Measures of Distribution

Measures of distribution describe the spread or variability of data, helping to explain how individual data points relate to the overall dataset. The main concepts to know are range, interquartile range (IQR), standard deviation, and outliers.

Key Takeaways

• The range (max − min) is the simplest spread measure but highly sensitive to outliers.

• The interquartile range (IQR) (Q3 − Q1) captures the middle 50% of the data and is used with the 1.5 × IQR rule to flag potential outliers.

• Standard deviation measures typical distance from the mean and underlies the 68-95-99.7 empirical rule.

• Outliers can arise from true anomalies, measurement error, or non-normal distributions — only measurement errors should be automatically excluded.

Range

The range is the simplest measure of distribution — the difference between the largest value (Xmax) and the smallest value (Xmin) in a dataset:

Range = Xmax − Xmin

MCAT Callout — Worked Example: Calculating Range: Dataset: {3, 7, 10, 15}. Range = 15 − 3 = 12.

The range provides a quick way to gauge the spread of data, but it's highly sensitive to outliers. If this dataset included an outlier of 50, the range would jump to 50 − 3 = 47, even though most of the data points are much closer together.

Interquartile Range (IQR)

The interquartile range (IQR) measures the spread of the middle 50% of the data — the difference between the third quartile (Q3) and the first quartile (Q1):

IQR = Q3 − Q1

To calculate quartiles, the data must first be sorted in ascending order. Q1 is the value below which 25% of the data falls; Q3 is the value below which 75% of the data falls.

• If n (the number of data points) is odd, find the positions of Q1 and Q3 using (n + 1)/4 and 3(n + 1)/4, respectively.

• If n is even, take the average of the data values at those positions.

The IQR is also used to flag outliers: data points below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR are considered potential outliers.

Standard Deviation

Standard deviation (σ) is a widely used measure of variability that quantifies how spread out the data points are around the mean.

Standard formula (not spoken aloud in the source video, shown on-screen): σ = √( Σ(xᵢ − μ)² / N ), where xᵢ is each data value, μ is the mean, and N is the number of data points.

Standard deviation is particularly important because it relates directly to the normal distribution: ~68% of data in a normal distribution falls within one standard deviation of the mean, 95% within two, and ~99.7% within three.

Outliers

Outliers are data points that deviate significantly from the rest of the dataset. They can distort measures like the mean and range, leading to misleading conclusions. Outliers typically result from one of three causes:

Not all outliers should be removed automatically. Researchers must determine whether an outlier represents a legitimate data point or a measurement error before deciding how to handle it — only a confirmed measurement error should be excluded from analysis; the other two situations are less clear-cut.

Common MCAT Mistakes

• Confusing range with IQR. Range uses the entire dataset (max − min) and is highly sensitive to outliers; IQR uses only the middle 50% and stays stable even when extreme values are present.

• Forgetting to sort the data first. Quartile positions — (n + 1)/4 for Q1, 3(n + 1)/4 for Q3 — are only meaningful once the dataset is arranged in ascending order.

• Misapplying the 1.5 × IQR rule. A point is a potential outlier only if it falls below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR — not simply outside the IQR itself.

• Assuming every outlier should be deleted. Only a confirmed measurement error should be automatically excluded; a true statistical anomaly or a point from a skewed distribution requires judgment, not automatic removal.

MCAT-Style Concept Check

Question: A sorted dataset contains the values 4, 8, 12, 16, and 100. Which measure of this dataset's spread is LEAST affected by the value 100?

• A) Range

• B) Standard deviation

• C) Interquartile range (IQR)

• D) Mean

Answer: C

Explanation: IQR is calculated from only the middle 50% of the data (Q3 − Q1), so an extreme value like 100 falls outside that middle portion and has no effect on it. Range (max − min) is defined directly by the extreme value, and standard deviation is pulled upward by any point far from the mean. Mean is not a measure of spread at all, and its value would also be skewed by 100.

FAQ

What's the difference between range and interquartile range (IQR)?

Range is the full spread of a dataset (max − min) and is highly sensitive to outliers. IQR (Q3 − Q1) captures only the middle 50% of the data, so it stays stable even when extreme values are present.

How do you use IQR to identify an outlier?

Flag any data point below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR as a potential outlier.

What does standard deviation tell you about a normal distribution?

Standard deviation measures the typical distance of data points from the mean. Under the empirical rule, about 68% of data in a normal distribution falls within one standard deviation of the mean, about 95% within two, and about 99.7% within three.

Should every outlier be removed from a dataset?

No. Only outliers confirmed to be measurement errors should be automatically excluded — true statistical anomalies and points from non-normal or skewed distributions require judgment rather than automatic removal.