8.3: Presenting Quantitative Data Graphically

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Page ID: 139294

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

Example $\PageIndex{1}$

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:

Solution

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

Using this table, it seems like we could create a standard bar chart from this summary, like we did for categorical data:

$clipboard_eaf1f6fa54bc1cfdf7ff3989bbb586a18.png$

However, since the scores are numerical values, the bar graph above 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.

Histogram

A histogram is like a bar graph, but where the horizontal axis is a number line.

The vertical axis still represents frequency, and the horizontal axis still represents the variable of interest.

Example $\PageIndex{2}$

For the quiz score values in example 9, a histogram would look like:

$clipboard_e3394c918e10c5553e82471ac751af4fb.png$

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.

Unfortunately, not a lot of common software packages can correctly graph a histogram. Fortunately, your graphing calculator can quickly create an accurate histogram using the
STAT PLOT option.

$clipboard_ec08ab5375afe84aa39b543ffb22802a3.png$

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.

Class Intervals

Class intervals are groupings of the data. In general, we define class intervals so that:

Each interval is equal in size. For example, if the first class contains values from 120-129, the second class should include values from 130-139.
We have somewhere between 5 and 20 classes, typically, depending upon the number of data we’re working with.

Example $\PageIndex{3}$

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:

$clipboard_ec31fb713c0eb44d48a2106d5842d1483.png$

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.

$clipboard_e194ebc18e299a7a1b734583e7c290cad.png$

Exercise $\PageIndex{1}$

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

Solution

Calculator Instructions for Drawing a Histogram Using a TI-83/84:
1. Turn on the calculator
2. Press the “STAT” key
3. Hit “Enter” on option 1: “Edit”
This will bring you to a screen that contains lists: L1, L2, L3, etc.
3. Enter the data values (one value per row) into L1. For any negative values you need to use the (-) key, not the subtraction key. Continue until all data is entered into L1.
4. Press the 2nd key and then “STAT PLOT” (top left of calculator)
5. Press “Enter” on 1: “Plot 1”
6. Use the arrows to “ON” to turn on Plot 1. Hit “Enter”.
7. Use the arrows to go to the histogram picture and hit “Enter”.
8. Make sure that Xlist displays L1
9. Press the 2nd key and then “STAT PLOT” again.
Make sure to turn off all other stat plots, and clear all equations before graphing the histogram.
10. Press the ZOOM menu.
11. Scroll down to option 9: “ZoomStat” and hit “Enter”.

Answer: Answers will vary depending on how the histogram is built

Search

Text Color

Text Size

Margin Size

Font Type

Solution

Solution

Example \(\PageIndex{1}\)

Solution

Histogram

Example \(\PageIndex{2}\)

Class Intervals

Example \(\PageIndex{3}\)

Exercise \(\PageIndex{1}\)

Solution