How much spread does one extreme value create?

Two columns can have similar typical values but very different tails. Before choosing a preprocessing rule, how would you tell whether variation is spread throughout the records or concentrated in one unusual value?

Start with 8, 9, 9, 10, 10, 11, 11, 12. The preceding lesson introduced mean and median. Here we examine spread while keeping seven records fixed.

Change one tail value

Use last value 12 and histogram bin width 2 as the reference. Predict the mean, range, and general histogram shape, then record the run. Change only the last value to 40. Predict whether the range, standard deviation, and middle-half spread will react equally.

Eight synthetic measurements. The first seven values stay 8, 9, 9, 10, 10, 11, 11. Sample standard deviation uses n−1. Quartiles interpolate at index (n−1)p. Histogram bins start at multiples of the chosen width and include their left edge only.

Last record value1216244080Histogram bin width12410

Prediction and reason

Record runReference settings

0/16 runs. Each run keeps its own inputs and results. Work stays in this tab across reloads; download a report before closing it.

Write a prediction to record your reference.

Observations: cite runs and valuesExplanation: what accounts for the change?Your next test, prediction, and a result that would challenge it

Download report

Compare both runs. Cite one quantity that changed strongly and one that stayed fixed before reading the explanation.

Compare the center and the tail

The reference mean and median are 10, the range is 4, and sample standard deviation is $\sqrt{12/7}\approx1.3093$. With last value 40, the mean becomes 13.5, the range becomes 32, and standard deviation becomes $\sqrt{810/7}\approx10.7571$.

The quartiles remain 9 and 11, so the interquartile range stays 2. Seven unchanged observations still occupy a narrow band. The much larger standard deviation reflects the tail rather than a uniform spreading of every record.

Name the measurements

Range is maximum minus minimum. Sample standard deviation is

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

Squaring deviations makes distant values influential; the square root restores the measurement’s original units. The interquartile range, or IQR, is $Q_{0.75}-Q_{0.25}$. Our quartiles linearly interpolate at zero-based position $(n-1)p$ in the sorted data. Other conventions may differ on small samples. NIST explains measures of scale.

A histogram counts records in intervals. Ours uses intervals closed on the left and open on the right, so a value exactly on a boundary belongs to the bin starting there. Histogram shape depends on the bins.

Design a comparison

Choose another tail value. Predict which summaries change, record it, and explain whether the record deserves investigation. An extreme observation is a reason to check provenance, not sufficient evidence to delete it. NIST distinguishes unusual observations from confirmed errors.

Then hold the records fixed and choose two bin widths. Predict whether the numerical summaries will change. Select a baseline with only the bin width different and download your report.

Check the display-only intervention

Bin width changes how the same records are grouped visually. It does not change mean, standard deviation, IQR, or range. Counts across all bins still sum to eight. A smoother histogram alone is not evidence that the data became less variable.

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