# Run this cell to set up packages for lecture.
from lec07_imports import *

Lecture 7 – Distributions and Histograms

DSC 10, Spring 2024

Announcements

• Quiz 1 scores are released!
  • Remember, quizzes are a place where it's okay to fail and learn from your mistakes. Your lowest two quiz scores are dropped!
  • Make sure to attend discussion section so you have experience solving the types of problems you'll encounter on quizzes and exams.
  • Meet with a course tutor to review your quiz performance and study strategies!
• Homework 1 is due tomorrow at 11:59PM.
  • For question 7.5, check out BPD 10.4 Searching for a Substring.
• Lab 2 is due Thursday at 11:59PM.
• The class is really picking up with quizzes, labs, and homeworks - start assignments early to not fall behind! 🏃‍♀️🏃‍

Agenda

• Line plots.
• Bar charts.
• Distributions.
• Density histograms.
• For your reference: Overlaid plots.

Density histograms are quite theoretical – you can practice with this material in this week's discussion section.

Line plots 📉

Line plots

• How has the 'Magnitude' of newly discovered exoplanets 🪐 changed over time?

# There were multiple exoplanets discovered each year.
# What operation can we apply to this DataFrame so that there is one row per year?
exo = bpd.read_csv('data/exoplanets.csv').set_index('Name')
exo

	Distance	Magnitude	Type	Year	Detection	Mass	Radius
Name
11 Comae Berenices b	304.0	4.72	Gas Giant	2007	Radial Velocity	6165.90	11.88
11 Ursae Minoris b	409.0	5.01	Gas Giant	2009	Radial Velocity	4684.81	11.99
14 Andromedae b	246.0	5.23	Gas Giant	2008	Radial Velocity	1525.58	12.65
...	...	...	...	...	...	...	...
YZ Ceti b	12.0	12.07	Terrestrial	2017	Radial Velocity	0.70	0.91
YZ Ceti c	12.0	12.07	Super Earth	2017	Radial Velocity	1.14	1.05
YZ Ceti d	12.0	12.07	Super Earth	2017	Radial Velocity	1.09	1.03

5043 rows × 7 columns

• Let's calculate the average 'Magnitude' of all exoplanets discovered in each 'Year'.

exo.groupby('Year').mean()

	Distance	Magnitude	Mass	Radius
Year
1995	50.00	5.45	146.20	13.97
1996	51.33	5.12	1020.67	13.09
1997	57.00	5.41	332.10	13.53
...	...	...	...	...
2021	1944.22	13.01	255.42	4.44
2022	508.61	10.62	943.16	6.77
2023	451.89	12.09	162.78	7.12

29 rows × 4 columns

exo.groupby('Year').mean().plot(kind='line', y='Magnitude');

• It looks like the brightest planets were discovered first, which makes sense.

• NASA's Kepler space telescope began its nine-year mission in 2009, leading to a boom in the discovery of exoplanets.

Line plots

• Line plots show trends in numerical variables over time.
• To create one from a DataFrame df, use

df.plot(
    kind='line', 
    x=x_column_for_horizontal, 
    y=y_column_for_vertical
)

• To use the index as the x-axis, omit the x= argument.
  • This doesn't work for scatterplots, but it works for most other plot types.

Bar charts 📊

Bar charts

• How big are each of the different 'Type's of exoplanets, on average?

types = exo.groupby('Type').mean()
types

	Distance	Magnitude	Year	Mass	Radius
Type
Gas Giant	1096.40	10.30	2013.73	1472.39	12.74
Neptune-like	2189.02	13.52	2016.59	15.28	3.11
Super Earth	1916.26	13.85	2016.43	5.81	1.58
Terrestrial	1373.60	13.45	2016.37	1.62	0.85

types.plot(kind='barh', y='Radius');

types.plot(kind='barh', y='Mass');

• It looks like the 'Gas Giant's are aptly named!

Bar charts

• Bar charts visualize the relationship between a categorical variable and a numerical variable.
• In a bar chart...
  • The thickness and spacing of bars is arbitrary.
  • The order of the categorical labels doesn't matter.
• To create one from a DataFrame df, use

df.plot(
    kind='barh', 
    x=categorical_column_name, 
    y=numerical_column_name
)

• The "h" in 'barh' stands for "horizontal".
  • It's easier to read labels this way.
• Note that in the previous chart, we set y='Mass' even though mass is measured by x-axis length.

Bar charts and sorting

What are the most popular 'Detection' methods for discovering exoplanets?

# Count how many exoplanets are discovered by each detection method.
popular_detection = exo.groupby('Detection').count()
popular_detection

	Distance	Magnitude	Type	Year	Mass	Radius
Detection
Astrometry	1	1	1	1	1	1
Direct Imaging	50	50	50	50	50	50
Disk Kinematics	1	1	1	1	1	1
...	...	...	...	...	...	...
Radial Velocity	1019	1019	1019	1019	1019	1019
Transit	3914	3914	3914	3914	3914	3914
Transit Timing Variations	23	23	23	23	23	23

11 rows × 6 columns

# Give columns more meaningful names and eliminate redundancy.
popular_detection = (popular_detection.assign(Count=popular_detection.get('Distance'))
                                      .get(['Count'])
                                      .sort_values(by='Count', ascending=False)
                    )
popular_detection

	Count
Detection
Transit	3914
Radial Velocity	1019
Direct Imaging	50
...	...
Astrometry	1
Disk Kinematics	1
Pulsar Timing	1

11 rows × 1 columns

# Notice that the bars appear in the opposite order relative to the DataFrame.
popular_detection.plot(kind='barh', y='Count');

# Change "barh" to "bar" to get a vertical bar chart. 
# These are harder to read, but the bars do appear in the same order as the DataFrame.
popular_detection.plot(kind='bar', y='Count');

Distributions

What is the distribution of a variable?

• The distribution of a variable consists of all values of the variable that occur in the data, along with their frequencies.
• Distributions help you understand:

  How often does a variable take on a certain value?

• Both categorical and numerical variables have distributions.

Categorical variables

The distribution of a categorical variable can be displayed as a table or bar chart, among other ways!

For example, let's look at the distribution of exoplanet 'Type's. To do so, we'll need to group.

exo

	Distance	Magnitude	Type	Year	Detection	Mass	Radius
Name
11 Comae Berenices b	304.0	4.72	Gas Giant	2007	Radial Velocity	6165.90	11.88
11 Ursae Minoris b	409.0	5.01	Gas Giant	2009	Radial Velocity	4684.81	11.99
14 Andromedae b	246.0	5.23	Gas Giant	2008	Radial Velocity	1525.58	12.65
...	...	...	...	...	...	...	...
YZ Ceti b	12.0	12.07	Terrestrial	2017	Radial Velocity	0.70	0.91
YZ Ceti c	12.0	12.07	Super Earth	2017	Radial Velocity	1.14	1.05
YZ Ceti d	12.0	12.07	Super Earth	2017	Radial Velocity	1.09	1.03

5043 rows × 7 columns

# Remember, when we group and use .count(), the column names aren't meaningful.
type_counts = exo.groupby('Type').count()
type_counts

	Distance	Magnitude	Year	Detection	Mass	Radius
Type
Gas Giant	1480	1480	1480	1480	1480	1480
Neptune-like	1793	1793	1793	1793	1793	1793
Super Earth	1577	1577	1577	1577	1577	1577
Terrestrial	193	193	193	193	193	193

# As a result, we could have set y='Magnitude', for example, and gotten the same plot.
type_counts.plot(kind='barh', y='Distance', 
                 legend=False, xlabel='Count', title='Distribution of Exoplanet Types');

Notice the optional title argument. Some other useful optional arguments are legend, figsize, xlabel, and ylabel. There are many optional arguments.

It looks like terrestrial exoplanets are the most rare in the dataset. They also have the smallest average radius of any 'Type'.

exo.groupby('Type').mean().get('Radius')

Type
Gas Giant       12.74
Neptune-like     3.11
Super Earth      1.58
Terrestrial      0.85
Name: Radius, dtype: float64

Let's look into them further!

Terrestrial exoplanets 🌑

terr = exo[exo.get('Type') == 'Terrestrial']
terr

	Distance	Magnitude	Type	Year	Detection	Mass	Radius
Name
EPIC 201497682 b	825.0	13.95	Terrestrial	2019	Transit	0.26	0.69
EPIC 201757695.02	1884.0	14.97	Terrestrial	2020	Transit	0.69	0.91
EPIC 201833600 c	840.0	14.71	Terrestrial	2019	Transit	0.97	1.00
...	...	...	...	...	...	...	...
TRAPPIST-1 e	41.0	17.02	Terrestrial	2017	Transit	0.69	0.92
TRAPPIST-1 h	41.0	17.02	Terrestrial	2017	Transit	0.33	0.76
YZ Ceti b	12.0	12.07	Terrestrial	2017	Radial Velocity	0.70	0.91

193 rows × 7 columns

Let's focus on the 'Radius' column of terr. To learn more about it, we can use the .describe() method.

terr.get('Radius').describe()

count    193.00
mean       0.85
std        0.26
          ...  
50%        0.86
75%        0.92
max        3.13
Name: Radius, Length: 8, dtype: float64

But how do we visualize its distribution?

Visualizing the distribution of 'Radius', a numerical variable

• A few slides ago, we looked at the distribution of 'Type', which is a categorical variable.
• Now, we'll look at the distribution of 'Radius', which is a numerical variable.
• As we'll see, a bar chart is not the right choice of visualization for the distribution of a numerical variable.

To try and see the distribution of 'Radius', we need to group by that column and count how many terrestrial planets there are of each radius.

terr_radius = terr.groupby('Radius').count()
terr_radius = (terr_radius
                 .assign(Count=terr_radius.get('Distance'))
                 .get(['Count'])
              )
terr_radius

	Count
Radius
0.37	1
0.40	1
0.47	1
...	...
1.80	1
2.85	1
3.13	1

85 rows × 1 columns

terr_radius.plot(kind='bar', y='Count', figsize=(15, 5));

The horizontal axis should be numerical (like a number line), not categorical. There should be more space between certain bars than others.

For instance, the planet with 'Radius' 1.8 is 80% larger than the planet with 'Radius' 1, but they appear to be about the same size here.

Density histograms

Density histograms show the distribution of numerical variables

Instead of a bar chart, we'll visualize the distribution of a numerical variable with a density histogram. Let's see what a density histogram for 'Radius' looks like. What do you notice about this visualization?

# Ignore the code for right now.
terr.plot(kind='hist', y='Radius', density=True, bins = np.arange(0, 3.5, 0.25), ec='w');

# There are 7 terrestrial exoplanets with a radius of exactly 1.0,
# but the height of the bar starting at 1.0 is not 7!
terr[terr.get('Radius') == 1]

	Distance	Magnitude	Type	Year	Detection	Mass	Radius
Name
EPIC 201833600 c	840.0	14.71	Terrestrial	2019	Transit	0.97	1.0
EPIC 206215704 b	358.0	17.83	Terrestrial	2019	Transit	0.97	1.0
K2-157 b	973.0	12.94	Terrestrial	2018	Transit	0.97	1.0
K2-239 c	101.0	14.63	Terrestrial	2018	Transit	0.97	1.0
Kepler-1417 b	3235.0	14.04	Terrestrial	2016	Transit	0.97	1.0
Kepler-1464 c	3757.0	14.36	Terrestrial	2016	Transit	0.97	1.0
Kepler-392 b	2223.0	13.53	Terrestrial	2014	Transit	0.97	1.0

First key idea behind histograms: Binning 🗑️

• Binning is the act of counting the number of numerical values that fall within ranges defined by two endpoints. These ranges are called “bins”.
• A value falls in a bin if it is greater than or equal to the left endpoint and less than the right endpoint.
  • [a, b): a is included, b is not.
• The width of a bin is its right endpoint minus its left endpoint.

binning_animation()

Plotting a density histogram

• Density histograms (not bar charts!) visualize the distribution of a single numerical variable by placing numbers into bins.
• To create one from a DataFrame df, use

df.plot(
    kind='hist', 
    y=column_name,
    density=True
)

• Optional but recommended: Use ec='w' to see where bins start and end more clearly.

Customizing the bins

• By default, Python will bin your data into 10 equally sized bins.
• You can specify another number of equally sized bins by setting the optional argument bins equal to some other integer value.
• You can also specify custom bin start and endpoints by setting bins equal to a list or array of bin endpoints.

# There are 10 bins by default, some of which are empty.
terr.plot(kind='hist', y='Radius', density=True, ec='w');

terr.plot(kind='hist', y='Radius', density=True, bins=20, ec='w');

terr.plot(kind='hist', y='Radius', density=True, bins=[0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5], ec='w');

In the three histograms above, what is different and what is the same?

Observations

• The general shape of all three histograms is the same, regardless of the bins.
• More bins gives a finer, more granular picture of the distribution of the variable 'Radius'.
• The $y$-axis values seem to change a lot when we change the bins. Hang onto that thought; we'll see why shortly.

Bin details

• In a histogram, only the last bin is inclusive of the right endpoint!
• The bins you specify don't have to include all data values; data values not in any bin won't be shown in the histogram.
• For equally sized bins, use np.arange.
  • Be very careful with the endpoints.
  • For example, bins=np.arange(4) creates the bins [0, 1), [1, 2), [2, 3].
• Bins can have different sizes!

terr.plot(kind='hist', y='Radius', density=True,
            bins=np.arange(0, 3.5, 0.5),
            ec='w');

terr.sort_values('Radius', ascending=False)

	Distance	Magnitude	Type	Year	Detection	Mass	Radius
Name
Kepler-33 c	3944.0	14.10	Terrestrial	2011	Transit	0.39	3.13
K2-138 f	661.0	12.25	Terrestrial	2017	Transit	1.63	2.85
Kepler-11 b	2108.0	13.82	Terrestrial	2010	Transit	1.90	1.80
...	...	...	...	...	...	...	...
Kepler-102 b	352.0	12.07	Terrestrial	2014	Transit	4.30	0.47
Kepler-444 b	119.0	8.87	Terrestrial	2015	Transit	0.04	0.40
Kepler-37 e	209.0	9.77	Terrestrial	2014	Transit Timing Variations	0.03	0.37

193 rows × 7 columns

In the above example, the terrestrial exoplanet with the largest radius (Kepler-33 c) is not included because the rightmost bin is [2.5, 3.0] and Kepler-33 c has a 'Radius' of 3.13.

terr.plot(kind='hist', y='Radius', density=True,
          bins=[0, 0.25, 0.5, 0.75, 2, 4], ec='w');

In the above example, the bins have different widths!

Second key idea behind histograms: Total area is 1

• In a density histogram, the $y$-axis can be hard to interpret, but it's designed to give the histogram a very nice property:
  
  The bars of a density histogram
  have a combined total area of 1.

• Important: The area of a bar is equal to the proportion of all data points that fall into that bin.

• Recall from the pretest, proportions and percentages represent the same thing.
  • A proportion is a decimal between 0 and 1, a percentage is between 0% and 100%.
  • The proportion 0.34 means 34%.

Example calculation

terr.plot(kind='hist', y='Radius', density=True,
          bins=[0, 0.25, 0.5, 0.75, 2, 4], ec='w');

Based on this histogram, what proportion of terrestrial exoplanets have a 'Radius' between 0.5 and 0.75?

Example calculation

• The height of the [0.5, 0.75) bar looks to be around 0.8.

• The width of the bin is 0.75 - 0.5 = 0.25.

• Therefore, using the formula for the area of a rectangle,

$$\begin{align}\text{Area} &= \text{Height} \times \text{Width} \\ &= 0.8 \times 0.25 \\ &= 0.2 \end{align}$$

• Since areas represent proportions, this means that the proportion of terrestrial exoplanets with a radius between 0.5 and 0.75 is about 0.2 (or 20%).

Check the math 🧮

in_range = terr[(terr.get('Radius') >= 0.5) & (terr.get('Radius') < 0.75)].shape[0]
in_range

39

in_range / terr.shape[0]

0.20207253886010362

This matches the result we got. (Not exactly, since we made an estimate for the height.)

Calculating heights in a density histogram

Since a bar of a histogram is a rectangle, its area is given by

$$\text{Area} = \text{Height} \times \text{Width}$$

That means

$$\text{Height} = \frac{\text{Area}}{\text{Width}} = \frac{\text{Proportion (or Percentage)}}{\text{Width}}$$

This implies that the units for height are "proportion per ($x$-axis unit)". The $y$-axis represents a sort of density, which is why we call it a density histogram.

terr.plot(kind='hist', y='Radius', density=True,
          bins=[0, 0.25, 0.5, 0.75, 2, 4], ec='w');

The $y$-axis units here are "proportion per radius", since the $x$-axis represents radius.

• Unfortunately, the $y$-axis units on the histogram always displays as "Frequency". This is wrong!
• We can fix this with the optional argument ylabel but we usually don't.

Concept Check ✅ – Answer at cc.dsc10.com

Suppose we created a density histogram of people's shoe sizes. 👟 Below are the bins we chose along with their heights.

Bin	Height of Bar
[3, 7)	0.05
[7, 10)	0.1
[10, 12)	0.15
[12, 16]	$X$

What should the value of $X$ be so that this is a valid histogram?

A. 0.02              B. 0.05              C. 0.2              D. 0.5              E. 0.7             

✅ Click here to see an explanation after you've answered.

From the provided bins, we can calculate the bin widths, and then multiply each bin's width by its height to find its area. The bin $[3, 7)$ has a width of $7-3=4$ and a height of $0.05$, so its area is $4*0.05 = 0.2$. Similarly, the bin $[3, 7)$ has an area of $3*0.1 = 0.3$ and the bin $[10, 12)$ has an area of $2*0.15 = 0.3$.

Adding these up, the total area of the first three bins is $0.2+0.3+0.3=0.8$, and since the total area of all bins in a histogram is always $1$, the fourth bin must have an area of $0.2$. This bin has a width of $4$, so its height must be $0.05$ to make its area $0.2$.

Review: Types of visualizations

The type of visualization we create depends on the kinds of variables we're visualizing.

• Scatter plot: Numerical vs. numerical.
  • Example: Weight vs. height.
• Line plot: Sequential numerical (time) vs. numerical.
  • Example: Height vs. time.
• Bar chart: Categorical vs. numerical.
  • Example: Heights of different family members.
• Histogram: Distribution of numerical.

We may interchange the words "plot", "chart", and "graph"; they all mean the same thing.

Bar charts vs. histograms

Bar chart	Histogram
Shows the distribution of a categorical variable	Shows the distribution of a numerical variable
Plotted from 2 columns of a DataFrame	Plotted from 1 column of a DataFrame
1 categorical axis, 1 numerical axis	2 numerical axes
Bars have arbitrary, but equal, widths and spacing	Horizontal axis is numerical and to scale
Lengths of bars are proportional to the numerical quantity of interest	Height measures density; areas are proportional to the proportion (percent) of individuals

🌟 Important 🌟

In this class, "histogram" will always mean a "density histogram". We will only use density histograms.

Note: It's possible to create what's called a frequency histogram where the $y$-axis simply represents a count of the number of values in each bin.

While easier to interpret, frequency histograms don't have the important property that the total area is 1, so they can't be connected to probability in the same way that density histograms can. This property will be useful to us later on in the course.

Overlaid plots

Multiple plots on the same axes

Let's look back at the exo DataFrame.

exo

	Distance	Magnitude	Type	Year	Detection	Mass	Radius
Name
11 Comae Berenices b	304.0	4.72	Gas Giant	2007	Radial Velocity	6165.90	11.88
11 Ursae Minoris b	409.0	5.01	Gas Giant	2009	Radial Velocity	4684.81	11.99
14 Andromedae b	246.0	5.23	Gas Giant	2008	Radial Velocity	1525.58	12.65
...	...	...	...	...	...	...	...
YZ Ceti b	12.0	12.07	Terrestrial	2017	Radial Velocity	0.70	0.91
YZ Ceti c	12.0	12.07	Super Earth	2017	Radial Velocity	1.14	1.05
YZ Ceti d	12.0	12.07	Super Earth	2017	Radial Velocity	1.09	1.03

5043 rows × 7 columns

Can we look at both the average 'Magnitude' and the average 'Radius' for each 'Type' at the same time?

types = exo.groupby('Type').mean()
types

	Distance	Magnitude	Year	Mass	Radius
Type
Gas Giant	1096.40	10.30	2013.73	1472.39	12.74
Neptune-like	2189.02	13.52	2016.59	15.28	3.11
Super Earth	1916.26	13.85	2016.43	5.81	1.58
Terrestrial	1373.60	13.45	2016.37	1.62	0.85

types.get(['Magnitude', 'Radius']).plot(kind='barh');

How did we do that?

Overlaying plots

When calling .plot, if we omit the y=column_name argument, all other columns are plotted.

types

	Distance	Magnitude	Year	Mass	Radius
Type
Gas Giant	1096.40	10.30	2013.73	1472.39	12.74
Neptune-like	2189.02	13.52	2016.59	15.28	3.11
Super Earth	1916.26	13.85	2016.43	5.81	1.58
Terrestrial	1373.60	13.45	2016.37	1.62	0.85

types.plot(kind='barh');

Selecting multiple columns at once

• To select multiple columns, use .get([column_1, ..., column_k]).
• Passing a list of column labels to .get returns a DataFrame.
  • .get([column_name]) will return a DataFrame with just one column!

types

	Distance	Magnitude	Year	Mass	Radius
Type
Gas Giant	1096.40	10.30	2013.73	1472.39	12.74
Neptune-like	2189.02	13.52	2016.59	15.28	3.11
Super Earth	1916.26	13.85	2016.43	5.81	1.58
Terrestrial	1373.60	13.45	2016.37	1.62	0.85

types.get(['Magnitude', 'Radius'])

	Magnitude	Radius
Type
Gas Giant	10.30	12.74
Neptune-like	13.52	3.11
Super Earth	13.85	1.58
Terrestrial	13.45	0.85

types.get(['Magnitude', 'Radius']).plot(kind='barh');

Plotting multiple graphs at once

Recipe:

1. .get only the columns that contain information relevant to your plot (or, equivalently, .drop all extraneous columns).
2. Specify the column for the $x$-axis (if not the index) in .plot(x=column_name).
3. Omit the y argument. Then all other columns will be plotted on a shared $y$-axis.

The same thing works for 'barh', 'bar', and 'hist', but not 'scatter'.

Another example: Heights of children and their parents 👪 📏

• The data below was collected in the late 1800s by Francis Galton.
  • He was a eugenicist and proponent of scientific racism, which is why he collected this data.
  • Today, we understand that eugenics is immoral, and that there is no scientific evidence or any other justification for racism.
• We will revisit this dataset later on in the course.
• For now, we'll only need the 'mother', and 'childHeight' columns.

mother_child = bpd.read_csv('data/galton.csv').get(['mother', 'child'])
mother_child

	mother	child
0	67.0	73.2
1	67.0	69.2
2	67.0	69.0
...	...	...
931	66.0	61.0
932	63.0	66.5
933	63.0	57.0

934 rows × 2 columns

Plotting overlaid histograms

alpha controls how transparent the bars are (alpha=1 is opaque, alpha=0 is transparent).

height_bins = np.arange(55, 80, 2.5)
mother_child.plot(kind='hist', density=True, ec='w',
                  alpha=0.65, bins=height_bins);

Why do children seem so much taller than their mothers?

Extra Practice

Try to answer these questions based on the overlaid histogram.

1. What proportion of children were between 70 and 75 inches tall?

2. What proportion of mothers were between 60 and 63 inches tall?

✅ Click here to see the answers to the problems above after you've tried them on your own.

Question 1

The height of the $[70, 72.5)$ bar is around $0.08$, meaning that $0.08 \cdot 2.5 = 0.2$ of children had heights in that interval. The height of the $[72.5, 75)$ bar is around $0.02$, meaning $0.02 \cdot 2.5 = 0.05$ of children had heights in that interval. Thus, the overall proportion of children who were between $70$ and $75$ inches tall was around $0.20 + 0.05 = 0.25$, or $25\%$.

To verify our answer, we can run

mother_child[(mother_child.get('child') >= 70) & (mother_child.get('child') < 75)].shape[0] / mother_child.shape[0]

Question 2

We can't tell. We could try and breaking it up into the proportion of mothers in $[60, 62.5)$ and $[62.5, 63)$, but we don't know the latter. In the absence of any additional information, we can't infer about the distribution of values within a bin. For example, it could be that everyone in the interval $[62.5, 65)$ actually falls in the interval $[62.5, 63)$ - or it could be that no one does!

Summary, next time

Summary

• Histograms (not bar charts!) are used to display the distribution of a numerical variable.
• We will always use density histograms in this course.
  • In a density histogram, the area of a bar represents the proportion (percentage) of values within its bin.
  • The total area of all bars is 1 (100%).
• We can overlay multiple line plots, bar charts, and histograms on top of one another to look at multiple relationships or distributions.

Next time

• Writing our own functions.
• Applying functions to the data in a DataFrame.