We have seen what are quartiles and how can be useful in quickly presenting the main characteristics of a group of data.
Let’s see how to visualise them.
I will use as example the age of the Nobel Prize winners – a discrete values set – from the Nobel Prize official site.
The best way to chart a data set with its quartiles is to use a box plot:
a box that goes from the upper to the lower quartile, plus optionally lines (the whiskers) extending from the box that go until a specified multiplier of the Inter-Quartile Range (IQR = upper – lower quartiles), while any other point outside this range is considered an outlier data point and displayed as a point.
Inside the box the median and the mean can be displayed, as lines or points.
The matplotlib function to draw a boxplot is appropriately called boxplot()
The function requires to pass as input data an array or a list of vectors, for example:
agePhysics = [ 25, 31, 31, 31, ... ] # goes on for almost 200 values
The basic plot would be:
import matplotlib.pyplot as plt # basic plot plt.boxplot(agePhysics) plt.show()
The defaults used in the other parameters of the boxplot function are:
notch = False : draw a rectangular box, not notched
vert = True: the box is vertical, not horizontal
sym = None: no fliers displayed
whis = 1.5 : the multipliers from the whiskers variability, they go until whis * IQR
Now let’s print how much are the quartiles and the mean before plotting and display the mean by using the parameter showmeans (default is False), by adding/changing these lines:
from datascience import stats print(stats.summary(agePhysics)) print("range = ", stats.range(agePhysics)) plt.boxplot(agePhysics, showmeans=True, whis = 99)
Output printed is:
Summary statistics Min: 25 Lower Qu.: 45.0 Median: 54.0 Mean: 54.955 Upper Qu.: 64.0 Max: 88 That's all
And the lines above display a box-and-whiskers chart like this:
As you see the box itself goes from the upper to the lower quartile (45 and 64 in this case), while the whiskers (the bars extending from the box) go from the minimum to the maximum (25 and 88 in this case) because whis is set to a very high number (99) therefore including all the data points.
The red line is the median (54) while the mean (similar value) is a red square but can be changed through the parameter meanprops.
Now to add bit more fun, let’s add two more boxplots, respectively for the Literature and the Economics winners. Assuming we have the ages in two arrays called ageLiterature and ageEconomics, the first thing to do is to concatenate all the arrays and pass them to the boxplot function:
ages=[agePhysics, ageLiterature, ageEconomics] box = plt.boxplot(ages, showmeans=True, whis=99)
Each boxplot can have its own colours, this can be set through the pyplot function setp():
# add colours # physics = green plt.setp(box['boxes'], color='green') plt.setp(box['caps'], color='green') plt.setp(box['whiskers'], color='green')
and so on for the other boxplots …
As for the other plots, you can add titles, labels and a grid:
plt.ylim([20, 95]) # y axis gets more space at the extremes plt.grid(True, axis='y') # let's add a grid on y-axis plt.title('Distribution of the Nobel Prize winner ages', fontsize=18) # chart title plt.ylabel('Age (years) at winning time') # y axis title plt.xticks([1,2,3], ['Physics','Literature','Economics']) # x axis labels
this is the final graph:
So, it seems that you have almost no chance to win a Nobel in Literature before you are 40 and more likely before you’re 55 years old but it’s even worse for Economics: nobody won it till now before age 50 and the mean/median are 65 …