Section 3.2 ~ Picturing Distributions of Data

Introduction to Probability and StatisticsMs. Young

Objective

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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

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

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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

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Example 1 Create a vertical bar graph from the essay grade data in section

3.1.

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Grade Frequency

A 4

B 7

C 9

D 3

F 2

Total 25

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

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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)

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Bar graph Pareto chart

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

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Example 2 Create a pie chart from the essay grade data in section 3.1.

Each sector is 10 degrees.

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Grade Frequency

A 4

B 7

C 9

D 3

F 2

Total 25

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)

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Refer to table 3.4 on p.91

Example 3 Create a histogram from the data below that shows the ages of

Academy Award-winning actresses from 1970 to 2007.

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Age Number of Actresses

20-29 8

30-39 18

40-49 8

50-59 0

60-69 2

70-79 1

80-89 1

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

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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

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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.

32 37 36 32 51 53 33 61 35 45

55 39 76 37 42 40 32 60 38 56

48 48 40 43 62 43 42 44 41 56

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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

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Time-Series Diagrams A time-series diagram is a histogram or line chart in

which the horizontal axis represents time

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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

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