Some define statistics as the field that focuses on turning information into knowledge. The first step in that process is to summarize and describe the raw information – the data. In this lab we explore flights, specifically a random sample of domestic flights that departed from the three major New York City airports in 2013. We will generate simple graphical and numerical summaries of data on these flights and explore delay times. Since this is a large data set, along the way you’ll also learn the indispensable skills of data processing and subsetting.

Getting started

Load packages

In this lab, we will explore and visualize the data using the tidyverse suite of packages. The data can be found in the companion package for OpenIntro labs, openintro.

Let’s load the packages.

library(tidyverse)
library(openintro)

The data

The Bureau of Transportation Statistics (BTS) is a statistical agency that is a part of the Research and Innovative Technology Administration (RITA). As its name implies, BTS collects and makes transportation data available, such as the flights data we will be working with in this lab.

First, we’ll view the nycflights data frame. Type the following in your console to load the data:

data(nycflights)

The data set nycflights that shows up in your workspace is a data matrix, with each row representing an observation and each column representing a variable. R calls this data format a data frame, which is a term that will be used throughout the labs. For this data set, each observation is a single flight.

To view the names of the variables, type the command

names(nycflights)
##  [1] "year"      "month"     "day"       "dep_time"  "dep_delay" "arr_time" 
##  [7] "arr_delay" "carrier"   "tailnum"   "flight"    "origin"    "dest"     
## [13] "air_time"  "distance"  "hour"      "minute"

This returns the names of the variables in this data frame. The codebook (description of the variables) can be accessed by pulling up the help file:

?nycflights

One of the variables refers to the carrier (i.e. airline) of the flight, which is coded according to the following system.

  • carrier: Two letter carrier abbreviation.
    • 9E: Endeavor Air Inc.
    • AA: American Airlines Inc.
    • AS: Alaska Airlines Inc.
    • B6: JetBlue Airways
    • DL: Delta Air Lines Inc.
    • EV: ExpressJet Airlines Inc.
    • F9: Frontier Airlines Inc.
    • FL: AirTran Airways Corporation
    • HA: Hawaiian Airlines Inc.
    • MQ: Envoy Air
    • OO: SkyWest Airlines Inc.
    • UA: United Air Lines Inc.
    • US: US Airways Inc.
    • VX: Virgin America
    • WN: Southwest Airlines Co.
    • YV: Mesa Airlines Inc.

Remember that you can use glimpse to take a quick peek at your data to understand its contents better.

glimpse(nycflights)
## Rows: 32,735
## Columns: 16
## $ year      <int> 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, …
## $ month     <int> 6, 5, 12, 5, 7, 1, 12, 8, 9, 4, 6, 11, 4, 3, 10, 1, 2, 8, 10…
## $ day       <int> 30, 7, 8, 14, 21, 1, 9, 13, 26, 30, 17, 22, 26, 25, 21, 23, …
## $ dep_time  <int> 940, 1657, 859, 1841, 1102, 1817, 1259, 1920, 725, 1323, 940…
## $ dep_delay <dbl> 15, -3, -1, -4, -3, -3, 14, 85, -10, 62, 5, 5, -2, 115, -4, …
## $ arr_time  <int> 1216, 2104, 1238, 2122, 1230, 2008, 1617, 2032, 1027, 1549, …
## $ arr_delay <dbl> -4, 10, 11, -34, -8, 3, 22, 71, -8, 60, -4, -2, 22, 91, -6, …
## $ carrier   <chr> "VX", "DL", "DL", "DL", "9E", "AA", "WN", "B6", "AA", "EV", …
## $ tailnum   <chr> "N626VA", "N3760C", "N712TW", "N914DL", "N823AY", "N3AXAA", …
## $ flight    <int> 407, 329, 422, 2391, 3652, 353, 1428, 1407, 2279, 4162, 20, …
## $ origin    <chr> "JFK", "JFK", "JFK", "JFK", "LGA", "LGA", "EWR", "JFK", "LGA…
## $ dest      <chr> "LAX", "SJU", "LAX", "TPA", "ORF", "ORD", "HOU", "IAD", "MIA…
## $ air_time  <dbl> 313, 216, 376, 135, 50, 138, 240, 48, 148, 110, 50, 161, 87,…
## $ distance  <dbl> 2475, 1598, 2475, 1005, 296, 733, 1411, 228, 1096, 820, 264,…
## $ hour      <dbl> 9, 16, 8, 18, 11, 18, 12, 19, 7, 13, 9, 13, 8, 20, 12, 20, 6…
## $ minute    <dbl> 40, 57, 59, 41, 2, 17, 59, 20, 25, 23, 40, 20, 9, 54, 17, 24…

The nycflights data frame is a massive trove of information. Let’s think about some questions we might want to answer with these data:

  • How delayed were flights that were headed to Los Angeles?
  • How do departure delays vary by month?
  • Which of the three major NYC airports has the best on time percentage for departing flights?

Analysis

Departure delays

Let’s start by examing the distribution of departure delays of all flights with a histogram.

ggplot(data = nycflights, aes(x = dep_delay)) +
  geom_histogram()

This function says to plot the dep_delay variable from the nycflights data frame on the x-axis. It also defines a geom (short for geometric object), which describes the type of plot you will produce.

Histograms are generally a very good way to see the shape of a single distribution of numerical data, but that shape can change depending on how the data is split between the different bins. You can easily define the binwidth you want to use:

ggplot(data = nycflights, aes(x = dep_delay)) +
  geom_histogram(binwidth = 15)

ggplot(data = nycflights, aes(x = dep_delay)) +
  geom_histogram(binwidth = 150)

  1. Look carefully at these three histograms. How do they compare? Are features revealed in one that are obscured in another?

Insert your answer here

Farhod’s answer:

Each of these histograms is unimodal, has different count modes affected by different bin sizes. In first histogram the mode is ~26500, second histogram has the mode ~20500, third histogram’s mode is ~30000. Also these histograms are showing outliers (very long departure delays) on x-axis.

The first histogram shows that distribution skewed to the right and I can say that mean is larger than median of distribution

The second histogram has more bins and gives better understanding that distribution is skewed to the right.

I wouldn’t use third histogram for analysis, because all values are stacked in just three bins.

I thought about to apply log transformation to get a better histogram, but dep_delay variable has some negative values.

End of the answer.

If you want to visualize only on delays of flights headed to Los Angeles, you need to first filter the data for flights with that destination (dest == "LAX") and then make a histogram of the departure delays of only those flights.

lax_flights <- nycflights %>%
  filter(dest == "LAX")
ggplot(data = lax_flights, aes(x = dep_delay)) +
  geom_histogram()

Let’s decipher these two commands (OK, so it might look like four lines, but the first two physical lines of code are actually part of the same command. It’s common to add a break to a new line after %>% to help readability).

  • Command 1: Take the nycflights data frame, filter for flights headed to LAX, and save the result as a new data frame called lax_flights.
    • == means “if it’s equal to”.
    • LAX is in quotation marks since it is a character string.
  • Command 2: Basically the same ggplot call from earlier for making a histogram, except that it uses the smaller data frame for flights headed to LAX instead of all flights.

Logical operators: Filtering for certain observations (e.g. flights from a particular airport) is often of interest in data frames where we might want to examine observations with certain characteristics separately from the rest of the data. To do so, you can use the filter function and a series of logical operators. The most commonly used logical operators for data analysis are as follows:

  • == means “equal to”
  • != means “not equal to”
  • > or < means “greater than” or “less than”
  • >= or <= means “greater than or equal to” or “less than or equal to”

You can also obtain numerical summaries for these flights:

lax_flights %>%
  summarise(mean_dd   = mean(dep_delay), 
            median_dd = median(dep_delay), 
            n         = n())
## # A tibble: 1 × 3
##   mean_dd median_dd     n
##     <dbl>     <dbl> <int>
## 1    9.78        -1  1583

Note that in the summarise function you created a list of three different numerical summaries that you were interested in. The names of these elements are user defined, like mean_dd, median_dd, n, and you can customize these names as you like (just don’t use spaces in your names). Calculating these summary statistics also requires that you know the function calls. Note that n() reports the sample size.

Summary statistics: Some useful function calls for summary statistics for a single numerical variable are as follows:

  • mean
  • median
  • sd
  • var
  • IQR
  • min
  • max

Note that each of these functions takes a single vector as an argument and returns a single value.

You can also filter based on multiple criteria. Suppose you are interested in flights headed to San Francisco (SFO) in February:

sfo_feb_flights <- nycflights %>%
  filter(dest == "SFO", month == 2)

Note that you can separate the conditions using commas if you want flights that are both headed to SFO and in February. If you are interested in either flights headed to SFO or in February, you can use the | instead of the comma.

  1. Create a new data frame that includes flights headed to SFO in February, and save this data frame as sfo_feb_flights. How many flights meet these criteria?

Insert your answer here

Farhod’s answer:

sfo_feb_flights <- nycflights |>
  filter(dest == "SFO", month == 2 )

sfo_feb_flights_n <- sfo_feb_flights |>
  summarise(n = n())

print(paste("Total flights to SFO in February:", sfo_feb_flights_n))
## [1] "Total flights to SFO in February: 68"

The code above filters all flights to SFO in February into sfo_feb_flights dataframe. After that, it filters summary of observations (n()) into sfo_feb_flights_n dataframe.Because each observation is for a single flight to SFO in February, total count of observations is equal total flights to SFO (68).

End of the answer.

  1. Describe the distribution of the arrival delays of these flights using a histogram and appropriate summary statistics. Hint: The summary statistics you use should depend on the shape of the distribution.

Insert your answer here

Farhod’s answer:

Let’s build a histogram of arrival delays:

ggplot(sfo_feb_flights, aes(x = arr_delay)) +
  geom_histogram(binwidth = 7, fill = "blue", color = "black")+
  geom_vline(xintercept = median(sfo_feb_flights$arr_delay),
             color = "red",
             linetype = "dashed",
             size = 1) +
  labs(title = "Histogram of SFO arrival delays with median line")

This histogram shows that the distribution is skewed to the right and there are few outliers on the right. Because of that I included the median line in histogram. We also can say if distribution is skewed looking at summary statistic of the distribution:

summary(sfo_feb_flights$arr_delay)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  -66.00  -21.25  -11.00   -4.50    2.00  196.00

Because the mean (-4.50) is larger than median (-11), the arirval delays distribution is skewed to the right.

End of answer.

Another useful technique is quickly calculating summary statistics for various groups in your data frame. For example, we can modify the above command using the group_by function to get the same summary stats for each origin airport:

sfo_feb_flights %>%
  group_by(origin) %>%
  summarise(median_dd = median(dep_delay), iqr_dd = IQR(dep_delay), n_flights = n())
## # A tibble: 2 × 4
##   origin median_dd iqr_dd n_flights
##   <chr>      <dbl>  <dbl>     <int>
## 1 EWR          0.5   5.75         8
## 2 JFK         -2.5  15.2         60

Here, we first grouped the data by origin and then calculated the summary statistics.

  1. Calculate the median and interquartile range for arr_delays of flights in in the sfo_feb_flights data frame, grouped by carrier. Which carrier has the most variable arrival delays?

Insert your answer here

Farhod’s answer:

This code returns the highest five medians and IQRs of arrival delays in SFO in February for each airline.

IQR_sfo_feb_flights <- sfo_feb_flights |>
  group_by(carrier) |>
  summarise(median_arr_delay = median(arr_delay, na.rm = TRUE),
            IQR_arr_delay = IQR(arr_delay, na.rm = TRUE),
            ) |>
  arrange(desc(IQR_arr_delay)) |>
  slice(1:5)
print(IQR_sfo_feb_flights)
## # A tibble: 5 × 3
##   carrier median_arr_delay IQR_arr_delay
##   <chr>              <dbl>         <dbl>
## 1 DL                 -15            22  
## 2 UA                 -10            22  
## 3 VX                 -22.5          21.2
## 4 AA                   5            17.5
## 5 B6                 -10.5          12.2

The interquartile range (IQR) is the measure for the middle 50% of the data and IQR’s length indicates how data values are spread out. Larger IQR’s have a wider spread of the data, or more variable. From the result of the code above, we can see that Delta Airlines (DL) has the largest IQR (22) in February, meaning it has the most variable arrival delays.

End of answer.

Departure delays by month

Which month would you expect to have the highest average delay departing from an NYC airport?

Let’s think about how you could answer this question:

  • First, calculate monthly averages for departure delays. With the new language you are learning, you could
    • group_by months, then
    • summarise mean departure delays.
  • Then, you could to arrange these average delays in descending order
nycflights %>%
  group_by(month) %>%
  summarise(mean_dd = mean(dep_delay)) %>%
  arrange(desc(mean_dd))
## # A tibble: 12 × 2
##    month mean_dd
##    <int>   <dbl>
##  1     7   20.8 
##  2     6   20.4 
##  3    12   17.4 
##  4     4   14.6 
##  5     3   13.5 
##  6     5   13.3 
##  7     8   12.6 
##  8     2   10.7 
##  9     1   10.2 
## 10     9    6.87
## 11    11    6.10
## 12    10    5.88
  1. Suppose you really dislike departure delays and you want to schedule your travel in a month that minimizes your potential departure delay leaving NYC. One option is to choose the month with the lowest mean departure delay. Another option is to choose the month with the lowest median departure delay. What are the pros and cons of these two choices?

Insert your answer here

Farhod’s answer:

Let’s calculate means and medians of departure delays for each month:

nycflights |>
  group_by(month) |>
  summarise(
    median_dd = median(dep_delay),
    mean_dd = mean(dep_delay),
    difference_m_mu = abs(median_dd - mean_dd))|>
  arrange(desc(median_dd))
## # A tibble: 12 × 4
##    month median_dd mean_dd difference_m_mu
##    <int>     <dbl>   <dbl>           <dbl>
##  1    12         1   17.4            16.4 
##  2     6         0   20.4            20.4 
##  3     7         0   20.8            20.8 
##  4     3        -1   13.5            14.5 
##  5     5        -1   13.3            14.3 
##  6     8        -1   12.6            13.6 
##  7     1        -2   10.2            12.2 
##  8     2        -2   10.7            12.7 
##  9     4        -2   14.6            16.6 
## 10    11        -2    6.10            8.10
## 11     9        -3    6.87            9.87
## 12    10        -3    5.88            8.88

Mean of departure delays is calculated by adding all departure delays of the month and divide by the number of delays.

  • Pros: Mean provides with a general idea of the overall delays in a given month .

  • Cons: The problem with the mean is that if there are even a few very long departure delays (outliers). they can make the mean much higher, even if the most flights were slightly delayed or on time.

Median of departure delays is the middle value of the ascending (from lowest to highest) line of departure delay values.

  • Pros: Median is usually considered as “typical” value of variable and is robust to outliers (few very long delays).

  • Cons: Since median are usually not affected much by outliers, it usually doesn’t tell about possibility of very long delays (outliers).

If someone doesn’t like departure delays and wants to schedule to travel in a month that minimizes his potential departure delay, the lowest median is likely a better choice. Because a very few extreme or unusual delays can increase the mean, even if most of the flights were departed close to the scheduled departure times.

The best choice would be to look for the months with lowest median. After that, choose the month with the smallest difference between median and mean in those months, since the smallest difference indicates that there is very small amount of very long delays (outliers) in that month.

The code above calculates difference between median and mean (difference_m_mu).We can see that November (month = 11) has the lowest difference (8.10), and that means that flying in November has the lowest chance to experience very long delays.

End of the answer.

On time departure rate for NYC airports

Suppose you will be flying out of NYC and want to know which of the three major NYC airports has the best on time departure rate of departing flights. Also supposed that for you, a flight that is delayed for less than 5 minutes is basically “on time.”” You consider any flight delayed for 5 minutes of more to be “delayed”.

In order to determine which airport has the best on time departure rate, you can

  • first classify each flight as “on time” or “delayed”,
  • then group flights by origin airport,
  • then calculate on time departure rates for each origin airport,
  • and finally arrange the airports in descending order for on time departure percentage.

Let’s start with classifying each flight as “on time” or “delayed” by creating a new variable with the mutate function.

nycflights <- nycflights %>%
  mutate(dep_type = ifelse(dep_delay < 5, "on time", "delayed"))

The first argument in the mutate function is the name of the new variable we want to create, in this case dep_type. Then if dep_delay < 5, we classify the flight as "on time" and "delayed" if not, i.e. if the flight is delayed for 5 or more minutes.

Note that we are also overwriting the nycflights data frame with the new version of this data frame that includes the new dep_type variable.

We can handle all of the remaining steps in one code chunk:

nycflights %>%
  group_by(origin) %>%
  summarise(ot_dep_rate = sum(dep_type == "on time") / n()) %>%
  arrange(desc(ot_dep_rate))
## # A tibble: 3 × 2
##   origin ot_dep_rate
##   <chr>        <dbl>
## 1 LGA          0.728
## 2 JFK          0.694
## 3 EWR          0.637
  1. If you were selecting an airport simply based on on time departure percentage, which NYC airport would you choose to fly out of?

You can also visualize the distribution of on on time departure rate across the three airports using a segmented bar plot.

ggplot(data = nycflights, aes(x = origin, fill = dep_type)) +
  geom_bar()

Insert your answer here

Farhod’s answer:

I would choose LGA airport to fly out because LGA has highest percentage of on-time departures (72.8%).

End of answer.

More Practice

  1. Mutate the data frame so that it includes a new variable that contains the average speed, avg_speed traveled by the plane for each flight (in mph). Hint: Average speed can be calculated as distance divided by number of hours of travel, and note that air_time is given in minutes.

Insert your answer here

Farhod’s answer:

nycflights <- nycflights |>
  mutate(avg_speed = distance / (air_time / 60))
glimpse(nycflights)
## Rows: 32,735
## Columns: 18
## $ year      <int> 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, …
## $ month     <int> 6, 5, 12, 5, 7, 1, 12, 8, 9, 4, 6, 11, 4, 3, 10, 1, 2, 8, 10…
## $ day       <int> 30, 7, 8, 14, 21, 1, 9, 13, 26, 30, 17, 22, 26, 25, 21, 23, …
## $ dep_time  <int> 940, 1657, 859, 1841, 1102, 1817, 1259, 1920, 725, 1323, 940…
## $ dep_delay <dbl> 15, -3, -1, -4, -3, -3, 14, 85, -10, 62, 5, 5, -2, 115, -4, …
## $ arr_time  <int> 1216, 2104, 1238, 2122, 1230, 2008, 1617, 2032, 1027, 1549, …
## $ arr_delay <dbl> -4, 10, 11, -34, -8, 3, 22, 71, -8, 60, -4, -2, 22, 91, -6, …
## $ carrier   <chr> "VX", "DL", "DL", "DL", "9E", "AA", "WN", "B6", "AA", "EV", …
## $ tailnum   <chr> "N626VA", "N3760C", "N712TW", "N914DL", "N823AY", "N3AXAA", …
## $ flight    <int> 407, 329, 422, 2391, 3652, 353, 1428, 1407, 2279, 4162, 20, …
## $ origin    <chr> "JFK", "JFK", "JFK", "JFK", "LGA", "LGA", "EWR", "JFK", "LGA…
## $ dest      <chr> "LAX", "SJU", "LAX", "TPA", "ORF", "ORD", "HOU", "IAD", "MIA…
## $ air_time  <dbl> 313, 216, 376, 135, 50, 138, 240, 48, 148, 110, 50, 161, 87,…
## $ distance  <dbl> 2475, 1598, 2475, 1005, 296, 733, 1411, 228, 1096, 820, 264,…
## $ hour      <dbl> 9, 16, 8, 18, 11, 18, 12, 19, 7, 13, 9, 13, 8, 20, 12, 20, 6…
## $ minute    <dbl> 40, 57, 59, 41, 2, 17, 59, 20, 25, 23, 40, 20, 9, 54, 17, 24…
## $ dep_type  <chr> "delayed", "on time", "on time", "on time", "on time", "on t…
## $ avg_speed <dbl> 474.4409, 443.8889, 394.9468, 446.6667, 355.2000, 318.6957, …

The code above creates new variable column avg_speed, and its values for each flight calculated by the avg_speed = distance / (air_time / 60)

  1. Make a scatterplot of avg_speed vs. distance. Describe the relationship between average speed and distance. Hint: Use geom_point().

Insert your answer here

Farhod’s answer:

ggplot(nycflights, aes(x = avg_speed, y = distance)) +
  geom_point() +
  labs(
    title = "Flight Speed vs. Distance",
    x = "Average Speed (mph)", 
    y = "Distance (miles)"
    ) +
  theme_bw()

In this scatterplot we can see that longer distance flights has higher average speeds.

It’s interesting to see why shorter distance flights has lower speeds.

After some research, I found out that lower average speed of shorter flights is primarily due to the proportionally larger amount of time spent in lower-speed phases of flight (takeoff, landing, climb, descent), potentially lower cruising altitudes, the type of aircraft used, and the direct mathematical relationship between distance and average speed.

End of the answer.

  1. Replicate the following plot. Hint: The data frame plotted only contains flights from American Airlines, Delta Airlines, and United Airlines, and the points are colored by carrier. Once you replicate the plot, determine (roughly) what the cutoff point is for departure delays where you can still expect to get to your destination on time.

Insert your answer here

The code replicates the plot above. Once I analyzed the scatterplot below, I determined that cutoff point for departure delay where you can still expect to get to your destination on time is roughly 65 minutes. After that point arrival delay start to become positive

selected_carriers <- nycflights |>
  filter(carrier %in% c("AA", "DL", "UA"))

ggplot(selected_carriers, aes(x = dep_delay, y = arr_delay, color = carrier)) +
  geom_point()