Describing Data with Numbers—Spread

Two students might both have an 85% average in a class, but their journeys could be completely different. One student consistently scores between 83% and 87% on every assignment—steady and predictable. The other swings wildly: 65% one week, 100% the next, then 72%, then 95%. Same average, completely different stories. If you were betting on who would score at least 80% on the next test, which student would you trust more?

This is exactly why knowing the center of your data is not enough. You also need to know how spread out the data is—whether values cluster tightly around the center or scatter widely. This chapter gives you the tools to measure and describe that spread. Whether you are evaluating the consistency of a manufacturing process, comparing the volatility of investments, or understanding why some test score distributions are “tighter” than others, measures of spread tell the part of the story that averages leave out.

Core Concepts

Why Center Alone Is Not Enough

Imagine two pizza delivery services. Both advertise an average delivery time of 30 minutes. But Service A consistently delivers in 28–32 minutes, while Service B ranges from 15 to 45 minutes. Which would you rather order from?

The average tells you what to expect “on average,” but the spread tells you how reliable that average is. A small spread means consistency—you can count on getting something close to the average. A large spread means variability—the average might be misleading because individual values can be far from it.

Statisticians use several measures to quantify spread, each with its own strengths. We will start with the simplest and work our way up to more sophisticated tools.

Range: Simple but Limited

The range is the easiest measure of spread to calculate: just subtract the smallest value from the largest.

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

If test scores in a class range from 62 to 98, the range is $98 - 62 = 36$ points.

The range gives you a quick sense of the total spread, but it has a major weakness: it depends entirely on just two values—the extreme ones. One unusual score (an outlier) can dramatically change the range even if most of the data is tightly clustered.

The strength of the range: It is fast and easy to calculate.

The weakness of the range: It is extremely sensitive to outliers and ignores everything between the minimum and maximum.

Quartiles: Dividing Data into Fourths

To get a better picture of how data spreads, we can divide it into four equal parts using quartiles. Think of quartiles as checkpoints that tell you where 25%, 50%, and 75% of your data falls.

• $Q_1$ (First Quartile): 25% of the data falls below this value
• $Q_2$ (Second Quartile): 50% of the data falls below this value (this is the median)
• $Q_3$ (Third Quartile): 75% of the data falls below this value

To find quartiles:

1. Arrange your data in order from smallest to largest
2. Find the median ($Q_2$)—this splits the data in half
3. Find $Q_1$ as the median of the lower half (values below $Q_2$)
4. Find $Q_3$ as the median of the upper half (values above $Q_2$)

If you have ever seen a standardized test report that says “You scored in the 75th percentile,” you scored at or above $Q_3$—better than 75% of test-takers.

The Five-Number Summary

The five-number summary gives you a quick snapshot of your data’s distribution using five key values:

1. Minimum: The smallest value
2. $Q_1$: The first quartile (25th percentile)
3. Median ($Q_2$): The middle value (50th percentile)
4. $Q_3$: The third quartile (75th percentile)
5. Maximum: The largest value

These five numbers tell you where the data starts, where the middle 50% lies, and where it ends. They form the basis of box plots (also called box-and-whisker plots), which are powerful visual tools for comparing distributions.

Interquartile Range (IQR): The Middle 50%

The Interquartile Range (IQR) measures the spread of the middle 50% of your data—the range between the first and third quartiles.

$$\text{IQR} = Q_3 - Q_1$$

Why focus on the middle 50%? Because it ignores the extreme values at both ends, making the IQR resistant to outliers. No matter how extreme your minimum or maximum values are, the IQR stays stable.

The strength of the IQR: It is resistant to outliers—extreme values do not affect it.

The weakness of the IQR: It ignores 50% of your data (the upper and lower quarters).

The IQR tells you the range you would expect a “typical” middle observation to fall within.

Identifying Outliers Using the 1.5×IQR Rule

How do you know if a value is an outlier? While there is no absolute definition, the 1.5×IQR rule provides a commonly used standard.

A value is considered a potential outlier if it falls:

• Below $Q_1 - 1.5 \times \text{IQR}$, or
• Above $Q_3 + 1.5 \times \text{IQR}$

These boundaries are sometimes called the fences. Any value outside the fences is flagged as unusually extreme.

For example, if $Q_1 = 20$, $Q_3 = 40$, then $\text{IQR} = 20$.

• Lower fence: $20 - 1.5(20) = 20 - 30 = -10$
• Upper fence: $40 + 1.5(20) = 40 + 30 = 70$

Any value below $-10$ or above $70$ would be considered an outlier.

This rule is not about declaring values “wrong”—outliers can be perfectly valid data points. But identifying them helps you understand your data better and decide how to handle unusual observations.

Variance: Average Squared Distance from the Mean

While range and IQR give useful information, they do not use every data point. Variance does—it measures how far, on average, each value is from the mean.

The idea is simple: find how far each value deviates from the mean, then average those deviations. But there is a catch—some deviations are positive (above the mean) and some are negative (below the mean), and they would cancel out if we just added them. The solution? Square the deviations first, which makes them all positive.

Sample variance formula:

$$s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}$$

Here is what each part means:

• $x_i$ is each individual data value
• $\bar{x}$ is the mean of the data
• $(x_i - \bar{x})$ is the deviation of each value from the mean
• $(x_i - \bar{x})^2$ is the squared deviation
• $\sum$ means add up all the squared deviations
• $n-1$ is the number of values minus one (we use $n-1$ instead of $n$ for samples to get a better estimate—this is called Bessel’s correction)

Why divide by $n-1$ instead of $n$? When working with a sample from a larger population, dividing by $n-1$ gives us a better estimate of the true population variance. This technical adjustment compensates for the fact that a sample tends to underestimate variability. For now, just remember: sample variance uses $n-1$ in the denominator.

A larger variance means data is more spread out; a smaller variance means data clusters more tightly around the mean.

Standard Deviation: The Typical Distance from the Mean

Variance has one awkward property: because we squared the deviations, the units are squared too. If your data is in dollars, the variance is in “square dollars”—which is hard to interpret.

Standard deviation solves this by taking the square root of the variance, bringing us back to the original units.

$$s = \sqrt{s^2} = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$$

The standard deviation tells you, roughly, the “typical” distance of a data point from the mean. If the mean is 100 and the standard deviation is 15, most values will be within about 15 units of 100.

The strength of standard deviation: It uses every data point and is in the same units as your data.

The weakness of standard deviation: Like the mean, it is sensitive to outliers.

Standard deviation is the most commonly used measure of spread in statistics. You will encounter it everywhere—test scores, scientific measurements, financial analysis, quality control, and more.

Interpreting Standard Deviation

What does a standard deviation of 10 actually mean? It depends on context. A standard deviation of 10 is large if the mean is 20, but small if the mean is 1000.

One useful rule of thumb for roughly bell-shaped (normal) distributions is the Empirical Rule (also called the 68-95-99.7 Rule):

• About 68% of values fall within 1 standard deviation of the mean
• About 95% of values fall within 2 standard deviations of the mean
• About 99.7% of values fall within 3 standard deviations of the mean

So if exam scores have a mean of 75 and a standard deviation of 10:

• About 68% of students score between 65 and 85
• About 95% of students score between 55 and 95
• Nearly all students score between 45 and 105

This rule helps you judge whether a particular value is typical, somewhat unusual, or extremely rare.

Comparing Spread Across Datasets

When comparing the spread of different data sets, keep these principles in mind:

Direct comparison: If two data sets have the same units and similar means, you can compare their standard deviations directly. A larger standard deviation means more spread.

Different scales: If data sets have very different means, the coefficient of variation (CV) allows for fair comparison:

$$\text{CV} = \frac{s}{\bar{x}} \times 100%$$

The CV expresses the standard deviation as a percentage of the mean. This lets you compare variability between data sets with different units or magnitudes. For example, is a standard deviation of $5,000 in salaries more or less variable than a standard deviation of 2 cm in heights? The CV helps answer this.

When to Use IQR vs. Standard Deviation

Both IQR and standard deviation measure spread, but they suit different situations:

Use IQR when:

• Your data is skewed
• There are outliers you want to downplay
• You are working with the median as your measure of center
• You want a resistant (robust) measure

Use standard deviation when:

• Your data is roughly symmetric
• There are no extreme outliers
• You are working with the mean as your measure of center
• You need to do further statistical calculations (most advanced statistics use standard deviation)

Here is a helpful pairing:

• Median + IQR for skewed data or data with outliers
• Mean + Standard deviation for symmetric data

Just as you would not use the mean for highly skewed data, you generally would not use standard deviation in that situation either.

Notation and Terminology

Term	Meaning	Example
Range	Maximum − Minimum	Easy but sensitive to outliers
$Q_1$ (First Quartile)	25th percentile	25% of data below
$Q_2$ (Median)	50th percentile	Middle value
$Q_3$ (Third Quartile)	75th percentile	75% of data below
IQR	$Q_3 - Q_1$	Middle 50% spread
Variance ($s^2$)	Average of squared deviations	$s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}$
Standard deviation ($s$)	Square root of variance	Same units as data
Outlier rule	Values below $Q_1 - 1.5 \times \text{IQR}$ or above $Q_3 + 1.5 \times \text{IQR}$
Five-number summary	Min, $Q_1$, Median, $Q_3$, Max	Quick data snapshot
Resistant measure	Not affected by outliers	IQR is resistant
Empirical Rule	68-95-99.7 rule for normal data	Interprets standard deviation
Coefficient of variation	$\frac{s}{\bar{x}} \times 100%$	Compares relative spread

Examples

Key Properties and Rules

Formulas for Measures of Spread

Range: $$\text{Range} = \text{Maximum} - \text{Minimum}$$

Interquartile Range: $$\text{IQR} = Q_3 - Q_1$$

Sample Variance: $$s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}$$

Sample Standard Deviation: $$s = \sqrt{s^2} = \sqrt{\frac{\sum(x_i - \bar{x})^2}{n-1}}$$

Outlier Detection:

• Lower fence: $Q_1 - 1.5 \times \text{IQR}$
• Upper fence: $Q_3 + 1.5 \times \text{IQR}$
• Values outside these fences are potential outliers

Coefficient of Variation: $$\text{CV} = \frac{s}{\bar{x}} \times 100%$$

Properties of Standard Deviation

1. Always non-negative: $s \geq 0$, and $s = 0$ only when all values are identical.
2. Same units as data: Unlike variance, standard deviation is in the original measurement units.
3. Sensitive to outliers: Like the mean, extreme values can inflate the standard deviation.
4. Affected by linear transformations:
   • Adding a constant to all values does not change $s$
   • Multiplying all values by a constant $c$ multiplies $s$ by $|c|$

Properties of IQR

1. Resistant to outliers: Extreme values do not affect $Q_1$ or $Q_3$.
2. Based on position: Only uses the 25th and 75th percentile values.
3. Captures middle 50%: Ignores the upper and lower 25% of data.

Comparing Measures: Quick Reference

Situation	Best Measure of Spread
Symmetric data, no outliers	Standard deviation
Skewed data	IQR
Data with outliers	IQR
Need for further calculations	Standard deviation
Comparing different scales	Coefficient of variation
Quick rough estimate	Range

The Empirical Rule (68-95-99.7)

For approximately bell-shaped (normal) distributions:

• About 68% of data falls within $\bar{x} \pm 1s$
• About 95% of data falls within $\bar{x} \pm 2s$
• About 99.7% of data falls within $\bar{x} \pm 3s$

Real-World Applications

Quality Control in Manufacturing

Manufacturing relies heavily on measures of spread. When a factory produces bolts that should be exactly 10 mm in diameter, the mean might be right at 10 mm—but that is not enough. If the standard deviation is too large, too many bolts will be too big or too small to fit properly.

Quality control uses standard deviation to set acceptable tolerance limits. A “six sigma” process—one where defects are more than 6 standard deviations from the mean—allows only about 3.4 defects per million items. This extreme consistency is what makes modern manufacturing reliable.

Weather Variability

When comparing climates, average temperature tells only part of the story. San Francisco and St. Louis might have similar average annual temperatures, but their variability is completely different. San Francisco’s temperatures stay mild year-round (small standard deviation), while St. Louis swings from freezing winters to sweltering summers (large standard deviation).

Farmers, city planners, and anyone considering where to live benefits from understanding not just average weather, but how much it varies.

Investment Risk and Volatility

In finance, standard deviation is the standard measure of risk—called volatility. Two investments might have the same average return over 10 years, but one might swing wildly (high volatility) while the other grows steadily (low volatility).

A retirement account probably should not be heavily invested in high-volatility stocks, because a big swing downward at the wrong time could be devastating. Younger investors with time to recover might accept higher volatility for potentially higher returns. Understanding spread helps investors match their risk tolerance to their investments.

Test Score Distributions

Standardized tests like the SAT and ACT are designed to have specific means and standard deviations. The SAT, for example, is designed so that the mean is around 1000-1050 with a standard deviation of about 200 points.

This standardization allows fair comparisons across years and versions. If you know the mean and standard deviation, you can use the Empirical Rule to estimate what percentage of students scored above or below any given score. A score of 1200 (about one standard deviation above the mean) puts you roughly in the top 16% of test-takers.

Medical Research

Clinical trials carefully track the spread of responses to treatments. Two medications might lower blood pressure by the same average amount, but one might work consistently while the other shows high variability—working great for some patients but barely for others. The more consistent medication is generally preferred because doctors can predict its effects more reliably.

Outlier detection also matters: if a few patients have extreme reactions, those outliers need to be investigated for safety reasons.

Self-Test Problems

Summary

• Range (Maximum − Minimum) is the simplest measure of spread, but is highly sensitive to outliers.

• Quartiles divide ordered data into four equal parts. $Q_1$ is the 25th percentile, $Q_2$ (median) is the 50th percentile, and $Q_3$ is the 75th percentile.

• The five-number summary (Min, $Q_1$, Median, $Q_3$, Max) provides a quick snapshot of data distribution.

• Interquartile Range (IQR) = $Q_3 - Q_1$ measures the spread of the middle 50% of data. It is resistant to outliers.

• The 1.5×IQR rule identifies potential outliers: values below $Q_1 - 1.5 \times \text{IQR}$ or above $Q_3 + 1.5 \times \text{IQR}$.

• Variance ($s^2$) measures average squared deviation from the mean: $s^2 = \frac{\sum(x_i - \bar{x})^2}{n-1}$.

• Standard deviation ($s$) is the square root of variance, giving a measure of spread in the original units. It represents the “typical” distance from the mean.

• The Empirical Rule (68-95-99.7) helps interpret standard deviation for bell-shaped distributions: about 68% of data falls within 1 SD, 95% within 2 SD, and 99.7% within 3 SD of the mean.

• Coefficient of variation ($\text{CV} = \frac{s}{\bar{x}} \times 100%$) allows comparison of variability across different scales.

• Choose your measure wisely: Use standard deviation with the mean for symmetric data; use IQR with the median for skewed data or when outliers are present.

• Two datasets can have identical centers but very different spreads—the spread tells you about consistency and reliability, which the center alone cannot reveal.