Mathematics

Frequency tables and grouped data

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This article explores grouped data, frequency tables and how to construct them.
When you have a very large amount of data especially where the variables repeat you can simplify the data by using frequency tables or grouped frequency tables. For example the data below can be simplified by using a frequency table.

Frequency tables

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The observations 5, 30 and 15 repeat at least two times instead of repeating all the values in a list we can just write down their frequency. Below is an example of the kind of table we can construct.
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In the table above we have simplified the data by avoiding a very long list of observations we get from collecting the data.

Example: Joyce measures the shoe size, x, of the female students in her class. She obtains the following results from the data she records.
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  • Find the number of female students with a shoe size 38.
  • Find the shoe size taken by the smallest number of female students
  • Find the shoe size with greatest number of students.
  • Find the total number of students
Answer:

  • 35 students have a shoe size 38.
  • The shoe size taken by the smallest number of students is 35.
  • The shoe size taken by the greatest number of students is 38
  • The total number of female students is 100.
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Total number of students was found by adding the frequency. You find the total number of observations by adding the frequency.

You may add another column to the table known as cumulative frequency. The column shows the running total of the frequencies at each stage.

Using the previous table add a column frequency column.

Answer:
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Grouped data

Data may be placed in a grouped data presentation. But there are some disadvantages, the specific values of the data are last since the variables are now available in a range.
Below are important facts that you must be aware of when working with grouped data.

  • Groups are known as classes
  • You use class boundaries for the lower and upper range of a group.
  • We use the mid-point of a class when working with a group.
  • You can find a class width of a group.
The length, x mm, to the nearest mm, of the random sample of adult grass hoppers is measured and shown in the table below. Find the class boundaries, mid-point and class width of the class 34 – 36.
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Answer:

  • For the class boundaries we must use half way between the data or group gaps. The data has gaps. The lower boundary is between 33 and 34, therefore the lower boundary is 33.5 and the upper boundary is between 36 and 37 therefore the boundary is 36.5
    Class boundaries are; 33.5mm, 36.5mm

  • Mid-point

    To find the mid-point we add the class boundaries and divide by two i.e; [IMAGE]

  • Class width

    The class width is the difference between the class boundaries.
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You will have cases where the classes appear to overlap especially when working with continuous data.

The time, x seconds, taken by a random sample of females to run 500m is measured and shown in the table below. Write down the class boundaries, mid-point and class width for 65-70.
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Notice this data has no gaps therefore the class boundaries are the lowest or biggest numbers of a class.
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