Maths › Statistics › Representing and interpreting data
Representing and interpreting data
A good chart is an argument you can see. Histograms make area mean frequency, box plots put five numbers on display, and a fence built from the quartiles decides, without sentiment, which values are outliers.
Builds on Measures of location and spread.
IN THIS TOPIC
- Construct and read histograms using frequency density, including finding frequencies from areas.
- Draw and compare box plots, and identify outliers with the quartile fence rule.
- Interpret cumulative frequency diagrams, and clean data before summarising it.
WHAT YOU PROBABLY THINK
In a histogram, the height of a bar tells you its frequency.
Histograms: area is frequency
A bar chart's bars have equal widths, so height alone carries the information. A histogram handles unequal class widths, and the price is that height must be recalculated as frequency density:
Then area equals frequency, and every histogram question is an area question. Given a bar, multiply height by width to recover its frequency; given a frequency, divide by the width to set the height. A histogram with a key (“one square represents 5 observations”) just rescales the same idea.
WORKED EXAMPLE
Reading a histogram
In a histogram, the class 15 ≤ x < 30 has frequency 30. The class 10 ≤ x < 15 is drawn with 1.5 times the height of that bar. Find its frequency.
The first bar's density is 30/15 = 2, so the second bar's density is 1.5 × 2 = 3.
Its frequency is density times width: 3 × 5 = 15.
Check by areas: a bar half as wide but one-and-a-half times as tall carries three quarters of half the area, 15 against 30, which matches.
Box plots and outliers
A box plot displays five numbers: the extremes, the quartiles and the median. The box spans the middle half of the data; the median line splits it; whiskers reach the smallest and largest values that are not outliers.
The 1.5 × IQR fences are the usual rule (a question may instead specify mean plus or minus two standard deviations; use whichever the question defines). An outlier is plotted as a separate point, and the whisker stops at the most extreme value inside the fence, not at the fence itself.
Comparing two box plots is a two-sentence exam answer with a shape: one sentence comparing location, medians, in context; one comparing spread, IQRs, in context. “The median mass is higher for brand A, and brand A's masses are more consistent since its IQR is smaller.”
Cumulative frequency
Plot cumulative frequency against the upper class boundary and join the points with a smooth increasing curve. Reading across from n/2, n/4 and 3n/4 gives the median and quartiles; reading between two values and subtracting counts the data in any interval. Every reading is an estimate, because grouping already blurred the exact values.
Cleaning the data
Real data arrives dirty, and the large data set is deliberately so: missing entries, “tr” for trace rainfall, wind directions recorded as bearings of text. Before any calculation, decide what to do with each, and say so. Removing a genuinely impossible value is cleaning; removing a merely inconvenient one is bias with better manners, so an anomaly should only go when there is a stated reason to doubt the measurement itself.
THE EXAM BIT
- Every histogram mark flows through frequency density: label the axis, and convert area to frequency or back deliberately.
- Learn the fence rule as a sentence: one and a half IQRs beyond either quartile.
- Compare box plots in context, one sentence for medians, one for IQRs; a comparison with no context scores the method mark at best.
- Cumulative frequency reads from the upper class boundaries; plotting at midpoints is a classic dropped mark.
CHECK YOURSELF
A data set has Q₁ = 15 and Q₃ = 27. Using the 1.5 × IQR rule, decide whether the value 48 is an outlier.
Show a hint
Build the upper fence first.
Show the answer
IQR = 27 − 15 = 12, so the upper fence is 27 + 1.5 × 12 = 45.
48 > 45, so 48 is an outlier and would be plotted as a separate point.
In a histogram, area is frequency; the vertical axis is frequency density.
Fences stand one and a half IQRs beyond the quartiles, and whatever is beyond the fence is plotted alone.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
Rate how confident you feel with each objective for this lesson. Ratings are saved in this browser, on this device only.
- Construct and read histograms using frequency density, including finding frequencies from areas.
- Draw and compare box plots, and identify outliers with the quartile fence rule.
- Interpret cumulative frequency diagrams, and clean data before summarising it.
No animated video for this topic yet; these notes stand alone.