1. Section 3.2 ~ Picturing Distributions of Data
Introduction to Probability and Statistics
Ms. Young

2. Sec. 3.2
Objective
In this section, we will look at the most common methods for displaying distributions of data
A distribution of data values refers to the way the values are spread out over the chosen categories
You will be able to create and interpret:
Basic bar graphs
Dotplots
Pie charts
Histograms
Stem-and-leaf plots
Line charts
Time-series diagrams

3. Sec. 3.2
Bar Graphs
A bar graph is typically used for qualitative data (categorical)
Each bar represents the frequency (or relative frequency of one category)
The bars can either be vertical or horizontal

4. Characteristics of Bar Graphs
Sec. 3.2
Characteristics of Bar Graphs
When the data is qualitative, the widths of the bars have no special meaning, so there is no reason for them to be touching and they should be drawn with uniform widths
The graph should have a title or caption that explains what is being shown
The vertical axis should be labeled and scaled appropriately
The tick marks should be evenly spaced and the range of values between each mark should be the same
The horizontal axis should be labeled and each category should be indicated (there is no need for tick marks if the data is qualitative)
The graph should include a legend if multiple data sets are displayed on a single graph

5. Sec. 3.2
Example 1
Create a vertical bar graph from the essay grade data in section 3.1.
Grade
Frequency A 4 B 7 C 9 D 3 F 2
Total 25

6. Sec. 3.2
Dotplots
A dotplot is a variation of a bar graph in which each dot represents one data value and the total number of dots represents the frequency
Dotplots are convenient when making graphs of raw data, because you can tally the data by making a dot for each value and then you can choose to convert the graph to a bar chart so it looks more formal

7. Sec. 3.2
Pareto Chart
A pareto chart is a bar graph with the bars arranged in frequency order (either high to low or low to high)
Pareto charts make sense only for data at the nominal level
Ex. ~ It wouldn’t make sense to create a pareto chart for the essay grade data because then it would put the grades out of order (C, B, A, D, F)
Bar graph
Pareto chart

8. Sec. 3.2
Pie Charts
A pie chart is a circle divided so that each wedge represents the relative frequency of a particular category
The wedge size is proportional to the relative frequency
The entire pie represents the total relative frequency of 100%
When the wedge sizes represent simple fractions, it’s easy to create a pie chart, but when there are numerous categories, a pie chart may not be the best way to represent the data

9. Sec. 3.2
Example 2
Create a pie chart from the essay grade data in section Each sector is 10 degrees.
Grade
Frequency A 4 B 7 C 9 D 3 F 2
Total 25 9

10. Sec. 3.2
Histograms
A histogram is a bar graph that shows a distribution of quantitative data (numerical)
Not only does the y-axis have numerical meaning, but so does the x-axis (therefore the bar widths have meaning)
Just like the tick marks on the y-axis, the tick marks on the x-axis must be evenly spaced and represent the same range of values between each one
The bars touch each other because there are no gaps in between the categories
The start of the bar includes the number (ex. the red bar is from 0 up to 20, but not including 20 and the pink bar is from 20 up to 40, but not including 40, and so on)
Refer to table 3.4 on p.91

11. Sec. 3.2
Example 3
Create a histogram from the data below that shows the ages of Academy Award-winning actresses from 1970 to 2007.
Age
Number of Actresses
20-29 8
30-39 18
40-49
50-59
60-69 2
70-79 1
80-89 11

12. Stem-and-leaf Plot for Qualitative Data
Sec. 3.2
Stem-and-leaf Plot for Qualitative Data
A stem-and-leaf plot looks somewhat like a histogram turned sideways, except in place of bars we see a listing of data
This gives us a more detailed look at the data
Ex. ~ The histogram for the state energy use only shows us how many states fall into each category, but the stem-and-leaf plot not only tells us how many fall into each category, but more specifically the name of the state

13. Stem-and-leaf Plot for Quantitative Data
Sec. 3.2
Stem-and-leaf Plot for Quantitative Data
A stem-and-leaf plot for quantitative data lists the data values using the first digit as the stem and the remaining digit(s) as the leaf
You must also include either a decimal point where it’s appropriate in the stem or a key stating the equivalency of the stem and the leaf put together

14. Sec. 3.2
Example 4
The following data values represent the ages of recent Academy Award-winning male actors at the time when they won the award. Make a stem-and-leaf plot for the data. 14

15. Sec. 3.2
Line Charts
A line chart shows a distribution of data using a series of dots connected by lines
When qualitative, the dot is positioned horizontally by placing it on the tick mark of the category and vertically by placing it at the appropriate value that corresponds to the frequency
When quantitative, the dot is positioned horizontally in the middle of the bin and vertically at the appropriate value that corresponds to the frequency

16. Time-Series Diagrams Sec. 3.2
A time-series diagram is a histogram or line chart in which the horizontal axis represents time

17. Sec. 3.2
Summary
Bar graph: each bar represents the frequency of a category
Dotplot: similar to a bar graph, but there is a dot for each piece of data that falls into a certain category
All the dots added up give the frequency for that category
Pareto chart: is a bar graph arranged in frequency order
Remember that this would only make sense for a nominal level of measurement
Pie chart: a circle that is divided into wedges that represent the relative frequency of a category
Histogram: is a bar graph in which the data is quantitative
Stem-and-leaf plot: is a table that represents either qualitative data or quantitative data by dividing that data into two parts
Line chart: a series of points connected by line segments in which the point represents the frequency of the category
Time-series diagram: is a histogram or line chart in which the x-axis represents time