Astrolemma logo

IOAA data analysis round

Turn tables of numbers into physics.

Everything the data analysis round tests, from error bars to O−C diagrams, in one study guide. Then put it to work on real IOAA papers, transcribed with their data tables and full solutions.

Start the study guidePractise past papersFitting tools
11
Guide chapters
12
IOAA problems
2007–2023
Papers covered
0/12
Solved by you

Study guide

Eleven chapters in the order you need them. Each ends with key formulas, common mistakes and the past problems that use it.

  1. 01The data analysis roundWhat the paper looks like, how it is marked, and how to spend the hours so that no easy marks are lost.8 min read
  2. 02Numbers, units and timeSignificant figures, the units astronomers actually use, Julian dates, and reading values off tables and figures.10 min read · 2 practice problems
  3. 03Uncertainties and error propagationRandom and systematic errors, propagating uncertainties through formulas, and the special rules for magnitudes and logarithms.14 min read · 7 practice problems
  4. 04Statistics of repeated measurementsMeans, standard deviations, standard errors, weighted means, outliers, and how to decide whether two results really differ.10 min read · 3 practice problems
  5. 05Graphing by handScales, labels, error bars, lines and curves of best fit, and reading slopes with honest uncertainties: everything the marking scheme looks for.12 min read · 8 practice problems
  6. 06Linearising a relationTurning power laws, exponentials and odd model functions into straight lines, so that a ruler (or least squares) gives the physics.11 min read · 4 practice problems
  7. 07Least squares fittingThe formulas for a best-fit line and its uncertainties, weighted fits, lines through the origin, and what r and χ² tell you.13 min read · 6 practice problems
  8. 08Periods, phases and O−C diagramsPhase folding, finding and refining periods, O−C diagrams, sinusoid fits to radial velocity curves, and reading light curves.13 min read · 4 practice problems
  9. 09Photometry toolkitMagnitudes and fluxes, distance moduli, extinction and reddening, bolometric corrections, AB magnitudes and reddening-free indices.13 min read · 6 practice problems
  10. 10Spectra and velocitiesDoppler shifts, rotation curves and dynamical masses, binary orbits and the mass function, and expanding shells.11 min read · 3 practice problems
  11. 11Distances and the cosmic ladderParallax, standard candles, geometric methods and the Hubble law, with the IOAA problems that use each rung.12 min read · 6 practice problems

Fitting tools

Check your hand calculations. Paste any two or three columns (the problem pages have a Copy CSV button on every table), or start from a real IOAA data set. In the exam you will do this with a ruler and a calculator, so use these to verify, not to replace, your own working.

Least squares line

Weighted or unweighted fits, lines through the origin, log and inverse transforms, residuals and χ².

Periods and V-band absolute magnitudes of five Galactic Cepheids with parallax errors. Fit M_V against log P: the slope should be close to −2.5.

Slope
-2.50 ± 0.25
Intercept
-1.44 ± 0.23
Correlation r
-0.9983
χ² / dof
0.666 / 3 = 0.222

Uncertainties come from the error bars. If χ²/dof is far from 1, the error bars or the straight-line model need another look.

Data with the least squares line0.60.811.21.41.6-5.5-5-4.5-4-3.5-3-2.5log₁₀(P)M_V
  • Data
  • Best fit
Residuals about the fitted line0.60.811.21.41.6-0.2-0.100.10.2log₁₀(P)Residual

Period finder

Scan trial periods with a sinusoid fit, see the periodogram, and fold the data on the best period.

TRES radial velocities of 2M05215658+4359220. The official solution reads P ≈ 83.3 d and K ≈ 45 km/s by eye.

Best period
83.091 d
Semi-amplitude
44.58
Mean level
-0.5332
Data folded on the period00.20.40.60.811.21.41.61.82-40-2002040PhaseRV (km/s)mean level
  • Data (shown twice)
  • Sinusoid
Periodogram: lower χ² means a better sinusoid2040608010012014016018020002e54e56e58e51e61.2e61.4e61.6e6Trial period (days)χ² of the sinusoid fitP = 83.091 d

For each trial period the tool fits mean + amplitude × sin(2π(t − t₀)/P) by least squares and records χ². Deep, narrow minima are good periods; several minima of similar depth are aliases, and more data are needed to choose.

Problem set: past IOAA papers

Transcribed from the official papers with their data tables. Each has a hint and the official solution, checked line by line; where the official solution contains an error, a note explains it.

Want the original PDFs? Every paper, including years not transcribed here, is in the official IOAA archive.

Exam day checklist

The habits that the marking schemes reward, on one card. Screenshot it the night before.

  1. 01Read the whole paper first and mark the parts that need no graph.
  2. 02Tabulate computed columns before plotting anything.
  3. 03Choose each scale before the first point: 1, 2 or 5 units per division, data over half the sheet.
  4. 04Label every axis with quantity and unit, and write the graph number.
  5. 05Draw error bars, then a single best-fit line or smooth curve.
  6. 06Read slopes from a large triangle on the line; get the error from steepest and shallowest lines.
  7. 07Carry extra digits through calculations; round only the final answer to match its error.
  8. 08Give every estimate an uncertainty, and say whether it is statistical or systematic.
  9. 09Answer YES/NO questions with the word and a one-line reason.
  10. 10Copy final values, with units, into the summary answer sheet in the last 15 minutes.