Grouped frequency tables
Every record must belong to exactly one class
- Cover every observation Choose non-overlapping intervals that include the minimum and maximum. Tally each observation once, including repeats
- State the convention For whole-number records, 20–29 includes 20 and 29. For continuous measurements, 20 ≤ x < 30 includes 20 but excludes 30.
- Check the total The frequencies must sum to the number of observations. Retain empty classes with frequency zero
- Percentage frequency Divide the class frequency by the total and multiply by 100. Round only at the end
Worked example
Group 3, 7, 8, 12, 14, 14, 18, 22, 24, 29 using whole-number classes of width 10 starting at 0.
\[\begin{array}{c|r|r}\text{Class}&\text{Count}&\%\\0\text{–}9&3&30\\10\text{–}19&4&40\\20\text{–}29&3&30\\\text{Total}&10&100\end{array}\]
- \(\displaystyle f=(3,4,3)\)Tally each value once. Keep both observations of 14.
- \(\displaystyle p=(30,40,30)\%\)Divide each count by 10 and multiply by 100.
- \(\displaystyle 3+4+3=10\)The frequency total accounts for every observation
Common mistakes
Double-counting a boundary
✕\(\displaystyle 20\in[10,20]\text{ and }[20,30]\)✓\(\displaystyle 20\in[20,30)\)
Use lower-inclusive, upper-exclusive boundaries for continuous valuesUsing the wrong denominator
✕\(\displaystyle 7/7\times100\)✓\(\displaystyle 7/40\times100=17.5\)
Divide by all observations, not the class frequencyDifficulty levels
introTally whole-number records
standardTally decimal records and calculate percentages
challengeChoose a scheme with a stated start and width
examSelect a scheme or read a table cell
Questions about this topic
Are equal widths compulsory?+
These tasks use equal widths. Unequal-width groupings are possible, but histogram height then needs special care
Should I omit a zero-frequency class?+
Keep it. Zero is useful information and the numerical intervals must remain in order
Can grouped data give an exact mean?+
Usually no. The individual values have been lost; a midpoint calculation is an estimate
What does 2.0–2.9 mean?+
Here it names recorded tenths from 2.0 through 2.9 inclusive. The next recorded tenth is 3.0
Why might percentages total 99.9%?+
Separately rounded percentages need not total exactly 100%, although their unrounded values do
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 9 Mathematics, chapter 9: Probability and statistics
In this chapter
Finite events, probability assignments and complementsDescribe and order probability values9B Venn diagrams and tables: Venn diagrams, two-way tables and set notation for two eventsComplete and use a two-way probability tableCompound events, the addition rule and mutual exclusivity9C Set notation: Event notation and shaded regionsFinite sets and set operationsBuild two-step sample spaces with and without replacementUse two-step probability treesExperimental probability from observed frequenciesDesign and interpret a probability simulationFind the mean and the median, and choose between themWork backwards from a mean or medianMode, bimodal and no-mode data, and the effect of an outlierConstruct an ordered stem plot from raw dataConstruct and read back-to-back stem-and-leaf plotsDescribe the shape of a distributionGroup numerical data into class intervalsConstruct a histogram from a frequency tableRead values from a histogramFind the range, the quartiles and the IQRProduce a five-number summary and construct a boxplotRead values and percentages from a boxplotCompare a numerical variable across groupsSurveys and statistical investigations