Read and interpret histograms
Check both axes, the class boundaries and the sample size first
- One interval Read its height against the vertical scale. State the whole interval, not just its midpoint
- Cumulative count Below a class boundary means all complete classes to its left. At or above means all classes starting at that boundary or later
- Percentage scale To recover a count, multiply the percentage by the known total and divide by 100
- Inference limits Grouped data generally do not reveal exact values, an exact median, or a count below a point inside a class
Worked example
A histogram has frequencies 6, 14, 17, 9, 4 for consecutive ten-minute classes starting at 0. Find the count and percentage below 30 minutes
- \(\displaystyle \begin{aligned}n&=6+14+17\\&\quad+9+4\\&=50\end{aligned}\)Find the total sample size
- \(\displaystyle N_{<30}=6+14+17=37\)Add all complete intervals below the boundary
- \(\displaystyle 37/50\times100=74\)Use the entire sample as denominator. The result is 74% of the observations
Common mistakes
Difficulty levels
Questions about this topic
Does at least 20 include 20?+
Yes. With lower-inclusive classes, include the class starting at 20 and all later classes
Can I split a class in half?+
Not exactly without raw data. Assuming observations are evenly spread is an estimate and must be stated
Is the tallest class midpoint the mode?+
No. The modal interval is known, but the most frequent exact value may not be
Can I always identify the median interval?+
No. If the two central observations lie in different classes, their average may not be locatable from the grouped counts alone
Why do I need the sample size with percentages?+
A percentage gives a proportion. Multiply that proportion by the sample size to recover a count
Which direction names skew?+
The direction of the longer tail, not the side with the tallest column
Can I still give bounds when the exact count is hidden?+
Yes. Complete classes below the threshold give a minimum. Add all observations in the class containing the threshold for a maximum. These are bounds, not a claim that values are evenly spread
Does an unchanged histogram mean the raw data are unchanged?+
No. Values can move within the same classes without changing any frequency. Exact medians, ranges and counts inside a class can still change
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, General Mathematics, chapter 1: Data distributions