Bolton analysis explained: ratios, formula and calculator

Simplified Orthodontics Academy · Published 5 October 2026

What the Bolton analysis measures

For the upper and lower teeth to fit together, their sizes need to be in proportion. The Bolton analysis checks this proportion. It adds up the mesiodistal widths of the teeth in each arch and divides the lower total by the upper total.

Wayne Bolton described the analysis in 1958, from casts with excellent occlusion [1], and its clinical use in 1962 [2]. His means show the ratio most likely to go with an excellent occlusion [3].

The two Bolton ratios

The overall ratio uses every tooth from the first molar on one side to the first molar on the other. The anterior ratio uses the canines and incisors only [4]. Second and third molars are not included.

How the tooth widths are measured

Each tooth is measured at its greatest mesiodistal width, between its contact points [4]. This can be done on plaster casts with a caliper or dividers, or on digital models.

Small errors add up over 12 teeth. In one study, four clinicians measured the same casts twice. Every one of them made at least one measurement error as large as a clinically significant Bolton result, on casts with 3 mm or more of crowding [5].

Bolton’s means and standard deviations

Source: Bolton [1, 2], as reported by Endo and colleagues [4].

A reading is usually described by its distance from the mean: within 1 SD, between 1 and 2 SD, or more than 2 SD. Many studies use 2 SD as the cut-off for a significant discrepancy [3, 4]. Endo and colleagues proposed using both measures together: more than 2 SD from Bolton’s mean and more than 2 mm of tooth material [4]. In their sample, anterior ratios beyond 2 SD did not always mean more than 2 mm.

The tooth-size excess in millimetres

A percentage is hard to picture. So the result is often also given in millimetres: how far one arch’s total is from the total that would give Bolton’s mean.

Use 91.3 as the mean for the overall ratio and 77.2 for the anterior ratio. The excess says which arch has relatively more tooth material. It does not say which tooth.

Worked example

These widths are invented for this example; they are not from a patient. All values are in millimetres.

Overall ratio

1. Sum of the 12 upper teeth: 93.9 mm. Sum of the 12 lower teeth: 87.1 mm.
2. Ratio: 87.1 ÷ 93.9 × 100 = 92.8%.
3. Distance from the mean: (92.8 − 91.3) ÷ 1.91 = 0.8 SD, so within 1 SD.
4. The ratio is above the mean, so the excess is in the lower arch: 87.1 − (93.9 × 91.3 ÷ 100) = 87.1 − 85.7 = 1.4 mm.

Anterior ratio

1. Sum of the 6 upper anterior teeth (13 to 23): 46.1 mm. Sum of the 6 lower anterior teeth (43 to 33): 36.8 mm.
2. Ratio: 36.8 ÷ 46.1 × 100 = 79.8%.
3. Distance from the mean: (79.8 − 77.2) ÷ 1.65 = 1.6 SD, so between 1 and 2 SD.
4. The ratio is above the mean, so the excess is in the lower arch: 36.8 − (46.1 × 77.2 ÷ 100) = 36.8 − 35.6 = 1.2 mm.

The calculator below gives the same figures; choose “Fill the worked example” to see them.

Limits of the Bolton analysis

• Populations differ. Smith and colleagues measured 180 patients from three populations and found significant differences in the ratios between ethnic groups and between males and females. They concluded that Bolton’s ratios apply to white females only [6]. A literature review found that sex and ethnic group seem unlikely to have a clinically significant effect, and that Class III malocclusions may have larger average ratios [3]. The studies do not agree, so read a ratio with the patient’s background in mind.
• Discrepancies are common. In the studies reviewed, 20 to 30% of people had a significant anterior discrepancy and 5 to 14% a significant overall discrepancy [3].
• Bolton’s sample shows the target, not the threshold. It was suited to showing which ratio goes with an excellent occlusion, but not to showing how large a discrepancy has to be to matter [3].
• The ratio does not point to a tooth. In one study, four teeth explained about half of the differences in the overall ratio between people: the lower second premolar, the upper lateral incisor, the upper second premolar and the lower central incisor [6]. Comparing single teeth is a separate step.
• Measurement error. Crowding makes contact points harder to measure, and errors add up over 12 teeth [5]. Computerised methods are faster [3].

Bolton calculator

Educational content for clinicians and students. Reference values vary between populations. This page does not replace clinical judgement.

References

1. Bolton WA. Disharmony in tooth size and its relation to the analysis and treatment of malocclusion. Angle Orthod. 1958;28(3):113-30. Publisher’s page
2. Bolton WA. The clinical application of a tooth-size analysis. Am J Orthod. 1962;48(7):504-29. doi: 10.1016/0002-9416(62)90129-X
3. Othman SA, Harradine NW. Tooth-size discrepancy and Bolton’s ratios: a literature review. J Orthod. 2006;33(1):45-51. doi: 10.1179/146531205225021384. PMID: 16514133
4. Endo T, Uchikura K, Ishida K, Shundo I, Sakaeda K, Shimooka S. Thresholds for clinically significant tooth-size discrepancy. Angle Orthod. 2009;79(4):740-6. doi: 10.2319/070208-339.1. PMID: 19537863
5. Shellhart WC, Lange DW, Kluemper GT, Hicks EP, Kaplan AL. Reliability of the Bolton tooth-size analysis when applied to crowded dentitions. Angle Orthod. 1995;65(5):327-34. PMID: 8526291
6. Smith SS, Buschang PH, Watanabe E. Interarch tooth size relationships of 3 populations: “does Bolton’s analysis apply?”. Am J Orthod Dentofacial Orthop. 2000;117(2):169-74. doi: 10.1016/s0889-5406(00)70228-9. PMID: 10672217

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