Measures of Dispersion
Master range, interquartile range (IQR), standard deviation, and box-and-whisker plots. Learn how adding or multiplying constants transforms your data.
Click the graph to explore it interactively →
The Core Concept
Spread vs. Middle: While central tendencies (Mean, Median, Mode) tell you where the "center" of your data lies, Measures of Dispersion describe how "spread out" or consistent your data is. A low dispersion means the data values are packed closely together; a high dispersion means they are scattered far apart.
DSE Measures of Dispersion Cheat Sheet — 5 Key Topics
Scan this compact reference table to review every essential concept, step, and shortcut in under 10 seconds.
| Topic | Key Concept & Steps | Exam Shortcuts & Tips |
|---|---|---|
| 1. Range | Range = Maximum − Minimum | Heavily affected by extreme outliers. Only use it when the distribution has no extreme values. |
| 2. Interquartile Range (IQR) | IQR = Q₃ − Q₁(where Q₁ = lower quartile and Q₃ = upper quartile) | Highly resistant to outliers. Displays the spread of the middle 50% of the data. |
| 3. Standard Deviation (SD) | SD (σ) = √(Variance) | Calculator Hack: Always use your Casio calculator's built-in SD Mode to calculate the mean (x̄) and standard deviation (σx) directly. |
| 4. Box-and-Whisker Plot | Visualizes the 5-number summary:Min, Q₁, Median, Q₃, Max | • IQR: Read as the width of the box (Q₃ − Q₁).• Skewness: If the right whisker is longer, the data is positively skewed. |
| 5. Data Transformations | If each data point xᵢ → a · xᵢ + b:• Mean / Med / Mode: Multiply by a, add b. • Range / IQR / SD: Multiply by |a| (adding b has no effect!). • Variance: Multiply by a². | Adding a constant never changes dispersion! Multiplying changes dispersion by the absolute value of the multiplier. |
💡 Exam Traps & Secrets
⚠️ The Shift-has-No-Effect TrapAdding or subtracting a constant to every data point never changes the dispersion. Rule: If every student gets +10 marks on an exam, the Mean increases by 10, but the Range, IQR, Standard Deviation, and Variance remain exactly the same because the spread of the marks has not changed.
⚠️ SD vs. Variance MultipliersIf every value in a dataset is multiplied by 3:
• The Standard Deviation is multiplied by 3.
• The Variance is multiplied by 3² = 9 (since
• The Standard Deviation is multiplied by 3.
• The Variance is multiplied by 3² = 9 (since
Variance = SD²).⚠️ Comparing Two Distributions (DSE Compulsory Written Questions)When asked to compare two distributions (e.g., Class A vs. Class B), you must write two distinct comparison statements:
1. Compare Performance (Average): Compare the Means or Medians. (e.g., "Since the mean of Class A is higher, Class A performed better on average.")
2. Compare Consistency (Spread): Compare the Standard Deviations or IQRs. (e.g., "Since the SD of Class B is lower, the performance of Class B is more consistent.")
1. Compare Performance (Average): Compare the Means or Medians. (e.g., "Since the mean of Class A is higher, Class A performed better on average.")
2. Compare Consistency (Spread): Compare the Standard Deviations or IQRs. (e.g., "Since the SD of Class B is lower, the performance of Class B is more consistent.")
✏️ Self-Test: Quick Interactive Practice
How to study: Click any question below to instantly load it on the coordinate grid and check your steps.
(Basic ungrouped data calculations)Given the dataset
{12, 15, 15, 18, 20, 22, 25, 30}, find the mean, median, IQR, and standard deviation.(Box plot interpretation)A box-and-whisker plot shows
Q₁ = 20, Median = 30, and Q₃ = 45. Find the IQR and describe the skewness of the distribution.(Applying data transformation laws)The standard deviation of a set of test scores is 4.5. If 5 marks are subtracted from each student's score, and the results are then doubled, find the new standard deviation and the new variance.
Measures of Dispersion Exercises: DSE-Style Practice
DSE-style practice sets from the SketchMath question bank, with solutions for every MC and long question.
Measures of Dispersion · Form 5
71 MC questions · 19 long questions
Frequently Asked Questions
Write two separate sentences: one comparing central tendency (which has a higher mean or median), and one comparing dispersion (which has a higher IQR or standard deviation). Award yourself one mark per correct statement. Always state which class performed better and which was more consistent.
Adding/subtracting a constant: Mean/Median/Mode shift by that constant, but Range/IQR/SD/Variance are unchanged. Multiplying by a constant a: Mean/Median/Mode multiply by a; Range/IQR/SD multiply by |a|; Variance multiplies by a². Remember: adding never changes spread!
Use your calculator's built-in SD Mode. Enter the data points, and the calculator directly outputs the mean (x̄), population standard deviation (σx), and sample standard deviation (sx). Always check whether the question asks for population or sample SD.