Distributions
In 1846 Adolphe Quetelet showed that the chest sizes of 5,738 Scottish soldiers piled up in a bell shape. Build frequency tables and histograms from real data, then use the normal curve, z-scores and the 68-95-99.7 rule, and see how sample size changes what a histogram shows.
Before you start
This course uses the mean, the median and the standard deviation from the start, so take Averages first if those are new. Histograms appear in every unit, and Reading charts covers how class width changes one. The lab computes normal-curve areas for you, so no table is needed.
When you finish
You will be able to build frequency, relative frequency and cumulative frequency tables, and describe a distribution as symmetric, skewed, uniform or bimodal. You can explain why a long tail pulls the mean below the median, use the 68-95-99.7 rule to find the middle 95% of heights, convert an SAT score of 325 to z = −1.5, and see why small samples look bumpy.
- 1Frequency tablesOf 50 students in an OpenStax example, 16 bought three books, so the relative frequency of "three books" is 16 ÷ 50 = 0.32.
- 2Shapes of distributionsThe 25 weekend gaming times in an OpenStax table have a median of 17.5 hours and a mean of 15.55: a long tail toward 0 pulls the mean down.
- 3The normal curveQuetelet's 5,738 chest sizes have mean 39.83 inches and standard deviation 2.05; a normal curve with those two numbers predicts 1,102 men at 40 inches, and the table has 1,079.
- 4The 68-95-99.7 ruleChilean boys aged 15 to 18 in 2009 to 2010 averaged 170 cm with a standard deviation of 6.28 cm, so about 95% were between 157.44 and 182.56 cm.
- 5z-scoresIn the 2012 SAT, verbal scores had mean 496 and standard deviation 114, so a score of 325 has z = (325 − 496) ÷ 114 = −1.5.
- 6Samples and sample sizeFive random heights from N(170, 6.28) can give a standard deviation of 3.4 cm; five thousand give a value within a few hundredths of 6.28.
Sources for this course (7)
- OpenStax, Introductory Statistics 2e by Barbara Illowsky and Susan Dean (, per the page notice on 2026-10-17): 2.2 frequency tables, relative and cumulative frequency, frequency polygons (calculus final test scores 5, 10, 30, 40, 15; ages of US presidents at inauguration in six classes)
- the 100 soccer player heights, the 50 students' books, the 25 classmates' weekend video game hours and the 22 New York restaurant staff counts (openstax.org/books/introductory-statistics-2e/pages/2-2-histograms-frequency-polygons-and-time-series-graphs)
- 6.1 the standard normal distribution, z-scores, the empirical rule, X ~ N(5, 6)
- heights of Chilean males aged 15 to 18 N(170, 6.28) for 2009 to 2010 and N(172.36, 6.34) for 1984 to 1985, SAT verbal 2012 N(496, 114) with 1,664,479 test takers, Jerome N(16, 4) (openstax.org/books/introductory-statistics-2e/pages/6-1-the-standard-normal-distribution)
- De Moivre and the normal approximation to coin flips: D. Lane, Online Statistics Education, 7.2 History of the Normal Distribution (stats.libretexts.org, Introductory Statistics (Lane) 7.2)
- EBSCO Research Starters, "De Moivre Describes the Bell-Shaped Curve" (1733)
- Quetelet's chest measurements of 5,738 Scottish militiamen (Quetelet 1846, from an 1817 source; Stigler 1986 corrections): R package HistData, ChestSizes and ChestStigler (vincentarelbundock.github.io/Rdatasets/doc/HistData/ChestSizes.html).
Checked: 2026-10-17