11.2 Presenting Quantitative Data Graphically

Quantitative, or numerical, data can also be summarized into frequency tables.

Example 1

A teacher records scores on a 20-point quiz for the 30 students in his class. The scores are:

19 20 18 18 17 18 19 17 20 18 20 16 20 15 17 12 18 19 18 19 17 20 18 16 15 18 20 5 0 0

These scores could be summarized into a frequency table by grouping like values:

Score	Frequency
0	2
5	1
12	1
15	2
16	2
17	4
18	8
19	4
20	6

Using this table, it would be possible to create a standard bar chart from this summary, like we did for categorical data:

Bar chart of quiz scores treated, wrongly, as categories. The horizontal axis Score carries the evenly spaced labels 0, 5, 12, 15, 16, 17, 18, 19, 20 and the vertical axis Frequency runs 0 to 8. Heights in that order: 2, 1, 1, 2, 2, 4, 8, 4, 6. Because the labels are evenly spaced, the gap from 0 to 5 occupies the same width as the gap from 17 to 18.

However, since the scores are numerical values, this chart doesn’t really make sense; the first and second bars are five values apart, while the later bars are only one value apart. It would be more correct to treat the horizontal axis as a number line. This type of graph is called a histogram.

Notice that in the histogram, a bar represents values on the horizontal axis from that on the left hand-side of the bar up to, but not including, the value on the right hand side of the bar. Some people choose to have bars start at ½ values to avoid this ambiguity.

The same quiz-score histogram redrawn with the bars starting and ending at half values so each bar is centred on its score, which removes the ambiguity about which endpoint belongs to which bar. Horizontal axis Score labelled 0 to 20 in steps of 2, vertical axis Frequency 0 to 8. Heights: 2 at score 0, 1 at 5, 1 at 12, 2 at 15, 2 at 16, 4 at 17, 8 at 18, 4 at 19, 6 at 20.

Unfortunately, not a lot of common software packages can correctly graph a histogram. About the best you can do in Excel or Word is a bar graph with no gap between the bars and spacing added to simulate a numerical horizontal axis.

If we have a large number of widely varying data values, creating a frequency table that lists every possible value as a category would lead to an exceptionally long frequency table, and probably would not reveal any patterns. For this reason, it is common with quantitative data to group data into class intervals.

Example 3

Suppose that we have collected weights from 100 male subjects as part of a nutrition study. For our weight data, we have values ranging from a low of 121 pounds to a high of 263 pounds, giving a total span of 263-121 = 142. We could create 7 intervals with a width of around 20, 14 intervals with a width of around 10, or somewhere in between. Often time we have to experiment with a few possibilities to find something that represents the data well. Let us try using an interval width of 15. We could start at 121, or at 120 since it is a nice round number.

Interval	Frequency
120-134	4
135-149	14
150-164	16
165-179	28
180-194	12
195-209	8
210-224	7
225-239	6
240-254	2
255-269	3

A histogram of this data would look like:

In many software packages, you can create a graph similar to a histogram by putting the class intervals as the labels on a bar chart.

The same weight data drawn as a bar chart with the class intervals used as category labels under the bars instead of a numerical axis, which is the closest most spreadsheet software gets to a histogram. Vertical axis Frequency 0 to 30. Heights: 120-134 is 4, 135-149 is 14, 150-164 is 16, 165-179 is 28, 180-194 is 12, 195-209 is 8, 210-224 is 7, 225-239 is 6, 240-254 is 2, 255-269 is 3.

Other graph types such as pie charts are possible for quantitative data. The usefulness of different graph types will vary depending upon the number of intervals and the type of data being represented. For example, a pie chart of our weight data is difficult to read because of the quantity of intervals we used.

Pie chart titled Weights (pounds) with one slice per class interval and a colour key at the right. Clockwise from the top the slices stand for 120-134 with 4, 135-149 with 14, 150-164 with 16, 165-179 with 28, 180-194 with 12, 195-209 with 8, 210-224 with 7, 225-239 with 6, 240-254 with 2 and 255-269 with 3, out of 100 subjects. With ten slices, several of them slivers, the chart is hard to read, which is the point being made.

Your Turn

Try it Now 3

The total cost of textbooks for the term was collected from 36 students. Create a histogram for this data.

import math
import matplotlib.pyplot as plt

# The book's textbook-cost data from 36 students.
costs = [140, 160, 160, 165, 180, 220, 235, 240, 250, 260, 280, 285, 285, 285,
         290, 300, 300, 305, 310, 310, 315, 315, 320, 320, 330, 340, 345, 350,
         355, 360, 360, 380, 395, 420, 460, 460]

WIDTH = 55       # <-- CHANGE ME. The book uses 55. Try 10, 25, 100, 200.
START = 140      # <-- CHANGE ME (the left edge of the first class interval)

def frequency_table(data, start, width):
    """Bins are [start, start+width) -- up to but NOT including the right edge."""
    top = max(data)
    edges = [start + i * width for i in range(math.ceil((top - start + 1) / width) + 1)]
    counts = [0] * (len(edges) - 1)
    for x in data:
        counts[min(int((x - start) // width), len(counts) - 1)] += 1
    return edges, counts

edges, counts = frequency_table(costs, START, WIDTH)
print("n = %d,  range $%d to $%d,  class width $%d,  %d classes"
      % (len(costs), min(costs), max(costs), WIDTH, len(counts)))
print()
print("%-18s %10s   %s" % ("cost interval", "Frequency", "bar"))
print("-" * 52)
for lo, hi, c in zip(edges[:-1], edges[1:], counts):
    print("%-18s %10d   %s" % ("$%d - $%d" % (lo, hi - 1), c, "#" * c))
print("-" * 52)
print("%-18s %10d" % ("total", sum(counts)))
print()
print("The book's rule of thumb: aim for 5 to 20 classes. Set WIDTH = 10 and the")
print("shape dissolves into noise; set WIDTH = 200 and it flattens into nothing.")

plt.figure(figsize=(7, 3.6))
plt.hist(costs, bins=edges, edgecolor="white", color="#4472c4")
plt.xticks(edges, rotation=45)
plt.xlabel("Textbook cost for the term ($)")
plt.ylabel("Frequency")
plt.title("Histogram, class width $%d" % WIDTH)
plt.tight_layout()
plt.show()

$140 $160 $160 $165 $180 $220 $235 $240 $250 $260 $280 $285

$285 $285 $290 $300 $300 $305 $310 $310 $315 $315 $320 $320

$330 $340 $345 $350 $355 $360 $360 $380 $395 $420 $460 $460

Answer

Using a class intervals of size 55, we can group our data into six intervals:

cost interval	Frequency
$$ 140 minus 194$	5
$$ 195 minus 249$	3
$$ 250 minus 304$	9
$$ 305 minus 359$	12
$$ 360 minus 414$	4
$$ 415 minus 469$	3

We can use the frequency distribution to generate the histogram.

When collecting data to compare two groups, it is desirable to create a graph that compares quantities.

Example 4

The data below came from a task in which the goal is to move a computer mouse to a target on the screen as fast as possible. On 20 of the trials, the target was a small rectangle; on the other 20, the target was a large rectangle. Time to reach the target was recorded on each trial.

Interval (milliseconds)	Frequency small target	Frequency large target
300-399	0	0
400-499	1	5
500-599	3	10
600-699	6	5
700-799	5	0
800-899	4	0
900-999	0	0
1000-1099	1	0
1100-1199	0	0

One option to represent this data would be a comparative histogram or bar chart, in which bars for the small target group and large target group are placed next to each other.

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.