Measures of Dispersion
Master range, interquartile range (IQR), standard deviation, and box-and-whisker plots. Learn how adding or multiplying constants transforms your data.
1. 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.
2. 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. |
Click the graph to explore it interactively →
💡 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.