-1]? What is
Given those answers, explain what
Creates a substring from index 1 up to (not including) the final index, effectively removing the first and last letters from ‘oxygen’
element[3:3]produces an empty string, i.e., a string that contains no characters. If
dataholds our array of patient data, what does
data[3:3, 4:4]produce? What about
Why do all of our plots stop just short of the upper end of our graph?
Because matplotlib normally sets x and y axes limits to the min and max of our data (depending on data range)
If we want to change this, we can use the
set_ylim(min, max)method of each ‘axes’, for example:
Update your plotting code to automatically set a more appropriate scale. (Hint: you can make use of the
minmethods to help.)
# One method axes3.set_ylabel('min') axes3.plot(numpy.min(data, axis=0)) axes3.set_ylim(0,6)
# A more automated approach min_data = numpy.min(data, axis=0) axes3.set_ylabel('min') axes3.plot(min_data) axes3.set_ylim(numpy.min(min_data), numpy.max(min_data) * 1.1)
Drawing Straight Lines
In the center and right subplots above, we expect all lines to look like step functions because non-integer value are not realistic for the minimum and maximum values. However, you can see that the lines are not always vertical or horizontal, and in particular the step function in the subplot on the right looks slanted. Why is this?
Because matplotlib interpolates (draws a straight line) between the points. One way to do avoid this is to use the Matplotlib
import numpy import matplotlib.pyplot data = numpy.loadtxt(fname='inflammation-01.csv', delimiter=',') fig = matplotlib.pyplot.figure(figsize=(10.0, 3.0)) axes1 = fig.add_subplot(1, 3, 1) axes2 = fig.add_subplot(1, 3, 2) axes3 = fig.add_subplot(1, 3, 3) axes1.set_ylabel('average') axes1.plot(numpy.mean(data, axis=0), drawstyle='steps-mid') axes2.set_ylabel('max') axes2.plot(numpy.max(data, axis=0), drawstyle='steps-mid') axes3.set_ylabel('min') axes3.plot(numpy.min(data, axis=0), drawstyle='steps-mid') fig.tight_layout() matplotlib.pyplot.show()
Make Your Own Plot
Create a plot showing the standard deviation (
numpy.std) of the inflammation data for each day across all patients.
std_plot = matplotlib.pyplot.plot(numpy.std(data, axis=0)) matplotlib.pyplot.show()
Moving Plots Around
Modify the program to display the three plots on top of one another instead of side by side.
import numpy import matplotlib.pyplot data = numpy.loadtxt(fname='inflammation-01.csv', delimiter=',') # change figsize (swap width and height) fig = matplotlib.pyplot.figure(figsize=(3.0, 10.0)) # change add_subplot (swap first two parameters) axes1 = fig.add_subplot(3, 1, 1) axes2 = fig.add_subplot(3, 1, 2) axes3 = fig.add_subplot(3, 1, 3) axes1.set_ylabel('average') axes1.plot(numpy.mean(data, axis=0)) axes2.set_ylabel('max') axes2.plot(numpy.max(data, axis=0)) axes3.set_ylabel('min') axes3.plot(numpy.min(data, axis=0)) fig.tight_layout() matplotlib.pyplot.show()
Arrays can be concatenated and stacked on top of one another, using NumPy’s
hstackfunctions for vertical and horizontal stacking, respectively.
import numpy A = numpy.array([[1,2,3], [4,5,6], [7, 8, 9]]) print('A = ') print(A) B = numpy.hstack([A, A]) print('B = ') print(B) C = numpy.vstack([A, A]) print('C = ') print(C)
A = [[1 2 3] [4 5 6] [7 8 9]] B = [[1 2 3 1 2 3] [4 5 6 4 5 6] [7 8 9 7 8 9]] C = [[1 2 3] [4 5 6] [7 8 9] [1 2 3] [4 5 6] [7 8 9]]
Write some additional code that slices the first and last columns of
A, and stacks them into a 3x2 array. Make sure to
A ‘gotcha’ with array indexing is that singleton dimensions are dropped by default. That means
A[:, 0]is a one dimensional array, which won’t stack as desired. To preserve singleton dimensions, the index itself can be a slice or array. For example,
A[:, :1]returns a two dimensional array with one singleton dimension (i.e. a column vector).
D = numpy.hstack((A[:, :1], A[:, -1:])) print('D = ') print(D)
D = [[1 3] [4 6] [7 9]]
An alternative way to achieve the same result is to use Numpy’s delete function to remove the second column of A.
D = numpy.delete(A, 1, 1) print('D = ') print(D)
D = [[1 3] [4 6] [7 9]]
Change In Inflammation
This patient data is longitudinal in the sense that each row represents a series of observations relating to one individual. This means that the change in inflammation over time is a meaningful concept.
numpy.diff()function takes a NumPy array and returns the differences between two successive values along a specified axis. For example, a NumPy array that looks like this:
npdiff = numpy.array([ 0, 2, 5, 9, 14])
numpy.diff(npdiff)would do the following calculations and put the answers in another array.
[ 2 - 0, 5 - 2, 9 - 5, 14 - 9 ]
array([2, 3, 4, 5])
Which axis would it make sense to use this function along?
Since the row axis (0) is patients, it does not make sense to get the difference between two arbitrary patients. The column axis (1) is in days, so the difference is the change in inflammation – a meaningful concept.
If the shape of an individual data file is
(60, 40)(60 rows and 40 columns), what would the shape of the array be after you run the
diff()function and why?
The shape will be
(60, 39)because there is one fewer difference between columns than there are columns in the data.
How would you find the largest change in inflammation for each patient? Does it matter if the change in inflammation is an increase or a decrease?
By using the
numpy.max()function after you apply the
numpy.diff()function, you will get the largest difference between days.
numpy.max(numpy.diff(data, axis=1), axis=1)
array([ 7., 12., 11., 10., 11., 13., 10., 8., 10., 10., 7., 7., 13., 7., 10., 10., 8., 10., 9., 10., 13., 7., 12., 9., 12., 11., 10., 10., 7., 10., 11., 10., 8., 11., 12., 10., 9., 10., 13., 10., 7., 7., 10., 13., 12., 8., 8., 10., 10., 9., 8., 13., 10., 7., 10., 8., 12., 10., 7., 12.])
If inflammation values decrease along an axis, then the difference from one element to the next will be negative. If you are interested in the magnitude of the change and not the direction, the
numpy.absolute()function will provide that.
Notice the difference if you get the largest absolute difference between readings.
numpy.max(numpy.absolute(numpy.diff(data, axis=1)), axis=1)
array([ 12., 14., 11., 13., 11., 13., 10., 12., 10., 10., 10., 12., 13., 10., 11., 10., 12., 13., 9., 10., 13., 9., 12., 9., 12., 11., 10., 13., 9., 13., 11., 11., 8., 11., 12., 13., 9., 10., 13., 11., 11., 13., 11., 13., 13., 10., 9., 10., 10., 9., 9., 13., 10., 9., 10., 11., 13., 10., 10., 12.])
Import a library into a program using
numpylibrary to work with arrays in Python.
variable = valueto assign a value to a variable in order to record it in memory.
Variables are created on demand whenever a value is assigned to them.
print(something)to display the value of
array.shapegives the shape of an array.
array[x, y]to select a single element from a 2D array.
Array indices start at 0, not 1.
low:highto specify a
slicethat includes the indices from
All the indexing and slicing that works on arrays also works on strings.
# some kind of explanationto add comments to programs.
numpy.min(array)to calculate simple statistics.
numpy.mean(array, axis=1)to calculate statistics across the specified axis.
matplotlibfor creating simple visualizations.