8.1 Dispersion
Measures of dispersion describe how values in a set of data are scattered or spread out. When data values are clustered closely together, the dispersion is small, whereas values that are widely spread have a large dispersion.
Learning Objectives
- Explain the meaning of dispersion in statistics.
- Determine the difference between the largest value and the smallest value in a data set.
- Compare and interpret the dispersion of two or more sets of data based on stem-and-leaf plots and dot plots, and make valid conclusions.
Visualising Dispersion with Stem-and-Leaf Plots and Dot Plots
See how data distribution, spread and comparison between two groups are structured and read on both plots.
Stem-and-Leaf Plot
A stem-and-leaf plot splits each value into a stem and a leaf. Leaves are arranged in ascending order and a key must be included.
| Stem | Leaf |
|---|---|
| 2 | 7 |
| 3 | 1 4 8 |
| 4 | 0 2 5 |
Key: \(2 \mid 7\) means \(27\) marks.
Dot Plot
A dot plot uses one dot for every observation on a uniform scale. A wider spread of dots indicates greater dispersion.
Dots concentrated towards the right indicate generally higher values.
Brief Explanation
What is the Meaning of Dispersion?
Dispersion gives an idea of how the values of a set of data are scattered. Dispersion is small if the data set has a small range, and large if it has a large range.
How is dispersion measured?
Measures of dispersion include quantitative measures such as range, interquartile range, variance and standard deviation.
A large difference indicates that data is widely dispersed, while a small difference indicates that data is closely concentrated.
Ethical Data Representation
The scale used in data representations must be uniform and must start from 0 to avoid confusion. The data displayed must be accurate.
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
Boiling points of four chemicals are \(60^\circ\text{C}\), \(75^\circ\text{C}\), \(150^\circ\text{C}\), and \(45^\circ\text{C}\). What is the difference between the highest and lowest boiling points?
In a stem-and-leaf plot, what does the key \(4 \mid 2\) represent if the stem represents tens and the leaf represents units?
When comparing two dot plots of test scores, how do you determine which group achieved better results?
Common Mistakes
- Forgetting to write the Key in a stem-and-leaf plot, making it impossible to know whether \(2 \mid 7\) means \(27\), \(2.7\), or \(270\).
- Writing the leaf digits in random order instead of arranging them in ascending order from smallest to largest.
- Confusing high dispersion with high performance; a wider spread simply means the values vary more, not that they are higher.
- Skipping repeated data values when plotting dots on a dot plot instead of stacking a dot for every single observation.
Misconceptions
A data set with a higher maximum value always has a wider dispersion.
Dispersion depends on the difference between the largest and smallest values (the spread), not solely on the maximum value.
On the left side of a back-to-back stem-and-leaf plot, leaves are read left-to-right.
Leaves on both sides are ordered starting from the stem outwards, meaning left-side leaves increase from right to left.
Summary
- Dispersion indicates how scattered data values are.
- Larger differences between extreme values reflect wider dispersion.
- Stem-and-leaf plots separate values into stems and leaves arranged in ascending order, accompanied by a key.
- Dot plots plot individual data points over a uniform scale to display spread, frequencies and concentrations.
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 Which of the following indicates that a set of data has a small dispersion?
2 In a dot plot, what does each plotted dot represent?
B. Fill in the Blanks
3 In a stem-and-leaf plot, leaf digits must be arranged in order from the smallest value to the ______ value.
4 To understand the dispersion of a data set, we calculate the difference between the largest value and the ______ value.
C. Drag and Drop: Match Each Term with Its Description
Drag each term to the correct description. On a phone, tap the answer choice first, then tap the description box.
