5: Plots

In the scatterplot above each green point is for one car in the dataset. See if you can locate the car with the highest mpg in the dataset, close to 50, which weighs a little above 2000 lbs.

The scatterplot shows a general pattern: the heavier the car is, the lower its fuel efficiency. This is just as we suspected, but the scatterplot makes it easy to see the relationship in the data.

When one feature generally increases or decreases with another feature, we say that there's an association between the two features. In this case, there is a negative association because as one feature increases, the other feature decreases. If both features increase together or decrease together, then it is a positive association.

We also note that the negative association in the plot above is a general trend and doesn't apply to all cars. For example, looking at the green dots above, we can find one car that weighs around 3000 lbs and is around 38 mpg, which is much higher than the 18 mpg of another car that weighs only around 2000 lbs.

The line plot shows that there was a big increase in the mpg between 1976 and 1980, and the overall trend is that fuel efficiency did indeed increase over time.

Example of quantitative data: mpg
Numbers of unique mpg values: [18.  15.  16.  17.  14.  24.  22.  21.  27.  26.  25.  10.  11.   9.
 28.  19.  12.  13.  23.  30.  31.  35.  20.  29.  32.  33.  17.5 15.5
 14.5 22.5 24.5 18.5 29.5 26.5 16.5 31.5 36.  25.5 33.5 20.5 30.5 21.5
 43.1 36.1 32.8 39.4 19.9 19.4 20.2 19.2 25.1 20.6 20.8 18.6 18.1 17.7
 27.5 27.2 30.9 21.1 23.2 23.8 23.9 20.3 21.6 16.2 19.8 22.3 17.6 18.2
 16.9 31.9 34.1 35.7 27.4 25.4 34.2 34.5 31.8 37.3 28.4 28.8 26.8 41.5
 38.1 32.1 37.2 26.4 24.3 19.1 34.3 29.8 31.3 37.  32.2 46.6 27.9 40.8
 44.3 43.4 36.4 44.6 33.8 32.7 23.7 32.4 26.6 25.8 23.5 39.1 39.  35.1
 32.3 37.7 34.7 34.4 29.9 33.7 32.9 31.6 28.1 30.7 24.2 22.4 34.  38.
 44. ]

Example of categorical data: cylinders
Numbers of unique cylinders values: [8 4 6 3 5]

The quantitative mpg data are within a range from 9 to 44.
The categorical cylinders data are in five distinct values or categories.

cylinders
3      4
4    199
5      3
6     83
8    103
Name: cylinders, dtype: int64

Now we use a bar chart to visually compare the count of each group.

From the output above, we can easily see that the dataset has the highest number of 4-cylinder cars, followed by 8-cylinder cars. There are very few 3-cylinder and 5-cylinder cars.

It is also possible to plot a horizontal bar chart instead of a vertical bar chart as above. We use most of the same code for plotting except we use barh() for horizontal bar chart, and we swap the x-axis and y-axis labels.

Generally, when the x-ticks are long strings, the bar chart is easier to read if it's a horizontal bar chart. In the first vertical bar chart, the cylinder counts along the x-axis are small numbers so they don't overlap each other in the plot, so both a vertical and horizontal bar chart can be used.

We now look at the acceleration column, which is the number of seconds it takes to accelerate from 0 to 60 mph. We plot the acceleration values in a histogram.

The histogram above has 10 bars or 10 bins that cover the continuous range of acceleration data along the x-axis.

We see that the fastest acceleration is just under 5 seconds, and the slowest acceleration is almost 25 seconds.

Since the range of each bin is 1/10 of the entire data range, the width of each bar is the same. Therefore the height of the bar corresponds to the count of data points in a particular bin, and we see that the tallest bar or the largest count is for an acceleration of around 15-16 seconds. This tallest bin is also around the middle of the data range, so we expect the mean or average acceleration of all cars is also between 15-16 seconds.

The histogram shows us the shape of the acceleration data. Looking at the three tallest bars in the middle, we can say that the majority of the cars in the dataset has an acceleration of around 13 to 18 seconds, and the faster and slower accelerations slope down symmetrically on the left and right of center. The shape of this data distribution is shaped like a bell, or is a bell curve: it is symmetric, with no skew on the left or right.

We also notice that there's no gap or no empty bin, and no outliers, which are data values that are far on the left or right on the x-axis from the rest of the data.

Count of each bin: [ 6. 15. 50. 85. 91. 78. 44. 12.  7.  4.]
Max value of bin edges: [ 8.    9.68 11.36 13.04 14.72 16.4  18.08 19.76 21.44 23.12 24.8 ]

plt.figure(figsize=(4,4))

# asking for twice the number of bins as the default:
(counts, edges, patches) = plt.hist(cars['acceleration'], color='brown', edgecolor='black', bins=20)

print("Count of each bin:", counts)
plt.xlabel('Acceleration')
plt.ylabel('Count of cars')
plt.grid()
plt.show()

Output?

Count of each bin: [ 3.  3.  5. 10. 21. 29. 29. 56. 57. 34. 50. 28. 19. 25.  6.  6.  7.  0.
  2.  2.]

The solution for the two cases above is to:

1. Scale the count in each bin to be a ratio of the count / total data points.
   This turns the count of each bin into a proportion of the dataset that falls into each bin.
   It normalizes the data so we can compare the distribution of two datasets that differ greatly in size (10,000 data points vs 100 data points).
2. The area of the bar represents this ratio.
   This means the ratio = height of the bar * width of the bar
   Or: height of the bar = ratio / width of the bar.
   Since the area of each bar is a ratio of count / total data points, when we add all the areas of all the bars, the total area is 1. Again, this normalizes the distribution so that different distributions can be compared together.

When the height of the bars is determined this way, it represents the density or concentration of data at a bin, and the histogram is called a density histogram.

Note that the two distribution shapes are identical. In the frequency histogram the y-axis is the actual count of cars in a bin, while in the density histogram, the y-axis is the density or how crowded with data a bin is.