Fix typos

README.md

 This repository consists of a collection of python examples intended as an
-introduction on the use of python in data analysis, especially for the
+introduction to the use of python in data analysis, especially for the
 advanced laboratories in physics at the University of Freiburg. In previous
 years, code examples for [ROOT](https://root.cern.ch/) have been provided.
 Material showing how to use python for the same task was missing. This document
 tries to fill the gap.
 
-If you think this tutorial is useful, lacks important information, or is
+If you think this tutorial is useful, lacks essential information, or is
 unclear, don't hesitate to give [feedback](mailto:frank@sauerburger.com).
 
 # Table of Contents
@@ -36,7 +36,7 @@ $ pip3 install --user numpy scipy matplotlib
 ```
 
 The `--user` argument for `pip` installs the python package in your home
-directory, which hides potentially older packages installed with `apt-get`.
+directory, which potentially hides older packages installed with `apt-get`.
 
 ## Windows
 Since I'm not using python on Windows myself, I don't have first-hand experience
@@ -46,7 +46,7 @@ good solution for windows users since it provides all required packages.
 # Prerequisites and About the Tutorial
 
 This tutorial assumes that you have some experience with python, which
-includes variable assignment, function calling and function definition, `if` statements and
+includes variable assignment, function calling, and function definition, `if` statements and
 `for` loops. The tutorial uses only the very basics, such as variable assignments
 and function calls, but it is certainly advisable to know about control
 structures.
@@ -113,7 +113,7 @@ The standard data structure to store numerical data is a numpy array. Numpy
 arrays are defined in the numpy package and are implemented in a very
 efficient way. 
 
-To get stared with numpy arrays, create a file `np_arrays.py` and add all lines
+To get started with numpy arrays, create a file `np_arrays.py` and add all lines
 listed in this section. The first line should be an import statement.
 <!-- write np_arrays.py -->
 ```python
@@ -167,7 +167,7 @@ The result is [-1.   0.5  2.   5.   6.5]
 ```
 
 Numpy offers many other functionalities which are beyond the scope of this basic
-introduction. It is definitely worth glancing at the
+introduction. It is worth glancing at the
 [documentation](https://docs.scipy.org/doc/numpy/index.html).
 
 # Plotting Functions
@@ -201,7 +201,7 @@ matplotlib.use('Agg')
 # Import the numpy library.
 import numpy as np
 
-# Import the powerfull plotting library.
+# Import the powerful plotting library.
 import matplotlib.pyplot as plt
 ```
 
@@ -215,7 +215,7 @@ x = np.linspace(-2.5, 3, 200)
 ```
 
 We can easily calculate the square of all these values with `x**2`. Cropping the
-right part is a bit more tricky. First we create an index array of `1`'s and
+right part is a bit more tricky. First, we create an index array of `1`'s and
 `0`'s, which
 indicate whether $`x \geq 2`$. This index array has the same length as our
 $`x`$-grid. The first elements of the index array are `0`'s, since the
@@ -225,7 +225,7 @@ The index array can be used to select a subset of
 $`y`$-values, namely all $`y`$-values, for which $`x\geq 2`$. Finally, we can
 assign the value $`4`$ to this subset, and therefore
 effectively crop the parabola. The implementation in python of the algorithm outlined
-above  is rather short.
+above is rather short.
 <!-- append func_plot.py -->
 ```python
 # Calculate the regular parabola.
@@ -237,7 +237,7 @@ idx = (x >= 2)
 # Set all y-values to 4, for which x >= 2.
 y[idx] = 4
 
-# One can get rid of the intermetdiate index array and combine both lines into
+# One can get rid of the intermediate index array and combine both lines into
 # the statement y[x >= 2] = 4
 ```
 
@@ -253,7 +253,7 @@ plt.plot(x, y)
 plt.xlabel("$x$") 
 plt.ylabel("Cropped Parabola")
 
-# Save the figure. Various different output formats are available.
+# Save the figure. Different output formats are available.
 plt.savefig("cropped_parabola.eps")
 ```
 <!-- append func_plot.py
@@ -288,7 +288,7 @@ follow $`f(x)`$.
 This example is based on the code from the previous example. Copy the file from
 the previous example to `data_plot.py`, such that we can append the following
 code snippets to `data_plot.py`. Keep
-the plotting code from the previous example as it is.
+the plotting code from the last example as it is.
 <!-- console
 ```bash
 $ cp func_plot.py data_plot.py
@@ -367,7 +367,7 @@ After running `data_plot.py`, you should have a plot similar to this.
 ![Plot of cropped parabola with data points](https://gitlab.sauerburger.com/frank/FP-python-examples/-/jobs/artifacts/master/raw/measurement.png?job=doxec_test)
 
 # Reading, Plotting and Fitting Experimental Data
-We are given with experimental data from a radioactive decay in this example.
+We are given with experimental data from radioactive decay in this example.
 The experimental setup consisted of a radioactive probe, a detector, and a
 multi-channel-analyzer. The recorded data in
 [`decay.txt`](https://gitlab.sauerburger.com/frank/FP-python-examples/blob/master/decay.txt)
@@ -471,13 +471,13 @@ def model(channel, m, s, A, y0, b):
 Please note that we are making an approximation with this definition.
 Strictly speaking, comparing the return values of our model to the measured
 count is not correct. The variable _channel_ corresponds to the energy
-measured with the setup. Lets assume channel $`c_i`$ corresponds to energy
+measured with the setup. Let's assume channel $`c_i`$ corresponds to the energy
 $`E_i`$. If we measure $`n_i`$ events in channel $`c_i`$, this means that we
 have measured $`n_i`$ in the energy interval $`[\frac{1}{2}(E_{i-1} + E_i),
 \frac{1}{2}(E_i + E_{i+1})]`$. The proper way is to integrate our continuous
 function $`n(c)`$ in each bin $`[c_{i} - \frac{1}{2}, c_{i} + \frac{1}{2}]`$ and
 compare these bin-wise integrals to the measured data. The
-procedure shown here is a good approximation, if the function can be considered
+procedure shown here is a good approximation if the function can be considered
 to be linear within each bin. However, the parameter $`A`$ and $`b`$ are not normalized to
 the bin width in this case.
 
@@ -486,10 +486,10 @@ To fit this model to our experimental data, we can use the function `curve_fit`
 provided by the scipy package. The function `curve_fit` performs a least square
 fit and returns the optimal parameters and the covariance matrix. The fit might
 not converge on its own. We can guide the optimization procedure by providing
-suitable start values of the free parameters. From the plot I read off a height
+suitable start values of the free parameters. From the plot, I read off a height
 $`A=50`$, a center $`m=60`$ and a width $`s = 10`$ for the Gaussian part and
 $`y_0 = 20`$ and $`b = 1`$ for the linear part. These values don't have to be
-accurate. They should be a rough estimation, this is usually enough to get a
+accurate. They should be a rough estimation. This is usually enough to get a
 stable fit result. More information on the fitting method can be found in the
 [documentation](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html).
 
@@ -600,9 +600,9 @@ print("  p-value = %g" % p)
 ```
 
 The additional `- 1` in the call of `chisquare` is necessary because the method
-implicitly assumes that the total number of events is fixed. The method
-therefore reduces the number of degrees of freedom by one. However, in our case,
-the number of decays is not fixed and we have an additional degree
+implicitly assumes that the total number of events is fixed. The method,
+therefore, reduces the number of degrees of freedom by one. However, in our case,
+the number of decays is not fixed, and we have an additional degree
 of freedom compared to what `chisquare` assumes. The `- 1` in the `ddof`
 corrects for this.