7.2 Measures of Dispersion
In grouped data, measures of dispersion describe how spread out or clustered data values are around a central value. Key measures include range, interquartile range (IQR), variance, and standard deviation. Furthermore, data distributions can be illustrated using box plots constructed from an ogive's five-number summary and used to compare different sets of data.
Learning Objectives
- Determine the range and interquartile range of grouped data.
- Calculate the variance and standard deviation of grouped data using formulae.
- Construct and interpret a box plot for a set of grouped data from an ogive.
- Compare and interpret two or more sets of grouped data based on measures of central tendency and measures of dispersion.
- Solve problems involving measures of dispersion for grouped data.
Summary Arena: Measures of Dispersion and Shapes of Distribution
Link the formulae for measures of dispersion with box plot distribution shapes.
Range
\(\text{Range} = \text{midpoint of the highest class} - \text{midpoint of the lowest class}\)
Interquartile Range
\(\text{IQR} = Q_3 - Q_1\)
Variance
\(\sigma^2 = \dfrac{\sum fx^2}{\sum f} - \bar{x}^2\)
Standard Deviation
\(\sigma = \sqrt{\dfrac{\sum fx^2}{\sum f} - \bar{x}^2}\)
Brief Explanation
Range and Interquartile Range
For grouped data, the range is the difference between the midpoints of the highest and lowest classes. The interquartile range is \(Q_3 - Q_1\), where \(Q_1\) and \(Q_3\) are read from the ogive at cumulative frequency positions \(\frac{1}{4}N\) and \(\frac{3}{4}N\).
Variance and Standard Deviation
The variance is \(\sigma^2 = \dfrac{\sum fx^2}{\sum f} - \bar{x}^2\). The standard deviation is \(\sigma = \sqrt{\sigma^2}\). A smaller standard deviation means the data are more consistent.
| Measure | Formula | Meaning |
|---|---|---|
| Range | \(\text{midpoint of highest class} - \text{midpoint of lowest class}\) | Difference between extreme midpoints |
| Interquartile range | \(Q_3 - Q_1\) | Spread of the middle 50% of the distribution |
| Variance | \(\sigma^2 = \dfrac{\sum fx^2}{\sum f} - \bar{x}^2\) | Average of squared differences from the mean |
| Standard deviation | \(\sigma = \sqrt{\sigma^2}\) | Spread relative to the mean in the original unit |
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
What is the formula used to calculate the range of a set of grouped data?
Between Brand A (mean = 14.35 cm, standard deviation = 0.64 cm) and Brand B (mean = 14.10 cm, standard deviation = 0.70 cm), which brand is more consistent in petal diameter and why?
If the distance between \(Q_1\) and \(Q_2\) in a box plot is longer than the distance between \(Q_2\) and \(Q_3\), what is the shape of the data distribution?
Common Mistakes
- Using the upper boundary minus lower boundary of the whole table to find the range, instead of subtracting the midpoint of the lowest class from the midpoint of the highest class.
- Forgetting to take the square root of variance when calculating the standard deviation.
- Forgetting that the unit of variance is squared (e.g., \(\text{litres}^2\)), while standard deviation retains the original unit (e.g., \(\text{litres}\)).
- Reading quartile values directly as the cumulative frequency positions (e.g., taking \(20\) instead of the value on the horizontal axis corresponding to cumulative frequency \(20\)).
Misconceptions
A larger standard deviation means better performance.
A larger standard deviation indicates greater spread or variability (less consistency), not necessarily better performance.
Skewness depends on where the whole box is placed along the axis.
Skewness depends on the position of the median relative to the quartiles (whisker lengths and box segments), not the absolute position of the box on the number line.
Summary
- Range = Midpoint of highest class − Midpoint of lowest class.
- IQR = \(Q_3 - Q_1\), where \(Q_1\) and \(Q_3\) are obtained from the ogive at cumulative frequencies \(\frac{1}{4}N\) and \(\frac{3}{4}N\).
- Variance: \(\sigma^2 = \dfrac{\sum fx^2}{\sum f} - \bar{x}^2\); Standard deviation: \(\sigma = \sqrt{\sigma^2}\).
- Consistency: Smaller standard deviation = more consistent data.
- Box Plot: Summarises minimum, \(Q_1\), \(Q_2\) (median), \(Q_3\), and maximum; reveals if data is symmetric, left-skewed, or right-skewed.
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 How is the range of a grouped frequency distribution determined?
2 In a box plot, the median line is closer to the third quartile (\(Q_3\)) than to the first quartile (\(Q_1\)), making the left side of the box longer. What shape of distribution does this represent?
B. Fill in the Blanks
3 The interquartile range of grouped data is calculated by subtracting \(Q_1\) from .
4 When comparing two data sets, the one with the smaller deviation is considered more consistent.
C. Drag and Drop: Match Each Statistical Measure
Drag each statistical measure to its correct description or formula. On a phone, tap the answer choice first, then tap the description box.
