Organise data

18.3 Organise data

After data has been collected, the next step is to organise the data – that is, to bring it together in a systematic way that makes it easier to read. We can organise data by using tallies, frequency tables and stem-and-leaf diagrams. We can also group data into intervals.

Tallies

Tallies are a way of counting by drawing a short vertical line to record each item as you count it. To make reading the tally marks easier, for every fifth mark, you draw a diagonal line through the first four lines.

For example:

Number	Description	Tally
\(8\)	\(1\) group of \(5\) and three vertical lines
\(12\)	\(2\) groups of \(5\) and two vertical lines
\(15\)	\(3\) groups of \(5\)

If we are given the tallies and have to find the number they represent, we count the groups of \(5\), then multiply the number of groups by \(5\) and add the individual strokes.

For example:

Tally	Calculation	Answer
	\((1 \times 5) + 2\)	\(5 + 2 = 7\)
	\((2 \times 5) + 3\)	\(10 + 3 = 13\)
	\((4 \times 5) + 4\)	\(20 + 4 = 24\)

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Frequency tables

When we have finished counting and adding up all the tally marks, the total number of tallies tell us how many times an event occurred. This is called the frequency.

frequency

how many times an event or observation occurred

A frequency table shows the list of different categories (for example, recyclable items), tallies for the number of times each item occurs, and the total number of tally marks.

For example:

The frequency table below shows the number of recycled items collected.

Recyclable item	Tally	Frequency
Plastic bottle		\(5\)
Cardboard box		\(7\)
Tin can		\(3\)
Glass bottle		\(10\)

So the frequency table shows that the frequency of plastic bottles was \(5\) and the frequency of glass bottles was \(10\).

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Stem-and-leaf plots

A stem-and-leaf plot is a special representation of numerical data. Each data point is split into two parts; the stem part and the leaf part. The stem part determines the row in which the leaf part will be recorded. The leaf part is always a single digit.

For example:

• \(27\) can be split into \(2 \ \vert \ 7\) where \(2\) is the stem part and \(7\) is the leaf part.
• \(58\) can be split into \(5 \ \vert \ 8\) where \(5\) is the stem part and \(8\) is the leaf part.
• \(104\) can be split into \(10 \ \vert \ 4\) where \(10\) is the stem part and \(4\) is the leaf part. The leaf part is always a single digit, and here it is \(4\), so the leaf part must be \(10\).

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Grouping data into intervals

We can also organise data by grouping data points into intervals (or classes). Intervals are a range of values that are consistently defined and equal in size. Intervals must not overlap, and there must be no gaps in the values between intervals.

The table below shows intervals of equal size in the first column. This is the correct method for defining the number range of intervals.
The second column shows intervals that overlap, and intervals that are also different in size. This is not correct.

Example: Correctly defined intervals	Example: Incorrectly defined intervals
\(20–29\)	\(20–30\)
\(30–39\)	\(30–39\)
\(40–49\)	\(41–49\)

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