Study guide
8.2 Measures of Spread: Range, IQR, Variance, & Standard Deviation
ACCUPLACER Quantitative Reasoning, Algebra, and Statistics Exam
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1 Chapter 1: Rational Numbers & Operations
1.1 Rational Number Arithmetic & Signed Numbers 1.2 Fractions, Decimals, & Mixed Numbers Operations 1.3 Absolute Value, Number Line, & Estimation
2 Chapter 2: Ratios, Proportions & Rates
2.1 Ratios, Unit Rates, & Dimensional Analysis 2.2 Proportions & Direct/Inverse Variation 2.3 Percent Applications, Markup, Discount, & Simple Interest
3 Chapter 3: Exponents & Scientific Notation
3.1 Integer Exponents & Laws of Exponents 3.2 Scientific Notation & Operations
4 Chapter 4: Algebraic Expressions
4.1 Evaluating & Translating Algebraic Expressions 4.2 Simplifying, Combining Like Terms, & Factoring
5 Chapter 5: Linear Equations & Inequalities
5.1 Solving Single-Variable Linear Equations 5.2 Solving & Graphing Linear Inequalities 5.3 Systems of Linear Equations
6 Chapter 6: Linear Applications & Graphing
6.1 Slope & Rate of Change 6.2 Forms of Linear Equations & Graphing 6.3 Linear Modeling & Inequality Graphs
7 Chapter 7: Probability & Set Theory
7.1 Fundamental Probability Concepts 7.2 Set Theory, Operations, & Venn Diagrams
8 Chapter 8: Descriptive Statistics & Data Displays
8.1 Measures of Center: Mean, Median, Mode, & Weighted Average 8.2 Measures of Spread: Range, IQR, Variance, & Standard Deviation 8.3 Data Displays: Histograms, Box Plots, & Scatter Plots
9 Chapter 9: Geometry Concepts & Measurement
9.1 2D Geometry: Perimeter, Area, & Circles 9.2 3D Geometry: Surface Area & Volume of Solids 9.3 Triangles, Angles, & the Pythagorean Theorem
8.2 Measures of Spread: Range, IQR, Variance, & Standard Deviation
Key Takeaways Explain This Topic AI
The range ( R = Maximum − Minimum R = \text{Maximum} - \text{Minimum} R = Maximum − Minimum ) quantifies the total span of a dataset in a single calculation but is exceptionally vulnerable to isolated outliers.
Quartiles divide an ordered dataset into four equal segments (25% each), establishing the Five-Number Summary: Minimum, Q_1 (25th percentile), Median/Q_2 (50th percentile), Q_3 (75th percentile), and Maximum.
The Interquartile Range ( IQR = Q 3 − Q 1 \text{IQR} = Q_3 - Q_1 IQR = Q 3 − Q 1 ) measures the spread of the middle 50% of the dataset and is resistant (robust) against extreme values.
The 1.5 × IQR 1.5 \times \text{IQR} 1.5 × IQR outlier rule establishes mathematical fences: Lower Fence = Q 1 − 1.5 × IQR \text{Lower Fence} = Q_1 - 1.5 \times \text{IQR} Lower Fence = Q 1 − 1.5 × IQR and Upper Fence = Q 3 + 1.5 × IQR \text{Upper Fence} = Q_3 + 1.5 \times \text{IQR} Upper Fence = Q 3 + 1.5 × IQR ; observations outside these fences are classified as outliers.
Standard deviation ( s = ∑ ( x i − x ˉ ) 2 n − 1 s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} s = n − 1 ∑ ( x i − x ˉ ) 2 ) measures average distance from the mean; shifting data ( x + c x + c x + c ) leaves s s s unchanged, while scaling data ( k ⋅ x k \cdot x k ⋅ x ) scales s s s by ∣ k ∣ |k| ∣ k ∣ .
Last updated: August 2026
The Critical Role of Measures of Dispersion (Spread)
While measures of central tendency identify the center or typical value of a dataset, measures of dispersion (spread) describe the variability, scatter, or degree of clustering among observations. Center and spread are complementary: knowing only the center of a distribution provides an incomplete picture.
Consider two manufacturing assembly lines producing steel bolts with an intended length of 50.0 mm 50.0\text{ mm} 50.0 mm :
Line A Bolt Lengths (mm) : { 49.9 , 50.0 , 50.0 , 50.0 , 50.1 } ⟹ x ˉ = 50.0 mm \{49.9, \; 50.0, \; 50.0, \; 50.0, \; 50.1\} \implies \bar{x} = 50.0\text{ mm} { 49.9 , 50.0 , 50.0 , 50.0 , 50.1 } ⟹ x ˉ = 50.0 mm (Extremely low dispersion; highly consistent quality control)
Line B Bolt Lengths (mm) : { 40.0 , 45.0 , 50.0 , 55.0 , 60.0 } ⟹ x ˉ = 50.0 mm \{40.0, \; 45.0, \; 50.0, \; 55.0, \; 60.0\} \implies \bar{x} = 50.0\text{ mm} { 40.0 , 45.0 , 50.0 , 55.0 , 60.0 } ⟹ x ˉ = 50.0 mm (Extremely high dispersion; severe manufacturing defect rate)
Both lines share the exact same mean ( 50.0 mm 50.0\text{ mm} 50.0 mm ) and median ( 50.0 mm 50.0\text{ mm} 50.0 mm ), but Line B exhibits extreme variability. The ACCUPLACER QAS exam tests four key measures of spread: Range , Quartiles & Interquartile Range (IQR) , Variance , and Standard Deviation .
Range: Measuring Total Span
The range ( R R R ) is the simplest measure of statistical dispersion, defined as the arithmetic difference between the maximum and minimum values in a dataset: R = Maximum − Minimum R = \text{Maximum} - \text{Minimum} R = Maximum − Minimum
Analytical Characteristics of Range
Ease of Computation : Calculated in a single subtraction step.
Extreme Sensitivity to Outliers : Because range depends entirely on the two extreme endpoints of the dataset, a single unusually large or small observation will artificially inflate the range, providing a distorted view of the dataset's overall variability.
Quartiles and the Five-Number Summary
Quartiles partition an ordered dataset into four equal segments, each containing exactly 25 % 25\% 25% of the observations.
First Quartile ( Q 1 Q_1 Q 1 / 25th Percentile) : The median of the lower half of the data; exactly 25 % 25\% 25% of observations lie at or below Q 1 Q_1 Q 1 , and 75 % 75\% 75% lie above it.
Second Quartile ( Q 2 Q_2 Q 2 / Median / 50th Percentile) : The overall median of the dataset; bisects the entire distribution into lower and upper halves ( 50 % 50\% 50% below, 50 % 50\% 50% above).
Third Quartile ( Q 3 Q_3 Q 3 / 75th Percentile) : The median of the upper half of the data; exactly 75 % 75\% 75% of observations lie at or below Q 3 Q_3 Q 3 , and 25 % 25\% 25% lie above it.
The Five-Number Summary
The Five-Number Summary provides a comprehensive numerical skeleton of any distribution: Five-Number Summary = ( Minimum , Q 1 , Median , Q 3 , Maximum ) \text{Five-Number Summary} = (\text{Minimum}, \; Q_1, \; \text{Median}, \; Q_3, \; \text{Maximum}) Five-Number Summary = ( Minimum , Q 1 , Median , Q 3 , Maximum )
Algorithm for Calculating Quartiles
Sort the entire dataset in ascending order.
Determine the overall Median ( Q 2 Q_2 Q 2 ) .
Divide the dataset into lower and upper halves :
When sample size n n n is ODD : Strictly exclude the median itself when forming the lower and upper halves.
When sample size n n n is EVEN : Split the dataset cleanly at the midpoint; include all observations up to the midpoint in the lower half and all observations above it in the upper half.
Calculate Q 1 Q_1 Q 1 as the median of the lower half.
Calculate Q 3 Q_3 Q 3 as the median of the upper half.
Worked Example 1: Odd Sample Size ( n = 11 n = 11 n = 11 )
Consider the sorted exam scores: 54 , 62 , 68 , 72 , 77 , 81 , 85 , 88 , 91 , 94 , 98 54, \; 62, \; 68, \; 72, \; 77, \; \mathbf{81}, \; 85, \; 88, \; 91, \; 94, \; 98 54 , 62 , 68 , 72 , 77 , 81 , 85 , 88 , 91 , 94 , 98
Median ( Q 2 Q_2 Q 2 ) : The 6th value is 81 81 81 .
Lower Half (5 values): { 54 , 62 , 68 , 72 , 77 } ⟹ Q 1 = 68 \{54, 62, \mathbf{68}, 72, 77\} \implies Q_1 = 68 { 54 , 62 , 68 , 72 , 77 } ⟹ Q 1 = 68 .
Upper Half (5 values): { 85 , 88 , 91 , 94 , 98 } ⟹ Q 3 = 91 \{85, 88, \mathbf{91}, 94, 98\} \implies Q_3 = 91 { 85 , 88 , 91 , 94 , 98 } ⟹ Q 3 = 91 .
Five-Number Summary : ( Min = 54 , Q 1 = 68 , Median = 81 , Q 3 = 91 , Max = 98 ) (\text{Min}=54, \; Q_1=68, \; \text{Median}=81, \; Q_3=91, \; \text{Max}=98) ( Min = 54 , Q 1 = 68 , Median = 81 , Q 3 = 91 , Max = 98 ) .
Worked Example 2: Even Sample Size ( n = 12 n = 12 n = 12 )
Consider the sorted response times (seconds): 12 , 15 , 18 , 21 , 24 , 27 , 31 , 36 , 40 , 45 , 49 , 58 12, \; 15, \; 18, \; 21, \; 24, \; \mathbf{27}, \; \mathbf{31}, \; 36, \; 40, \; 45, \; 49, \; 58 12 , 15 , 18 , 21 , 24 , 27 , 31 , 36 , 40 , 45 , 49 , 58
Median ( Q 2 Q_2 Q 2 ) : 27 + 31 2 = 29 seconds \frac{27 + 31}{2} = 29\text{ seconds} 2 27 + 31 = 29 seconds .
Lower Half (6 values): { 12 , 15 , 18 , 21 , 24 , 27 } ⟹ Q 1 = 18 + 21 2 = 19.5 \{12, 15, \mathbf{18}, \mathbf{21}, 24, 27\} \implies Q_1 = \frac{18 + 21}{2} = 19.5 { 12 , 15 , 18 , 21 , 24 , 27 } ⟹ Q 1 = 2 18 + 21 = 19.5 .
Upper Half (6 values): { 31 , 36 , 40 , 45 , 49 , 58 } ⟹ Q 3 = 40 + 45 2 = 42.5 \{31, 36, \mathbf{40}, \mathbf{45}, 49, 58\} \implies Q_3 = \frac{40 + 45}{2} = 42.5 { 31 , 36 , 40 , 45 , 49 , 58 } ⟹ Q 3 = 2 40 + 45 = 42.5 .
Five-Number Summary : ( Min = 12 , Q 1 = 19.5 , Median = 29 , Q 3 = 42.5 , Max = 58 ) (\text{Min}=12, \; Q_1=19.5, \; \text{Median}=29, \; Q_3=42.5, \; \text{Max}=58) ( Min = 12 , Q 1 = 19.5 , Median = 29 , Q 3 = 42.5 , Max = 58 ) .
The Interquartile Range (IQR): Robust Measure of Spread
The Interquartile Range (IQR) is the difference between the third and first quartiles, measuring the spread of the middle 50 % 50\% 50% of the dataset: IQR = Q 3 − Q 1 \text{IQR} = Q_3 - Q_1 IQR = Q 3 − Q 1
Analytical Properties of the IQR
Resistance to Outliers : Because the IQR focuses strictly on the central half of the distribution and completely discards the top 25 % 25\% 25% and bottom 25 % 25\% 25% of observations, extreme values do not affect its value.
Paired with the Median : In skewed distributions or datasets containing outliers, the median is the preferred measure of center and the IQR is the corresponding preferred measure of spread.
Outlier Identification: The 1.5 × IQR 1.5 \times \text{IQR} 1.5 × IQR Criterion (Tukey's Fences)
A data point is formally classified as a statistical outlier if it lies beyond either the lower or upper boundary fences: Lower Fence = Q 1 − 1.5 × IQR \text{Lower Fence} = Q_1 - 1.5 \times \text{IQR} Lower Fence = Q 1 − 1.5 × IQR Upper Fence = Q 3 + 1.5 × IQR \text{Upper Fence} = Q_3 + 1.5 \times \text{IQR} Upper Fence = Q 3 + 1.5 × IQR
Any observation x < Lower Fence x < \text{Lower Fence} x < Lower Fence is classified as a low outlier .
Any observation x > Upper Fence x > \text{Upper Fence} x > Upper Fence is classified as a high outlier . Visualizing Tukey's Outlier Fences:
- [Low Outliers] <--- |------- Q1 ========= Median ========= Q3 -------| ---> [High Outliers]
- Lower Fence Upper Fence
- (Q1 - 1.5*IQR) (Q3 + 1.5*IQR)
Step-by-Step Outlier Detection Algorithm:
Compute Q 1 Q_1 Q 1 and Q 3 Q_3 Q 3 .
Calculate IQR = Q 3 − Q 1 \text{IQR} = Q_3 - Q_1 IQR = Q 3 − Q 1 .
Multiply IQR by 1.5 1.5 1.5 to find the step value: Step = 1.5 × IQR \text{Step} = 1.5 \times \text{IQR} Step = 1.5 × IQR .
Subtract the step from Q 1 Q_1 Q 1 to establish the Lower Fence .
Add the step to Q 3 Q_3 Q 3 to establish the Upper Fence .
Flag any data point falling outside the interval [ Lower Fence , Upper Fence ] [\text{Lower Fence}, \text{Upper Fence}] [ Lower Fence , Upper Fence ] .
Variance and Standard Deviation ( s s s and σ \sigma σ )
Variance and standard deviation measure the typical or average squared distance that data points deviate from the arithmetic mean.
Mathematical Formulation
Sample Variance ( s 2 s^2 s 2 ) : s 2 = ∑ i = 1 n ( x i − x ˉ ) 2 n − 1 s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1} s 2 = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2
Sample Standard Deviation ( s s s ) : s = s 2 = ∑ i = 1 n ( x i − x ˉ ) 2 n − 1 s = \sqrt{s^2} = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}} s = s 2 = n − 1 ∑ i = 1 n ( x i − x ˉ ) 2
Population Standard Deviation ( σ \sigma σ ) : σ = ∑ i = 1 N ( x i − μ ) 2 N \sigma = \sqrt{\frac{\sum_{i=1}^N (x_i - \mu)^2}{N}} σ = N ∑ i = 1 N ( x i − μ ) 2
(Note: The denominator n − 1 n - 1 n − 1 in the sample standard deviation is known as Bessel's correction, which provides an unbiased estimator of the population variance.)
Conceptual Properties for the ACCUPLACER Test
Non-Negativity : s ≥ 0 s \ge 0 s ≥ 0 . Standard deviation can never be negative.
Zero Spread : s = 0 s = 0 s = 0 if and only if all values in the dataset are identical (e.g., in { 8 , 8 , 8 , 8 } \{8, 8, 8, 8\} { 8 , 8 , 8 , 8 } , s = 0 s = 0 s = 0 ).
Clustering vs. Dispersion :
Data points tightly clustered around the mean produce a small standard deviation .
Data points widely dispersed across a broad spectrum produce a large standard deviation .
Sensitivity to Outliers : Like the arithmetic mean, standard deviation is non-resistant; squaring deviations magnifies the influence of extreme values.
Comparing Spread Visually : On the ACCUPLACER exam, you are often asked to compare the standard deviations of two dot plots or histograms without computing formulas. Look at how far points are from the center: datasets with more points in the outer tails have larger standard deviations.
Linear Transformations and Their Effects on Spread
Understanding how arithmetic transformations affect summary statistics is a frequently tested conceptual topic:
1. Shift Transformation: Adding or Subtracting a Constant ( x new = x + c x_{\text{new}} = x + c x new = x + c )
Effect on Center : The mean and median increase by c c c ( x ˉ new = x ˉ + c \bar{x}_{\text{new}} = \bar{x} + c x ˉ new = x ˉ + c ).
Effect on Spread : The range, IQR, variance, and standard deviation remain completely unchanged ( Δ s = 0 , Δ IQR = 0 \Delta s = 0, \Delta \text{IQR} = 0 Δ s = 0 , Δ IQR = 0 ). Adding a constant shifts every point by the same amount, preserving the exact distances between all points.
2. Scale Transformation: Multiplying or Dividing by a Constant ( x new = k ⋅ x x_{\text{new}} = k \cdot x x new = k ⋅ x , where k > 0 k > 0 k > 0 )
Effect on Center : The mean and median are multiplied by k k k .
Effect on Spread : The range, IQR, and standard deviation are multiplied by k k k ( s new = k ⋅ s s_{\text{new}} = k \cdot s s new = k ⋅ s ). The variance is multiplied by k 2 k^2 k 2 .
Statistic Adding Constant c c c ( x + c x + c x + c ) Multiplying by Constant k > 0 k > 0 k > 0 ( k ⋅ x k \cdot x k ⋅ x )
- Mean ( x ˉ \bar{x} x ˉ ) x ˉ + c \bar{x} + c x ˉ + c k ⋅ x ˉ k \cdot \bar{x} k ⋅ x ˉ
- Median ( M M M ) M + c M + c M + c k ⋅ M k \cdot M k ⋅ M
- Range ( R R R ) Unchanged ( R R R ) k ⋅ R k \cdot R k ⋅ R
- IQR Unchanged ( IQR \text{IQR} IQR ) k ⋅ IQR k \cdot \text{IQR} k ⋅ IQR
- Standard Deviation ( s s s ) Unchanged ( s s s ) k ⋅ s k \cdot s k ⋅ s
- Variance ( s 2 s^2 s 2 ) Unchanged ( s 2 s^2 s 2 ) k 2 ⋅ s 2 k^2 \cdot s^2 k 2 ⋅ s 2
- The Empirical Rule (68–95–99.7 Rule) for Normal Distributions
For symmetric, bell-shaped (normal) distributions, the Empirical Rule defines the exact percentage of observations falling within standard deviation intervals around the mean:
≈ 68 % \approx 68\% ≈ 68% of observations fall within 1 1 1 standard deviation of the mean: [ x ˉ − 1 s , x ˉ + 1 s ] [\bar{x} - 1s, \; \bar{x} + 1s] [ x ˉ − 1 s , x ˉ + 1 s ]
≈ 95 % \approx 95\% ≈ 95% of observations fall within 2 2 2 standard deviations of the mean: [ x ˉ − 2 s , x ˉ + 2 s ] [\bar{x} - 2s, \; \bar{x} + 2s] [ x ˉ − 2 s , x ˉ + 2 s ]
≈ 99.7 % \approx 99.7\% ≈ 99.7% of observations fall within 3 3 3 standard deviations of the mean: [ x ˉ − 3 s , x ˉ + 3 s ] [\bar{x} - 3s, \; \bar{x} + 3s] [ x ˉ − 3 s , x ˉ + 3 s ] Normal Distribution Symmetric Partitions:
- |------- 68% -------|
- |--------------- 95% ---------------|
- |------------------- 99.7% -------------------|
- 2.35% 13.5% 34% 34% 13.5% 2.35%
- --|-------|-------|-------|-------|-------|-------|---
- μ-3σ μ-2σ μ-1σ μ μ+1σ μ+2σ μ+3σ
- Key Symmetric Sub-Intervals
- Between x ˉ \bar{x} x ˉ and x ˉ + 1 s \bar{x} + 1s x ˉ + 1 s : 34 % 34\% 34%
- Between x ˉ + 1 s \bar{x} + 1s x ˉ + 1 s and x ˉ + 2 s \bar{x} + 2s x ˉ + 2 s : 13.5 % 13.5\% 13.5%
- Between x ˉ + 2 s \bar{x} + 2s x ˉ + 2 s and x ˉ + 3 s \bar{x} + 3s x ˉ + 3 s : 2.35 % 2.35\% 2.35%
- Greater than x ˉ + 3 s \bar{x} + 3s x ˉ + 3 s : 0.15 % 0.15\% 0.15%
- Step-by-Step Multi-Step Worked Examples
- Worked Example 1: Full Five-Number Summary, IQR, and Outlier Analysis
An analytical laboratory measures the concentration of a chemical compound (in parts per million, ppm) across 12 12 12 samples: 14 , 18 , 22 , 25 , 28 , 31 , 33 , 37 , 41 , 44 , 48 , 88 14, \; 18, \; 22, \; 25, \; 28, \; 31, \; 33, \; 37, \; 41, \; 44, \; 48, \; 88 14 , 18 , 22 , 25 , 28 , 31 , 33 , 37 , 41 , 44 , 48 , 88
Step 1: Determine the Five-Number Summary
Ordered dataset ( n = 12 n = 12 n = 12 ):
- Minimum = 14 \text{Minimum} = 14 Minimum = 14
- Median = 31 + 33 2 = 32 \text{Median} = \frac{31 + 33}{2} = 32 Median = 2 31 + 33 = 32
Lower half: { 14 , 18 , 22 , 25 , 28 , 31 } ⟹ Q 1 = 22 + 25 2 = 23.5 \{14, 18, 22, 25, 28, 31\} \implies Q_1 = \frac{22 + 25}{2} = 23.5 { 14 , 18 , 22 , 25 , 28 , 31 } ⟹ Q 1 = 2 22 + 25 = 23.5
Upper half: { 33 , 37 , 41 , 44 , 48 , 88 } ⟹ Q 3 = 41 + 44 2 = 42.5 \{33, 37, 41, 44, 48, 88\} \implies Q_3 = \frac{41 + 44}{2} = 42.5 { 33 , 37 , 41 , 44 , 48 , 88 } ⟹ Q 3 = 2 41 + 44 = 42.5
Maximum = 88 \text{Maximum} = 88 Maximum = 88
Five-Number Summary : ( 14 , 23.5 , 32 , 42.5 , 88 ) (14, \; 23.5, \; 32, \; 42.5, \; 88) ( 14 , 23.5 , 32 , 42.5 , 88 )
Step 2: Calculate the Interquartile Range IQR = Q 3 − Q 1 = 42.5 − 23.5 = 19.0 ppm \text{IQR} = Q_3 - Q_1 = 42.5 - 23.5 = 19.0\text{ ppm} IQR = Q 3 − Q 1 = 42.5 − 23.5 = 19.0 ppm
Step 3: Calculate Outlier Fences Step = 1.5 × IQR = 1.5 × 19.0 = 28.5 \text{Step} = 1.5 \times \text{IQR} = 1.5 \times 19.0 = 28.5 Step = 1.5 × IQR = 1.5 × 19.0 = 28.5 Lower Fence = Q 1 − 28.5 = 23.5 − 28.5 = − 5.0 \text{Lower Fence} = Q_1 - 28.5 = 23.5 - 28.5 = -5.0 Lower Fence = Q 1 − 28.5 = 23.5 − 28.5 = − 5.0 Upper Fence = Q 3 + 28.5 = 42.5 + 28.5 = 71.0 \text{Upper Fence} = Q_3 + 28.5 = 42.5 + 28.5 = 71.0 Upper Fence = Q 3 + 28.5 = 42.5 + 28.5 = 71.0
Step 4: Identify Outliers
All observations are ≥ − 5.0 \ge -5.0 ≥ − 5.0 , so there are no low outliers.
The observation 88 88 88 exceeds the Upper Fence ( 88 > 71.0 88 > 71.0 88 > 71.0 ).
Thus, 88 ppm 88\text{ ppm} 88 ppm is a confirmed high outlier .
Worked Example 2: Outlier Screening in Hospital Wait Times
An emergency department evaluates triage wait times with Q 1 = 24 minutes Q_1 = 24\text{ minutes} Q 1 = 24 minutes and Q 3 = 56 minutes Q_3 = 56\text{ minutes} Q 3 = 56 minutes . An auditor investigates four specific patient wait times: 8 minutes 8\text{ minutes} 8 minutes , 75 minutes 75\text{ minutes} 75 minutes , 102 minutes 102\text{ minutes} 102 minutes , and 110 minutes 110\text{ minutes} 110 minutes . Which of these wait times are statistical outliers under the 1.5 × IQR 1.5 \times \text{IQR} 1.5 × IQR rule?
IQR = Q 3 − Q 1 = 56 − 24 = 32 minutes \text{IQR} = Q_3 - Q_1 = 56 - 24 = 32\text{ minutes} IQR = Q 3 − Q 1 = 56 − 24 = 32 minutes .
Step = 1.5 × 32 = 48 minutes \text{Step} = 1.5 \times 32 = 48\text{ minutes} Step = 1.5 × 32 = 48 minutes .
Lower Fence = 24 − 48 = − 24 minutes \text{Lower Fence} = 24 - 48 = -24\text{ minutes} Lower Fence = 24 − 48 = − 24 minutes (effective lower limit is 0 0 0 ).
Upper Fence = 56 + 48 = 104 minutes \text{Upper Fence} = 56 + 48 = 104\text{ minutes} Upper Fence = 56 + 48 = 104 minutes .
Evaluating candidates against the non-outlier interval [ 0 , 104 ] [0, 104] [ 0 , 104 ] :
8 min 8\text{ min} 8 min : Not an outlier ( 8 ≤ 104 8 \le 104 8 ≤ 104 ).
75 min 75\text{ min} 75 min : Not an outlier ( 75 ≤ 104 75 \le 104 75 ≤ 104 ).
102 min 102\text{ min} 102 min : Not an outlier ( 102 ≤ 104 102 \le 104 102 ≤ 104 ).
110 min 110\text{ min} 110 min : Outlier ( 110 > 104 110 > 104 110 > 104 ).
Only the 110 -minute 110\text{-minute} 110 -minute wait time is an outlier.
Worked Example 3: Empirical Rule Application in Standardized Quality Testing
A factory produces lightbulbs whose operational lifespans follow a normal distribution with a mean μ = 1 , 200 hours \mu = 1,200\text{ hours} μ = 1 , 200 hours and standard deviation σ = 80 hours \sigma = 80\text{ hours} σ = 80 hours . In a production batch of 10 , 000 10,000 10 , 000 lightbulbs, approximately how many lightbulbs are expected to last between 1 , 040 hours 1,040\text{ hours} 1 , 040 hours and 1 , 280 hours 1,280\text{ hours} 1 , 280 hours ?
Express the bounds in terms of standard deviations from the mean:
Lower bound: 1 , 040 = 1 , 200 − 160 = μ − 2 σ 1,040 = 1,200 - 160 = \mu - 2\sigma 1 , 040 = 1 , 200 − 160 = μ − 2 σ
Upper bound: 1 , 280 = 1 , 200 + 80 = μ + 1 σ 1,280 = 1,200 + 80 = \mu + 1\sigma 1 , 280 = 1 , 200 + 80 = μ + 1 σ
Sum the known Empirical Rule percentages for these regions:
- Between μ − 2 σ \mu - 2\sigma μ − 2 σ and μ − 1 σ \mu - 1\sigma μ − 1 σ : 13.5 % 13.5\% 13.5%
- Between μ − 1 σ \mu - 1\sigma μ − 1 σ and μ \mu μ : 34 % 34\% 34%
- Between μ \mu μ and μ + 1 σ \mu + 1\sigma μ + 1 σ : 34 % 34\% 34%
Total Percentage = 13.5 % + 34 % + 34 % = 81.5 % \text{Total Percentage} = 13.5\% + 34\% + 34\% = 81.5\% Total Percentage = 13.5% + 34% + 34% = 81.5%
Calculate expected bulb count: 10 , 000 × 0.815 = 8 , 150 lightbulbs 10,000 \times 0.815 = 8,150\text{ lightbulbs} 10 , 000 × 0.815 = 8 , 150 lightbulbs
Common Pitfalls & ACCUPLACER Exam Traps
Including the Median in Quartile Halves When n n n is Odd : Never include the central median element in either the lower or upper halves when finding Q 1 Q_1 Q 1 and Q 3 Q_3 Q 3 .
Believing Addition Shifts Standard Deviation : If every employee receives a $ 3 , 000 \$3,000 $3 , 000 cost-of-living salary increase, the mean salary increases by $ 3 , 000 \$3,000 $3 , 000 , but standard deviation and IQR remain completely unchanged .
Confusing Range with IQR : Range is Max − Min \text{Max} - \text{Min} Max − Min (all data), whereas IQR is Q 3 − Q 1 Q_3 - Q_1 Q 3 − Q 1 (middle 50 % 50\% 50% ).
Assuming Negative Standard Deviation : Standard deviation is the principal square root of non-negative squared deviations and can never be negative .