6.1 Graphical representation of data

Learning objectives

6.1A Different methods of presenting data Pictograms, baruse different methods of presenting data Pictograms, bar charts and pie charts, and only two-way tables 6.1B Tabulating data for statistical diagramsuse appropriate methods of tabulation to enable the construction of statistical diagrams 6.1C Statistical diagramsinterpret statistical diagrams 6.1HA Histogramsconstruct and interpret histograms For continuous variables with unequal class intervals 6.1HB Cumulative frequency diagrams from tabulated dataconstruct cumulative frequency diagrams from tabulated data 6.1HC Cumulative frequency diagramsuse cumulative frequency diagrams

Choose a display by matching it to the data and the comparison you need. Pictograms and bar charts show discrete categories; pie charts show parts of one whole; two-way tables classify each item by two categorical variables.

Display	Best use	Essential check
pictogram	small category frequencies	read the key, including partial symbols
bar chart	compare discrete categories	equal-width separated bars and a labelled scale
pie chart	compare proportions of one total	sectors total $360^\circ$
two-way table	cross-classify two categories	row and column totals agree with the grand total

For a pie chart, $\text{sector angle}=\dfrac{\text{frequency}}{\text{total}}\times360^\circ$ . Reverse the calculation with $\text{frequency}=\dfrac{\text{angle}}{360^\circ}\times\text{total}$ .

A taller bar may represent a larger frequency, but a wider bar in an ordinary bar chart does not. Do not use bar area unless the graph is a histogram.

A frequency table organises every observation into one category or class, records tallies in groups of five, and converts each tally into a frequency.

Step	Action
1	define categories or non-overlapping class intervals
2	process the raw list once, adding one tally per observation
3	convert each tally to a numerical frequency
4	add the frequencies and compare with the number of observations

Categories must be exhaustive and mutually exclusive: every value belongs somewhere and no value belongs twice. Interval notation such as $10<x\le20$ fixes where a boundary value goes.

Do not recount separately for each category; a single pass with tallies reduces omissions and double-counting.

Before reading a statistical diagram, identify its title, variable, units, category labels or class intervals, and the scale on each axis or in its key.

Task	Reliable method
read a value	trace from the mark or bar to the labelled scale
compare categories	read both values, then subtract or form the requested ratio
find a total	convert every symbol, bar or sector to frequency before adding
interpret a pie sector	use its fraction of $360^\circ$ , not its visual width alone

If one vertical grid step represents 2 games and a bar is 5 steps high, its frequency is $5\times2=10$ games. The scale must be decoded before the height is used.

A diagram supports statements about the displayed data, not an unexplained causal claim. Also check for truncated axes or unequal scales before comparing visual size.

A histogram displays grouped continuous data. Bars touch, class intervals determine bar widths, and the vertical axis is frequency density so that each bar's area represents frequency.

Quantity	Relationship
class width	upper boundary $-$ lower boundary
frequency density	$\dfrac{\text{frequency}}{\text{class width}}$
frequency	class width $\times$ frequency density

Compute every class width and frequency density, mark the continuous class boundaries on the horizontal axis, then draw touching rectangles. To estimate a frequency over part of a class, use the corresponding fraction of that bar's area.

When class widths differ, bar height is not frequency. Compare or add bar areas; treating a histogram as an ordinary bar chart gives the wrong result.

Cumulative frequency is a running total. Each plotted point shows how many observations are at or below an upper class boundary.

Class frequency	Cumulative frequency	Plot at
3	3	first upper boundary
16	$3+16=19$	second upper boundary
24	$19+24=43$	third upper boundary

Add frequencies successively, pair each running total with its upper class boundary, include the lower starting boundary with cumulative frequency 0 when appropriate, plot the points, and join them with a smooth increasing curve or line segments.

Do not plot at class midpoints and do not use the separate class frequencies as vertical coordinates. A cumulative-frequency graph must never decrease.

A cumulative frequency diagram converts between a value and the number at or below that value. Read up from a value to the curve and across to cumulative frequency, or reverse those steps to find a value for a chosen cumulative count.

Required estimate for total $N$	Cumulative position
lower quartile $Q_1$	$N/4$
median	$N/2$
upper quartile $Q_3$	$3N/4$
interquartile range	$Q_3-Q_1$

If the graph gives cumulative frequency $c$ at value $x$ , then about $c$ observations are $\le x$ and about $N-c$ are $>x$ . For an interval $a<x\le b$ , subtract the readings: $CF(b)-CF(a)$ .

Graph readings are estimates, so use sensible precision. When comparing two groups, state what the medians show about typical value and what the interquartile ranges show about spread.

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—6.2 Statistical measuresRead

6.1 Graphical representation of data

Learning objectives

Choose and use a suitable data display

Turn raw data into a frequency table

Read statistical diagrams accurately

Construct and interpret histograms

Build a cumulative frequency diagram

Use a cumulative frequency diagram

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