Grouping data into intervals
Grouped frequency tables replace many individual numerical values with a small number of class intervals. The intervals must cover the data once, with a clear start and width
- Cover the full range Choose a starting value and width so that the intervals cover the minimum and maximum without unnecessary extra intervals \(\displaystyle \min=12,\quad\max=47\Rightarrow10\text{--}19,20\text{--}29,30\text{--}39,40\text{--}49\)
- Use the interval start A value on a boundary belongs to the interval that starts with that value \(\displaystyle 20\Rightarrow\text{20--29, not 10--19}\)
- Tally once only Put each observation in exactly one interval and keep an empty interval as a zero row \(\displaystyle 2+0+4=6\)
- Convert counts to percentages Divide each interval frequency by the total and multiply by 100, rounding only the displayed percentage \(\displaystyle \frac{4}{11}\approx36.4\%\)
Worked example
Group the values 12, 17, 23, 28, 31, 36, 42, 47 and 49 into intervals of width 10 starting at 10. Find the frequency and percentage frequency for each interval
\[\text{data: }12,17,23,28,31,36,42,47,49\]
- \(\displaystyle \min=12,\quad\max=49\)The intervals must include every value from the minimum 12 to the maximum 49
- \(\displaystyle \text{10--19},\quad\text{20--29},\quad\text{30--39},\quad\text{40--49}\)These intervals have width 10, are consecutive and cover the full range
- \(\displaystyle f=(2,2,2,3),\quad2+2+2+3=9\)The tallies are 2, 2, 2 and 3, and they account for all 9 values
- \(\displaystyle \frac{2}{9}\approx22.2\%,\quad\frac{3}{9}\approx33.3\%,\quad3(22.2\%)+33.3\%=99.9\%\)The percentages are rounded separately, so the displayed total of 99.9% is a rounding artefact
Common mistakes
Double-counting a boundary
✕\(\displaystyle 20\in\text{10--19 and 20--29}\)✓\(\displaystyle 20\in\text{20--29}\quad\text{and not 10--19}\)
A boundary value belongs only to the interval that starts with that valueLeaving a gap between intervals
✕\(\displaystyle \text{10--19},\quad\text{21--30}\)✓\(\displaystyle \text{10--19},\quad\text{20--29}\)
Consecutive intervals must cover every value without gaps or overlapsUsing the wrong percentage denominator
✕\(\displaystyle \frac{3}{3}\times100=100\%\)✓\(\displaystyle \frac{3}{9}\times100\approx33.3\%\)
An interval percentage uses the total number of observations, not the interval frequency, as the denominatorDifficulty levels
introGroup discrete values into given intervals and complete the frequency column
standardGroup one-decimal measurements into given intervals and complete count and percentage columns
challengeChoose an admissible interval scheme and reject schemes with gaps, overlaps or an unsuitable count
examChoose an interval frequency or the admissible interval scheme from generated alternatives
Questions about this topic
How should a boundary value be assigned?+
Assign it to the interval whose starting value is that boundary. This gives each observation exactly one interval
How many class intervals should be used?+
A useful grouped table usually has roughly 5 to 15 intervals, with a width that is easy to read and enough range to include every value
Why keep an empty interval?+
An empty interval shows that no observations occurred there while keeping the numerical scale continuous and the other frequencies in their correct positions
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 12: Continuous probability distributions
In this chapter
12A Continuous random variables and density functions: Validate and normalise a continuous probability density function12A Continuous random variables and density functions: Estimate probabilities from grouped continuous frequency dataGroup numerical data into class intervalsConstruct a histogram from a frequency table12A Continuous random variables and density functions: Finite events, probability assignments and complements12A Continuous random variables and density functions: Conditional probability from exact representationsEvaluate and determine function rules12B Probability density validity and normalisation: Validate and normalise a continuous probability density function12B Probability density validity and normalisation: Estimate probabilities from grouped continuous frequency dataSolve a linear equation12B Probability density validity and normalisation: Evaluate a polynomial or recover a coefficient parameterWork with sets and interval notation12B Probability density validity and normalisation: Round to decimal places and significant figuresInterval and conditional probabilities from a densityCompound events, the addition rule and mutual exclusivity12C Interval and conditional probabilities: Finite events, probability assignments and complements12C Interval and conditional probabilities: Conditional probability from exact representationsMean, variance and standard deviation of a continuous random variableMedian, quartiles and percentiles of a continuous random variableExpand and factor polynomial expressions12D Mean, variance, standard deviation and percentiles: Evaluate a polynomial or recover a coefficient parameterMove between roots, factors, polynomial rules and equationsSolve a quadratic exactly with the quadratic formula12D Mean, variance, standard deviation and percentiles: Round to decimal places and significant figuresLinear transformations and totals of independent continuous random variables