Agenda

Learn to construct and use histograms to examine the underlying distribution of a continuous variable. Specifically

create a bare bones histogram
specify the number of bins/intervals
represent frequency density on the Y axis
add colors to the bars and the border
add labels to the bars

Introduction

A histogram is a plot that can be used to examine the shape and spread of continuous data. It looks very similar to a bar graph and can be used to detect outliers and skewness in data. The histogram graphically shows the following:

center (location) of the data
spread (dispersion) of the data
skewness
outliers
presence of multiple modes

Histograms

To construct a histogram

the data is split into intervals called bins
the intervals may or may not be equal sized
for each bin, the number of data points that fall into it are counted (frequency)
the Y axis of the histogram represents the frequency and
the X axis represents the variable

Histogram

h <- hist(mtcars$mpg)

Histogram

# display number of breaks
h$breaks

## [1] 10 15 20 25 30 35

# frequency of the intervals
h$counts

## [1]  6 12  8  2  4

# frequency density
h$density

## [1] 0.0375 0.0750 0.0500 0.0125 0.0250

# mid points of the intervals
h$mids

## [1] 12.5 17.5 22.5 27.5 32.5

# varible name
h$xname

## [1] "mtcars$mpg"

# whether intervals are of equal size
h$equidist

## [1] TRUE

Bins

hist(mtcars$mpg, breaks = 10)

Bins

Intervals

h <- hist(mtcars$mpg, breaks = c(10, 18, 24, 30, 35))

Frequency Density

frequency <- h$counts
class_width <- c(8, 6, 6, 5)
rel_freq <- frequency / length(mtcars$mpg)
freq_density <- rel_freq / class_width
d <- data.frame(frequency = frequency, class_width = class_width, 
  relative_frequency = rel_freq, frequency_density = freq_density)
d

##   frequency class_width relative_frequency frequency_density
## 1        13           8            0.40625        0.05078125
## 2        12           6            0.37500        0.06250000
## 3         3           6            0.09375        0.01562500
## 4         4           5            0.12500        0.02500000

Frequency Density

When multiplied by the class width, the product will always sum upto 1.

sum(d$frequency_density * d$class_width)

## [1] 1

Intervals

We will learn more about frequency density in a bit. Before we end this section, we need to learn about one more way to specify the intervals of the histogram, algorithms. The hist() function allows us to specify the following algorithms:

Sturges (default)
Scott
Freedman-Diaconis (FD)

Intervals

Frequency Distribution II

Probability

hist(mtcars$mpg, probability = TRUE)

Color

hist(mtcars$mpg, col = 'blue')

Border Color

hist(mtcars$mpg, border = c('red', 'blue', 'green', 'yellow', 'brown'))

Labels

hist(mtcars$mpg, labels = TRUE)

Labels

hist(mtcars$mpg, labels = c("6", "12", "8", "2", "4"))

Title & Axis Labels

hist(mtcars$mpg, labels = TRUE, prob = TRUE,
     ylim = c(0, 0.1), xlab = 'Miles Per Gallon',
     main = 'Distribution of Miles Per Gallon',
     col = rainbow(5))

Normal Distribution

Skewed Distribution

Case Study

Data

returns <- read_csv('https://raw.githubusercontent.com/rsquaredacademy/datasets/master/tickers.csv')

## # A tibble: 250 x 5
##         AAPL       AMZN        FB       GOOG      MSFT
##        <dbl>      <dbl>     <dbl>      <dbl>     <dbl>
##  1  1.377845  24.169983  2.119995  22.409973  1.120701
##  2  2.834412   3.250000 -0.860001   5.989990  0.766800
##  3 -0.039360   9.910034  1.450005   6.750000  0.973240
##  4  0.108261   3.759949 -0.770004 -10.690002 -0.285091
##  5  1.643570  19.840027  4.750000   8.660034  0.501365
##  6  0.068894   5.330017 -0.299996  -0.929992  0.255596
##  7 -0.560975  -5.210022 -0.630005  -7.280030 -0.707809
##  8  0.551140   0.250000 -0.459999   0.690003  0.127796
##  9 -0.216522 -13.599975  0.030007   6.559997  0.078648
## 10 -0.108253  -4.250000  0.459999   2.600037  0.471878
## # ... with 240 more rows

Daily Returns Summary: Apple

summary(returns$AAPL)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -5.9650 -0.4937  0.1578  0.2573  0.9687  7.2829

Daily Returns: Apple

hist(returns$AAPL)

Daily Returns Summary: Google

summary(returns$GOOG)

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -33.5800  -3.1826   1.3950   0.9718   5.8475  31.7100

Daily Returns: Google

hist(returns$GOOG)

Daily Returns Summary: Facebook

summary(returns$FB)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -7.6700 -0.5675  0.1300  0.2258  1.1800  4.8300

Daily Returns: Facebook

hist(returns$FB)

Daily Returns Summary: Microsoft

summary(returns$MSFT)

##     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
## -1.91978 -0.22750  0.04423  0.08851  0.43451  1.65121

Daily Returns: Microsoft

hist(returns$MSFT)

Daily Returns Summary: Amazon

summary(returns$AMZN)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -32.260  -4.230   0.950   1.126   8.040  24.170

Daily Returns: Amazon

hist(returns$AMZN)