8.2 Measures of Spread: Range, IQR, Variance, & Standard Deviation

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 $50.0\text{ mm}$:

• Line A Bolt Lengths (mm): $\{49.9, \; 50.0, \; 50.0, \; 50.0, \; 50.1\} \implies \bar{x} = 50.0\text{ mm}$ (Extremely low dispersion; highly consistent quality control)
• Line B Bolt Lengths (mm): $\{40.0, \; 45.0, \; 50.0, \; 55.0, \; 60.0\} \implies \bar{x} = 50.0\text{ mm}$ (Extremely high dispersion; severe manufacturing defect rate)

Both lines share the exact same mean ($50.0\text{ mm}$) and median ($50.0\text{ mm}$), 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.

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Range: Measuring Total Span

The range ($R$) is the simplest measure of statistical dispersion, defined as the arithmetic difference between the maximum and minimum values in a dataset:

$R = \text{Maximum} - \text{Minimum}$

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.

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Quartiles and the Five-Number Summary

Quartiles partition an ordered dataset into four equal segments, each containing exactly $25\%$ of the observations.

1. First Quartile ($Q_1$ / 25th Percentile): The median of the lower half of the data; exactly $25\%$ of observations lie at or below $Q_1$, and $75\%$ lie above it.
2. Second Quartile ($Q_2$ / Median / 50th Percentile): The overall median of the dataset; bisects the entire distribution into lower and upper halves ($50\%$ below, $50\%$ above).
3. Third Quartile ($Q_3$ / 75th Percentile): The median of the upper half of the data; exactly $75\%$ of observations lie at or below $Q_3$, and $25\%$ lie above it.

The Five-Number Summary

The Five-Number Summary provides a comprehensive numerical skeleton of any distribution:

$\text{Five-Number Summary} = (\text{Minimum}, \; Q_1, \; \text{Median}, \; Q_3, \; \text{Maximum})$

Algorithm for Calculating Quartiles

1. Sort the entire dataset in ascending order.
2. Determine the overall Median ($Q_2$).
3. Divide the dataset into lower and upper halves:
   • When sample size $n$ is ODD: Strictly exclude the median itself when forming the lower and upper halves.
   • When sample size $n$ 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.
4. Calculate $Q_1$ as the median of the lower half.
5. Calculate $Q_3$ as the median of the upper half.

Worked Example 1: Odd Sample Size ($n = 11$)

Consider the sorted exam scores: $54, \; 62, \; 68, \; 72, \; 77, \; \mathbf{81}, \; 85, \; 88, \; 91, \; 94, \; 98$

• Median ($Q_2$): The 6th value is $81$.
• Lower Half (5 values): $\{54, 62, \mathbf{68}, 72, 77\} \implies Q_1 = 68$.
• Upper Half (5 values): $\{85, 88, \mathbf{91}, 94, 98\} \implies Q_3 = 91$.
• Five-Number Summary: $(\text{Min}=54, \; Q_1=68, \; \text{Median}=81, \; Q_3=91, \; \text{Max}=98)$.

Worked Example 2: Even Sample Size ($n = 12$)

Consider the sorted response times (seconds): $12, \; 15, \; 18, \; 21, \; 24, \; \mathbf{27}, \; \mathbf{31}, \; 36, \; 40, \; 45, \; 49, \; 58$

• Median ($Q_2$): $\frac{27 + 31}{2} = 29\text{ seconds}$.
• Lower Half (6 values): $\{12, 15, \mathbf{18}, \mathbf{21}, 24, 27\} \implies Q_1 = \frac{18 + 21}{2} = 19.5$.
• Upper Half (6 values): $\{31, 36, \mathbf{40}, \mathbf{45}, 49, 58\} \implies Q_3 = \frac{40 + 45}{2} = 42.5$.
• Five-Number Summary: $(\text{Min}=12, \; Q_1=19.5, \; \text{Median}=29, \; Q_3=42.5, \; \text{Max}=58)$.

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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 $50\%$ of the dataset:

$\text{IQR} = Q_3 - Q_1$

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 $25\%$ and bottom $25\%$ 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.

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Outlier Identification: The $1.5 \times \text{IQR}$ 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:

$\text{Lower Fence} = Q_1 - 1.5 \times \text{IQR}$ $\text{Upper Fence} = Q_3 + 1.5 \times \text{IQR}$

• Any observation $x < \text{Lower Fence}$ is classified as a low outlier.
• Any observation $x > \text{Upper Fence}$ 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:

1. Compute $Q_1$ and $Q_3$.
2. Calculate $\text{IQR} = Q_3 - Q_1$.
3. Multiply IQR by $1.5$ to find the step value: $\text{Step} = 1.5 \times \text{IQR}$.
4. Subtract the step from $Q_1$ to establish the Lower Fence.
5. Add the step to $Q_3$ to establish the Upper Fence.
6. Flag any data point falling outside the interval $[\text{Lower Fence}, \text{Upper Fence}]$.

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Variance and Standard Deviation ($s$ and $\sigma$)

Variance and standard deviation measure the typical or average squared distance that data points deviate from the arithmetic mean.

Mathematical Formulation

• Sample Variance ($s^2$): $s^2 = \frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}$
• Sample Standard Deviation ($s$): $s = \sqrt{s^2} = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}}$
• Population Standard Deviation ($\sigma$): $\sigma = \sqrt{\frac{\sum_{i=1}^N (x_i - \mu)^2}{N}}$

(Note: The denominator $n - 1$ 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

1. Non-Negativity: $s \ge 0$. Standard deviation can never be negative.
2. Zero Spread: $s = 0$ if and only if all values in the dataset are identical (e.g., in $\{8, 8, 8, 8\}$, $s = 0$).
3. 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.
4. Sensitivity to Outliers: Like the arithmetic mean, standard deviation is non-resistant; squaring deviations magnifies the influence of extreme values.
5. 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.

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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 ($x_{\text{new}} = x + c$)

• Effect on Center: The mean and median increase by $c$ ($\bar{x}_{\text{new}} = \bar{x} + c$).
• Effect on Spread: The range, IQR, variance, and standard deviation remain completely unchanged ($\Delta s = 0, \Delta \text{IQR} = 0$). 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 ($x_{\text{new}} = k \cdot x$, where $k > 0$)

• Effect on Center: The mean and median are multiplied by $k$.
• Effect on Spread: The range, IQR, and standard deviation are multiplied by $k$ ($s_{\text{new}} = k \cdot s$). The variance is multiplied by $k^2$.

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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:

• $\approx 68\%$ of observations fall within $1$ standard deviation of the mean: $[\bar{x} - 1s, \; \bar{x} + 1s]$
• $\approx 95\%$ of observations fall within $2$ standard deviations of the mean: $[\bar{x} - 2s, \; \bar{x} + 2s]$
• $\approx 99.7\%$ of observations fall within $3$ standard deviations of the mean: $[\bar{x} - 3s, \; \bar{x} + 3s]$

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 $\bar{x}$ and $\bar{x} + 1s$: $34\%$
• Between $\bar{x} + 1s$ and $\bar{x} + 2s$: $13.5\%$
• Between $\bar{x} + 2s$ and $\bar{x} + 3s$: $2.35\%$
• Greater than $\bar{x} + 3s$: $0.15\%$

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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 $12$ samples: $14, \; 18, \; 22, \; 25, \; 28, \; 31, \; 33, \; 37, \; 41, \; 44, \; 48, \; 88$

Step 1: Determine the Five-Number Summary

• Ordered dataset ($n = 12$):
• $\text{Minimum} = 14$
• $\text{Median} = \frac{31 + 33}{2} = 32$
• Lower half: $\{14, 18, 22, 25, 28, 31\} \implies Q_1 = \frac{22 + 25}{2} = 23.5$
• Upper half: $\{33, 37, 41, 44, 48, 88\} \implies Q_3 = \frac{41 + 44}{2} = 42.5$
• $\text{Maximum} = 88$
• Five-Number Summary: $(14, \; 23.5, \; 32, \; 42.5, \; 88)$

Step 2: Calculate the Interquartile Range $\text{IQR} = Q_3 - Q_1 = 42.5 - 23.5 = 19.0\text{ ppm}$

Step 3: Calculate Outlier Fences $\text{Step} = 1.5 \times \text{IQR} = 1.5 \times 19.0 = 28.5$ $\text{Lower Fence} = Q_1 - 28.5 = 23.5 - 28.5 = -5.0$ $\text{Upper Fence} = Q_3 + 28.5 = 42.5 + 28.5 = 71.0$

Step 4: Identify Outliers

• All observations are $\ge -5.0$, so there are no low outliers.
• The observation $88$ exceeds the Upper Fence ($88 > 71.0$).
• Thus, $88\text{ ppm}$ is a confirmed high outlier.

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Worked Example 2: Outlier Screening in Hospital Wait Times

An emergency department evaluates triage wait times with $Q_1 = 24\text{ minutes}$ and $Q_3 = 56\text{ minutes}$. An auditor investigates four specific patient wait times: $8\text{ minutes}$, $75\text{ minutes}$, $102\text{ minutes}$, and $110\text{ minutes}$. Which of these wait times are statistical outliers under the $1.5 \times \text{IQR}$ rule?

1. $\text{IQR} = Q_3 - Q_1 = 56 - 24 = 32\text{ minutes}$.
2. $\text{Step} = 1.5 \times 32 = 48\text{ minutes}$.
3. $\text{Lower Fence} = 24 - 48 = -24\text{ minutes}$ (effective lower limit is $0$).
4. $\text{Upper Fence} = 56 + 48 = 104\text{ minutes}$.
5. Evaluating candidates against the non-outlier interval $[0, 104]$:
   • $8\text{ min}$: Not an outlier ($8 \le 104$).
   • $75\text{ min}$: Not an outlier ($75 \le 104$).
   • $102\text{ min}$: Not an outlier ($102 \le 104$).
   • $110\text{ min}$: Outlier ($110 > 104$).

Only the $110\text{-minute}$ wait time is an outlier.

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Worked Example 3: Empirical Rule Application in Standardized Quality Testing

A factory produces lightbulbs whose operational lifespans follow a normal distribution with a mean $\mu = 1,200\text{ hours}$ and standard deviation $\sigma = 80\text{ hours}$. In a production batch of $10,000$ lightbulbs, approximately how many lightbulbs are expected to last between $1,040\text{ hours}$ and $1,280\text{ hours}$?

1. Express the bounds in terms of standard deviations from the mean:
   • Lower bound: $1,040 = 1,200 - 160 = \mu - 2\sigma$
   • Upper bound: $1,280 = 1,200 + 80 = \mu + 1\sigma$
2. Sum the known Empirical Rule percentages for these regions:
   • Between $\mu - 2\sigma$ and $\mu - 1\sigma$: $13.5\%$
   • Between $\mu - 1\sigma$ and $\mu$: $34\%$
   • Between $\mu$ and $\mu + 1\sigma$: $34\%$
   • $\text{Total Percentage} = 13.5\% + 34\% + 34\% = 81.5\%$
3. Calculate expected bulb count: $10,000 \times 0.815 = 8,150\text{ lightbulbs}$

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Common Pitfalls & ACCUPLACER Exam Traps

1. Including the Median in Quartile Halves When $n$ is Odd: Never include the central median element in either the lower or upper halves when finding $Q_1$ and $Q_3$.
2. Believing Addition Shifts Standard Deviation: If every employee receives a $\$3,000$ cost-of-living salary increase, the mean salary increases by $\$3,000$, but standard deviation and IQR remain completely unchanged.
3. Confusing Range with IQR: Range is $\text{Max} - \text{Min}$ (all data), whereas IQR is $Q_3 - Q_1$ (middle $50\%$).
4. Assuming Negative Standard Deviation: Standard deviation is the principal square root of non-negative squared deviations and can never be negative.