1. Displaying Categorical Data
Drawing Bar Graphs

2. Drawing Bar Graphs A good bar graph should contain: A Graph Title
T. Serino
A good bar graph should contain:
A Graph Title
  A Title for each axis
  Clearly identified categories on the category axis
(and a key for a multiple bar graph)
  A consistent scale on the numeric (frequency) axis
Although most of us have drawn bar graphs before and the above aspects of the graph may be obvious, small errors can lead to misunderstandings.

3. Drawing Bar Graphs Constructing a bar graph Steps:
T. Serino
Constructing a bar graph
Steps:
Start by using graph paper or a grid. (This is much easier than measuring the distance between each set of tick marks.)
2. Determine:
  the graph title
  the label for each axis
  the units and names that will be used for each axis

4. Drawing Bar Graphs 3. Draw the axes
T. Serino
3. Draw the axes
  Determine which axis will be the frequency axis. (vertical or horizontal bar graph)
  Write your axis labels and graph title
  Determine scale for frequency axis
  Find the range of data you want to display.
  Divide the range by the number of tick marks available.
  Use a convenient number close to your result as your scale. (round up)
Use the data to draw bars of equal width.
(use a ruler, be neat)

5. Drawing Bar Graphs
T. Serino
Example draw a bar graph to display the following data.
Last January in New York City, of the 31 days, there were 5 rainy days, 3 snowy days, 10 sunny days, and 5 windy days.
Step 1: Start by using graph paper or a grid.
We will graph the data on the grid to the right.
Notice that there are 15 tick marks on the horizontal axis (starting the count at zero).
Notice that there are 6 tick marks on the vertical axis (starting the count at zero).
This is enough to make 5 “double” bars (each two tick marks wide). Remember that the bars have to be of equal width and must have a space between them.
We will use this number to determine the scale for our frequency axis.

6. Drawing Bar Graphs
T. Serino
Example draw a bar graph to display the following data.
Last January in New York City, of the 31 days, there were 5 rainy days, 3 snowy days, 10 sunny days, and 5 windy days.
Step 2: Determine:
  the graph title
  the label for each axis
  the units and names that will be used for each axis
The graph title would probably be “January Weather in New York City”.
The category axis would be labeled “weather” and the frequency axis would be labeled ”number of days”.
The category axis would include: Rainy, Snowy, Sunny, Windy, and Other.

7. January Weather in New York City
Drawing Bar Graphs
T. Serino
Example) Draw a bar graph to display the following data.
Last January in New York City, of the 31 days, there were 5 rainy days, 3 snowy days, 10 sunny days, and 5 windy days.
January Weather in New York City
Rainy
Snowy
Sunny
Windy
Other
Often times, it is easier to draw a bar graph when given a table. Complete the table to the right using the information given.
The “Other” category is necessary here, because there are 31 days in January and the information given only accounts for 23 days. What number corresponds to the “Other” category?

8. January Weather in New York City
Drawing Bar Graphs
T. Serino
Example draw a bar graph to display the following data.
Step 3: Draw the axes
  Determine which axis will be the frequency axis. (vertical or horizontal bar graph)
  Write your axis labels and graph title
January Weather in New York City
# of Days
January Weather in New York City
Rainy 5
Snowy 3
Sunny 10
Windy
Other 8
Rainy
Snowy
Sunny
Windy
Other
Weather

9. January Weather in New York City
Drawing Bar Graphs
T. Serino
Example draw a bar graph to display the following data.
Step 3: (continued)
  Determine scale for frequency axis
  Find the range of data you want to display.
  Divide the range by the number of tick marks available.
  Use a convenient number close to your result as your scale. (round up)
Remember that “range” is found my subtracting the minimum data value from the maximum data value.
Range = Max – Min
Be careful here, we want to find the range of data that we want to display. A broken graph will violate the area principle, so we need to include zero when we display our data. If all of the data values are positive, the minimum to display will always be zero. (The range to display is usually equal to the maximum data value.)
January Weather in New York City
Rainy 5
Snowy 3
Sunny 10
Windy
Other 8

10. January Weather in New York City
Drawing Bar Graphs
T. Serino
Example draw a bar graph to display the following data.
Step 3: (continued)
  Determine scale for frequency axis
  Find the range of data you want to display.
  Divide the range by the number of tick marks available.
  Use a convenient number close to your result as your scale. (round up)
Range = Max – Min
January Weather in New York City
Rainy 5
Snowy 3
Sunny 10
Windy
Other 8
The maximum value to display is 10
The minimum value to display will always be zero, unless there are negative values in the data.
Not the Min
Max
Range = 10 – 0
The Range to display here is 10.

11. January Weather in New York City
Drawing Bar Graphs
T. Serino
Example draw a bar graph to display the following data.
Step 3: (continued)
  Determine scale for frequency axis
  Find the range of data you want to display.
  Divide the range by the number of tick marks available.
  Use a convenient number close to your result as your scale. (round up)
When we counted earlier, we found that there were 6 tick marks available on our frequency axis and the range we wanted to display was 10.
January Weather in New York City
Rainy 5
Snowy 3
Sunny 10
Windy
Other 8

12. January Weather in New York City
Drawing Bar Graphs
T. Serino
Example draw a bar graph to display the following data.
Step 3: (continued)
  Determine scale for frequency axis
  Find the range of data you want to display.
  Divide the range by the number of tick marks available.
  Use a convenient number close to your result as your scale. (round up)
Convenient scales are scales that are easy to read…
For a convenient scale separate tick marks by
1, 2, 5, 10, 20, 25, 50, 100, 200, etc.
Fraction values like ½ and ¼ are also easy scales to create and read. We can even use 3 or 4, but they are not as nice looking as the scale examples above.
(Remember to always round up! Otherwise your data won’t fit on your graph.
January Weather in New York City
Rainy 5
Snowy 3
Sunny 10
Windy
Other 8

13. January Weather in New York City
Drawing Bar Graphs
T. Serino
Example draw a bar graph to display the following data.
Step 3: (continued)
  Determine scale for frequency axis
  Find the range of data you want to display.
  Divide the range by the number of tick marks available.
  Use a convenient number close to your result as your scale. (round up)
Range to display ÷ Tick marks available ≈
January Weather in New York City
Rainy 5
Snowy 3
Sunny 10
Windy
Other 8
Choose the closest “convenient number” that rounds up to:
Convenient numbers: 1, 2, 5, 10, 20, 25, 50, 100, 200, …
The scale for this example should be 2.

14. January Weather in New York City
Drawing Bar Graphs
T. Serino
Example draw a bar graph to display the following data.
The scale for our graph should be 2.
So the distance between each set of tick marks on the vertical axis should be 2.
January Weather in New York City
# of Days
January Weather in New York City
Rainy 5
Snowy 3
Sunny 10
Windy
Other 8
Rainy
Snowy
Sunny
Windy
Other
Weather

15. January Weather in New York City
Drawing Bar Graphs
T. Serino
Example draw a bar graph to display the following data.
Step 4: Use the data to draw bars of equal width.
(use a ruler, be neat)
January Weather in New York City
# of Days
January Weather in New York City
Rainy 5
Snowy 3
Sunny 10
Windy
Other 8
Rainy
Snowy
Sunny
Windy
Other
Weather

16. Drawing Bar Graphs
T. Serino
A double bar graph is constructed the same way, but requires a key.
A key to distinguish between the different types of each category.
A space between categories, but not between each bar.

17. Try this.
T. Serino
The table shows the life expectancies of people in three countries. Construct a double bar graph to display this data on the grid provided. (Show work)

18. Try this. (Hint) Hint: The scale and titles should be as follows…
T. Serino
Hint: The scale and titles should be as follows…
Country
Life Expectancy (yrs)
Although the frequency axis only shows every 10th unit, the scale for this graph is 5’s. If you inspect the graph, you will see that the distance between each set of tick marks is 5.
Be sure to show/explain how you determined the scale of 5.

19. What can go wrong! Use a ruler!
T. Serino
1. Be sure to use a ruler and draw bars of equal width.
When graphs are drawn by hand, you will most likely violate the area principle.
Although the thee bars in the graph to the right are all of equal height, it is obvious that they do not have the same area.
Use a ruler!

20. What can go wrong!
T. Serino
2. Don’t use a broken graph. All graphs must show zero otherwise they will violate the area principle.
Papers Delivered
Mon 22
Tue 23
Wed 20
Thurs 25
Fri 21
Mon
Tue
Wed
Thur
Fri
If you look at this graph, it seems that there were twice as many papers delivered on Thursday as there were Friday. Looking at the table, this is obviously not true. (25 is not twice as much as 21)

21. What can go wrong!
T. Serino
3. Many mistakes can be made when creating the scale. The value of the distance between each set of tick marks must be the same.
The person who made this graph obviously just entered data values on the frequency axis instead of creating a scale.
Windy Days
Jan 5
Feb 6
Mar 20
These distances must be the same!
Although the distance between the tick marks seems to be the same, the value of the distance between tick marks is not.
The bars seem to increase size at a constant rate, but they shouldn’t. The increase in windy days from Feb to Mar is much larger than the increase in windy days from Jan to Feb.
Distance from 6 to 20 = 14
Distance from 5 to 6 = 1
Distance from 0 to 5 = 5
Jan
Feb
Mar

22. What can go wrong!
T. Serino
A similar but more subtle error is shown in the graph below.
What is the scale of this graph?
The numbers go by 10’s, but remember, the scale is the distance between each set of tick marks.
The distance between each of these tick marks is 5.
Is the scale 5?
The mistake is here!
This distance is 10.
Mon
Tue
Wed
Thur
Fri
Remember, the distance between each tick mark must be exactly the same!

23. Try this.
T. Serino
Display the following data with a bar graph.

24. athematical M D
ecision
aking