8.2 Measures of Dispersion
Measures of dispersion describe how data is spread out or distributed relative to central values. In this topic, students learn how to determine range, interquartile range, variance, and standard deviation for ungrouped data, construct and interpret box plots, evaluate the effects of changing data sets, and compare data distributions to make justified conclusions.
Learning Objectives
- Determine the range, interquartile range, variance, and standard deviation as a measure to describe dispersion of ungrouped data.
- Explain the advantages and disadvantages of various measures of dispersion to describe ungrouped data.
- Construct and interpret the box plot for a set of ungrouped data.
- Determine the effect of data changes on dispersion based on the value of measure of dispersion and graphical representation.
- Compare and interpret two or more sets of ungrouped data, based on the appropriate measures of dispersion, and hence make conclusions.
- Solve problems involving measures of dispersion.
Summary of Measures of Dispersion & Box Plot Structure
See the five-number summary on a box plot and the effect of operations on range, IQR, variance and standard deviation.
Box Plot Structure
The box shows the middle 50% of data. The whiskers extend to the minimum and maximum values.
Effect of Operations
| Operation | Range | IQR | \(\sigma\) | \(\sigma^2\) |
|---|---|---|---|---|
| \(+k\) or \(-k\) | Unchanged | Unchanged | Unchanged | Unchanged |
| \(\times k\) | \(\times k\) | \(\times k\) | \(\times k\) | \(\times k^2\) |
| \(\div k\) | \(\div k\) | \(\div k\) | \(\div k\) | \(\div k^2\) |
Brief Explanation
Range and Interquartile Range
Range is the difference between the largest and smallest values. The interquartile range is \(Q_3 - Q_1\), measuring the spread of the middle 50% of data.
Variance and Standard Deviation
Variance (\(\sigma^2\)) and standard deviation (\(\sigma\)) measure how far data values deviate from the mean.
| Measure | Formula | Advantage | Disadvantage |
|---|---|---|---|
| Range | \(\text{Largest} - \text{Smallest}\) | Easiest to calculate | Sensitive to outliers |
| Interquartile range | \(Q_3 - Q_1\) | Not affected by extreme values | Ignores outer 50% of data |
| Standard deviation | \(\sigma = \sqrt{\dfrac{\sum (x - \bar{x})^2}{N}}\) | Uses all data values | More complex to calculate |
Key Formulas for Ungrouped Data
Mean: \(\bar{x} = \dfrac{\sum x}{N}\)
Variance: \(\sigma^2 = \dfrac{\sum (x - \bar{x})^2}{N} = \dfrac{\sum x^2}{N} - \bar{x}^2\)
Standard deviation: \(\sigma = \sqrt{\dfrac{\sum (x - \bar{x})^2}{N}} = \sqrt{\dfrac{\sum x^2}{N} - \bar{x}^2}\)
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
Given the data set \(4, 6, 7, 9, 10, 12, 15\), find the range and interquartile range.
A set of data has a standard deviation of \(2.5\) and a variance of \(6.25\). If each data value is multiplied by \(3\), what are the new standard deviation and the new variance?
Why is the interquartile range a more appropriate measure of dispersion than the range for a data set containing an extreme value?
Common Mistakes
- Subtracting frequencies instead of data values (scores) when calculating the range from a frequency table.
- Forgetting to arrange data in ascending order before determining quartiles and median.
- Multiplying variance by \(k\) instead of \(k^2\) when all data values are multiplied by a constant \(k\).
- Changing the measure of dispersion when a constant is added to or subtracted from each data value. Dispersion does not change under addition or subtraction.
Misconceptions
A higher standard deviation means better performance.
Standard deviation measures dispersion, not performance level; a lower standard deviation indicates higher consistency or stability around the mean.
Adding an outlier increases the interquartile range significantly.
The interquartile range is relatively unaffected by outliers because it only evaluates the middle \(50\%\) of the sorted data.
When each data value is multiplied by \(k\), variance also multiplies by \(k\).
Variance is measured in squared units, so multiplying each data value by \(k\) causes the variance to be multiplied by \(k^2\).
Summary
- Measures of dispersion quantify the spread of data.
- Range is the difference between maximum and minimum values.
- Interquartile range (\(Q_3 - Q_1\)) measures middle-half dispersion and resists outliers.
- Variance and standard deviation describe deviations from the mean; lower values denote greater consistency.
- Adding or subtracting a constant does not alter dispersion, whereas multiplying by \(k\) scales range, IQR, and standard deviation by \(k\), and scales variance by \(k^2\).
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 The variance of a set of data is \(4.5\). If each value in the data set is multiplied by \(2\), what is the new variance?
2 Which measure of dispersion is most appropriate to describe data distribution when an outlier is present?
B. Fill in the Blanks
3 When each value of a set of data is added by a constant, the measures of dispersion remain .
4 In a box plot, the difference between the third quartile and the first quartile represents the range.
C. Drag and Drop: Match Each Measure to Its Definition
Drag each measure to the correct definition. On a phone, tap the answer choice first, then tap the definition box.
