Lecture 8

Histograms

DATA 8

Spring 2022

Announcements

• Lab 3 is due today at 5pm PT
• HW3 due Thursday, 02/10
• Turn in on Wednesday for bonus points
• Lab 4 will be released on Monday
• Read this article about causality!

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Weekly Goals

• Monday
• Table review
• Working with Census data
• Wednesday
• Visualizing data
• Line plots, scatter plots, bar charts
• Today
• Visualizing two kinds of distributions
• Proportions as areas

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Distributions

Terminology

• Individuals: those whose attributes are recorded
• Variable: an attribute (column)
• can be numerical or categorical
• has different values
• each individual has exactly one value
• has a distribution:
• For each different value of the variable, the frequency of individuals that have that value

A Distribution

Source: Pew Research

Each individual is in exactly one category. Percents add up to 100.

Not a Distribution

Percents of survey respondents on “a major reason they would find it difficult to quarantine themselves for at least 14 days”

Source: Pew Research

Each respondent can pick more than one answer.

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The bars represent overlapping groups.

Categorical Distributions

(Demo)

Bar Chart

To display all the values of the variable along with all their frequencies

• Bar chart
• One bar for each category
• You can choose the order of the bars
• Length of bar is the percent (or count) of individuals in that category

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(Demo)

Numerical Distributions

Grouping Numerical Values: Binning

Binning is counting the number of numerical values that lie within ranges, called bins.

• Bins are defined by their lower bounds (inclusive)
• The upper bound is the lower bound of the next bin

188, 170, 189, 163, 183, 171, 185, 168, 173, ...

The [185,190) bin

(Demo)

Area Principle

What Is Wrong With This Picture?

Source

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Caption: The new iPad battery is 70% bigger than the previous iPad.

Area Principle

Areas should be proportional to the values they represent.

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For example

• If you represent 20% of a population by

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• Then 40% can be represented by:

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• But not by:

Drawing Histograms

Histogram

(Demo)

• Displays the distribution of a numerical variable
• One bar corresponding to each bin
• Uses the area principle:
• The area of each bar is the percent of individuals in the corresponding bin

• Later in the course, we will approximate histograms by smooth curves.

Areas will still represent percents.

68%

Density

Histogram Axes

• By default, hist uses a scale (normed=True) that ensures the area of the chart sums to 100%

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• The area of each bar is a percentage of the whole

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• The horizontal axis is a number line (e.g. years), and the bins sizes don’t have to be equal to each other

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• The vertical axis is a rate (e.g., percent per year)

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(Demo)

How to Calculate Height

The [40, 65) bin contains 56 out of 200 movies

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• “56 out of 200” is 28%
• The bin is 65 - 40 = 25 years wide

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28 percent

Height of bar = --------------------------

25 years

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= 1.12 percent per year

Height Measures Density

% in bin

Height = ---------------------

width of bin

• The height measures the percent of data in the bin relative to the amount of space in the bin.

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• Height measures crowdedness, or density.

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• Units: percent per unit on the horizontal axis

Area Measures Percent

Area of bar = % in bin = Height x width of bin

• “How many individuals in the bin?” Use area.

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• “How crowded is the bin?” Use height.

Discussion Questions

Compare the bins [10, 20) and [20, 40).

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• Which one has more movies?

Answer: [20, 40), bigger area

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• Which one is more crowded?

Answer: [10, 20), taller

Bar Chart or Histogram?

Bar Chart

• Distribution of categorical variable
• Bars have arbitrary �(but equal) widths and spacings; in any order
• height (or length) and area of bars proportional to the percent of individuals

Histogram

• Distribution of numerical variable
• Horizontal axis is numerical: drawn to scale, no gaps, bins can be unequal
• Area of bars proportional to the percent of individuals; height measures density

To display a distribution:

Discussion Questions

What is the height of each bar in this �histogram?

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my_bins = make_array(0, 15, 25, 85)

incomes.hist(1, bins = my_bins)

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What are the vertical axis units?

Answers

Vertical axis units: Percent per million

my_bins = make_array(0,15,25,85)

[0, 15): (45%)/(15 million)

= 3 % per million

[15, 25): (40%)/(10 million)

= 4 % per million

[25, 85): (15%)/(60 million)

= 0.25 % per million

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