Question 1

What is the range in statistics?

Accepted Answer

The range measures the total spread of a dataset by subtracting the minimum from the maximum. For the dataset {3, 7, 2, 9, 5}: minimum = 2, maximum = 9, range = 7. It is the simplest measure of dispersion and gives you a quick sense of how spread out the data is — but it only considers two values and is heavily influenced by outliers.

Question 2

What is the midrange and when is it useful?

Accepted Answer

The midrange is the average of the minimum and maximum: (min + max) / 2. It represents the midpoint of the data's span. Unlike the median (which is positionally the middle value), the midrange can be far from the bulk of the data if outliers are present. It is occasionally used in weather reporting (average of daily high and low temperatures) but is rarely used in formal statistics.

Question 3

Why is the range considered a weak measure of spread?

Accepted Answer

Because it only uses two values — the extremes — and ignores everything in between. Two datasets can have identical ranges but very different distributions. For example: {1, 5, 5, 5, 9} and {1, 2, 5, 8, 9} both have a range of 8, but the first is tightly clustered while the second is spread out. Standard deviation and IQR are generally preferred because they use all values (std dev) or are resistant to outliers (IQR).

Question 4

What is the five-number summary?

Accepted Answer

The five-number summary consists of: Minimum, Q1 (25th percentile), Median (Q2, 50th percentile), Q3 (75th percentile), and Maximum. Together they describe a dataset's center, spread, and shape. The five-number summary is the foundation of a box plot (box-and-whisker diagram) — the box spans Q1 to Q3, the line inside is the median, and the whiskers extend to the min and max (or to the Tukey fences if outliers are present).

Question 5

What are Tukey fences and how do they detect outliers?

Accepted Answer

Tukey fences define the boundaries beyond which values are considered outliers. Inner fences: [Q1 − 1.5×IQR, Q3 + 1.5×IQR] — values outside are mild outliers. Outer fences: [Q1 − 3×IQR, Q3 + 3×IQR] — values outside are extreme outliers. Invented by statistician John Tukey, this method is robust because the IQR itself is unaffected by the outliers being detected.

Question 6

How does range differ from variance and standard deviation?

Accepted Answer

Range uses only the two extreme values. Variance and standard deviation use every value in the dataset, measuring average squared deviation from the mean. For normal distributions: range ≈ 4–6 standard deviations. Standard deviation is usually preferred for formal analysis because it is mathematically tractable, plays well with statistical theory (confidence intervals, t-tests), and is less sensitive to single extreme values.

Measure	Formula	Outlier resistant?	Best when
Range	Max − Min	No — uses extremes	Quick overview, small clean data
IQR	Q3 − Q1	Yes — uses middle 50%	Skewed data, outliers present
Variance (σ²)	Σ(x−μ)² / n	No — squares outliers	Mathematical analysis, ANOVA
Std Dev (σ)	√Variance	No — inflated by outliers	Normal distributions, reporting
CV	σ / μ × 100%	Moderate	Comparing different-scale datasets

Range Calculator

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Measures of Spread — Comparison

Why Range Alone Isn't Enough

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