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Year 6 Maths Data

Mode, range & shape of data

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Theory

The mode is the value that appears most often. The range is the highest value take away the lowest, and tells how spread out the data is. The shape of the data describes where it clusters, where the gaps are and whether there is an outlier. This Year 6 maths page shows how to find and use all three.

The mode is the most common value. A data set can have two modes (\(3, 5, 5, 8, 8, 10\) has modes \(5\) and \(8\)) or no mode if every value appears once.

The mode also works for words: if most students choose banana, the mode is banana.

The range is the highest value minus the lowest value. A bigger range means the data is more spread out.

Shape: a cluster is where many values bunch together; a gap is a part with no values; an outlier is a value far away from the rest.

Dot plot of the number of siblings A dot plot from 0 to 6 of how many brothers and sisters 18 students have: three dots at 0, seven at 1, five at 2, two at 3, none at 4 or 5, and one at 6. 0123456 Number of siblings
Mode \(1\) (the tallest stack). Range \(6 - 0 = 6\). Cluster from \(0\) to \(3\), a gap at \(4\) and \(5\), and an outlier at \(6\).
Column graph of hours of sleep A column graph of the hours of sleep 23 students had last night: 7 hours, 1 student; 8 hours, 4; 9 hours, 10; 10 hours, 6; 11 hours, 2. 024681012 Number of students 7 8 9 10 11 Hours of sleep
The tallest column is \(9\) hours, so the mode is \(9\) hours. Range \(11 - 7 = 4\) hours.

Mode and range:

\[ \text{mode} = \text{most common value} \]
mode = most common value
\[ \text{range} = \text{highest} - \text{lowest} \]
range = highest − lowest
WordMeaning
clustermany values close together
gapno values in part of the scale
outliera value far from the rest
spread outvalues over a wide range
Order first. Sorting the data from smallest to largest makes the mode and range easy to see.
Mode is a value, not a count. In \(6, 9, 4, 9, 7, 3, 9, 5, 7\) the mode is \(9\), not \(3\) (the number of times it appears).

How to find the mode

  1. Sort or tally the data.
  2. Find the value (or values) with the highest count.
  3. No repeats: say there is no mode.

How to find the range

  1. Find the highest value and the lowest value.
  2. Subtract: highest − lowest. Include the units.

How to describe the shape

  1. Look for where the values cluster.
  2. Point out any gaps and outliers, and whether most values are high or low.
Example 1 — Mode of a list
Find the mode: \(6, 9, 4, 9, 7, 3, 9, 5, 7\).
Solution

Sorted: \(3, 4, 5, 6, 7, 7, 9, 9, 9\).

\(9\) appears three times, more than any other value. Mode \(= 9\).

Example 2 — Range of a list
Find the range: \(18, 33, 12, 27, 40, 22\) minutes.
Solution
\(40 - 12\)\(=\)\(28\)
40−12=28

\(28\) minutes.

Example 3 — Two modes or none
Find the mode of each set.
(i) \(3, 5, 5, 8, 8, 10\)
(ii) \(11, 14, 16, 19\)
Solution

(i) Two modes: \(5\) and \(8\)
(ii) No mode (no value repeats)

Example 4 — Mode of words
A tally of favourite fruit shows apple \(6\), banana \(9\), mango \(7\). What is the mode?
Solution

Banana: it has the highest count.

Example 5 — From a dot plot
Use the dot plot of siblings.
(i) What is the mode?
(ii) What is the range?
(iii) Describe the shape.
Solution
\(6 - 0\)\(=\)\(6\)
6−0=6

(i) \(1\) sibling
(ii) \(6\)
(iii) Most values cluster from \(0\) to \(3\); there is a gap at \(4\) and \(5\) and one outlier at \(6\).

Example 6 — Decimal range
Four high jumps were \(1.35\) m, \(1.62\) m, \(1.48\) m and \(1.29\) m. Find the range.
Solution
\(1.62 - 1.29\)\(=\)\(0.33\)
1.62−1.29=0.33

\(0.33\) m.

Example 7 — Work backwards
The lowest score in a game was \(15\) and the range was \(23\). What was the highest score?
Solution
\(15 + 23\)\(=\)\(38\)
15+23=38

\(38\).

Example 8 — Compare two classes
Spelling scores: 6A \(12, 15, 14, 15, 13\); 6B \(8, 19, 15, 10, 19\).
(i) Find each mode and range.
(ii) Which class has more spread-out scores?
Solution
\(15 - 12\)\(=\)\(3\)
\(19 - 8\)\(=\)\(11\)
6 A range 3,6 B range 11

(i) 6A: mode \(15\), range \(3\). 6B: mode \(19\), range \(11\).
(ii) 6B, because its range is much larger.

Example 9 — Find the mistake
For the data \(9, 4, 15, 7, 12\), Tom says the range is \(12 - 9 = 3\). What is right?
Solution
\(15 - 4\)\(=\)\(11\)
15−4=11

Tom used the last and first values. The range uses the highest and lowest: \(11\).

Common pitfalls

Using the first and last values for the range. Use the highest and the lowest, wherever they are in the list.
Giving the count as the mode. The mode is the value that appears most, not how many times it appears.
Forgetting there can be two modes or none. Check every value’s count before deciding.
Leaving out the outlier. The range includes every value, even an outlier.

Frequently asked questions

What is the mode?

The value that appears most often in a data set.

Can there be more than one mode?

Yes. If two values tie for the most, both are modes. If no value repeats, there is no mode.

What is the range?

The highest value minus the lowest value. It shows how spread out the data is.

What is an outlier?

A value much higher or much lower than the rest of the data.

How do I describe the shape of data?

Say where the values cluster, where any gaps are, and whether there are outliers.

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