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3.1 Frequency Tables
LEARNING GOAL
Be able to create and interpret frequency tables.
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Definition
A basic frequency table has two columns:
• One column lists all the categories of data.
• The other column lists the frequency of each category, which is the number of data values in the category.
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3. (bad taste) 1 2 3 4 5 (excellent taste) The 20 ratings are as follows:
EXAMPLE 1 Taste Test
The Rocky Mountain Beverage Company wants feedback on its new product, Coral Cola, and sets up a taste test with 20 people. Each individual is asked to rate the taste of the cola on a 5-point scale:
(bad taste) (excellent taste)
The 20 ratings are as follows:
Construct a frequency table for these data.
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EXAMPLE 1 Taste Test
Solution:
The variable of interest is taste, and this variable can take on five values: the taste categories 1 through 5. (Note that the data are qualitative and at the ordinal
level of measurement.)
We construct a table
with these five
categories in the left
column and their
frequencies in the
right column, as
shown in Table 3.2.
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Binning Data
Definition
When it is impossible or impractical to have a category for every value in a data set, we bin (or group) the data into categories (bins), each covering a range of possible data values.
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EXAMPLE 2 The Dow Stocks
For the 30 stocks of the Dow Jones Industrial Average, Table 3.5 (previous slide) shows the annual revenue (in billions of dollars), the one-year total return, and the rank on the Fortune 500 list of largest U.S. companies. Create a frequency table for the revenue. Discuss the pros and cons of the binning choices.
Solution: The revenue data range from $21.6 billion (McDonald’s) to $351.1 billion (Wal-Mart). There are many possible ways to bin data for this range; here’s one good way and the reasons for it:
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• We create bins spanning a range from $0 to $400 billion. This covers the full range of the data, with extra room below the lowest data value and above the highest data value.
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EXAMPLE 2 The Dow Stocks
Solution: (cont.)
• We give each bin a width of $50 billion so that we can span the $0 to $400 billion range with eight bins. Also, the width of $50 billion is a convenient number that helps make the table easy to read.
• Because the data values are given to the nearest tenth (of a billion dollars), we also define the bins to the nearest tenth so that they do not overlap. That is, bins go from $0 to $49.9 billion, from $50.0 to $99.9 billion, and so on.
Table 3.6 (on the next slide) shows the resulting frequency table.
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TIME OUT TO THINK
Consider three other possible ways of binning the data in Table 3.6: 4 bins spanning the range $0 to $400 billion, 11 bins spanning the range $0 to $375 billion, and 36 bins spanning the range $0 to $360 billion. Briefly discuss the pros and cons of each of these choices.
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Relative Frequency
Definition
The relative frequency of any category is the proportion or percentage of the data values that fall in that category:
relative frequency =
frequency in category
total frequency
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Cumulative Frequency
Definition
The cumulative frequency of any category is the number of data values in that category and all preceding categories.
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EXAMPLE 3 More on the Taste Test
Using the taste test data from Example 1, create a frequency table with columns for the relative and cumulative frequencies. What percentage of the respondents gave the cola the highest rating? What percentage gave the cola one of the three lowest ratings?
Solution: We find the relative frequencies by dividing the frequency in each category by the total frequency of 20. We find the cumulative frequencies by adding the frequency in each category to the sum of the frequencies in all preceding categories. Table 3.9 (on the next slide) shows the results.
The relative frequency column shows that 0.10, or 10%, of the respondents gave the cola the highest rating.
The cumulative frequency column shows that 14 out of 20 people, or 70%, gave the cola a rating of 3 or lower.
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The End
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