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5.1 Representation of data

Syllabus
9709–2028–2029
Topic
5.1
Level
A2

Learning objectives

5.1.1Data representation• select a suitable way of presenting raw statistical data, and discuss advantages and/ or disadvantages that particular representations may have5.1.2Charts and diagrams• draw and interpret stem-and-leaf diagrams, box-and-whisker plots, histograms and cumulative frequency graphs Including back-to-back stem-and-leaf diagrams.5.1.3Averages and spread• understand and use different measures of central tendency (mean, median, mode) and variation (range, interquartile range, standard deviation) e.g. in comparing and contrasting sets of data.5.1.4A cumulative frequency graph e• use a cumulative frequency graph e.g. to estimate medians, quartiles, percentiles, the proportion of a distribution above (or below) a given value, or between two values.5.1.5And use the mean and standard• calculate and use the mean and standard deviation of a set of data (including grouped data) either from the data itself or from given totals x/ and x2/, or coded totals xa-^h/ and xa 2-^h/, and use such totals in solving problems which may involve up to two data sets.

Choose a display by what must remain visible

Need/data Suitable representation Advantage Limitation
preserve small raw dataset stem-and-leaf retains individual values/order crowded for large range
compare centre/spread box plot compact five-number comparison hides shape/details
continuous class shape histogram area shows frequency/density loses raw values
percentiles/proportions cumulative-frequency graph reads cumulative ranks graph readings are estimates

Identify variable type, sample size, grouping and comparison question. Choose a display whose encoding answers that question, label scale/units/key and state one relevant advantage and disadvantage.

For raw numerical data, a stem-and-leaf diagram may preserve more information than grouped displays. Grouping improves compactness but loses within-class positions.

Use the same scale/class boundaries where direct comparison is intended; otherwise visual differences may be caused by encoding choices.

No display is universally “best”. Histograms are for continuous intervals, not separated categorical bars; scatter plots are not part of this objective’s named one-variable displays.

Construct and read each named statistical display

Display Construction invariant Main reading
stem-and-leaf / back-to-back ordered leaves, common stem, key raw values, shape, median/mode
box-and-whisker min, Q1Q_1, median, Q3Q_3, max on scale centre, IQR/range, skew comparison
histogram contiguous class boundaries; density=f/class width\text{density}=f/\text{class width} bar area = frequency
cumulative-frequency cumulative totals at upper class boundaries, monotone curve quantiles/percentiles/proportions

For back-to-back stems, use one shared key and order leaves away from the stem consistently on each side so both datasets can be reconstructed.

\text{frequency density}=\frac{\text{frequency}}{\text{class width}},\qquad \text{frequency}=\text{bar area}.

State comparisons using numerical features: median/centre, IQR/range, shape or modal class. Label axes, units and sample identity.

Histogram height is not frequency for unequal class widths. A cumulative-frequency graph uses running totals, not ordinary class frequencies.

Averages describe centre while spread describes variation around it

Mean, median and mode describe location. Range, interquartile range, variance and standard deviation describe spread; each responds differently to outliers and skew.

Use the mean when all values and squared deviations are meaningful, the median for skewed or ordinal data, and state which spread measure matches the centre.

One extreme value can raise the mean and standard deviation while leaving the median and IQR almost unchanged.

A larger mean does not imply greater variability, and “average” is not automatically the arithmetic mean.

Convert cumulative ranks, values and intervals in both directions

For total frequency $N$: median rank $N/2$, quartiles $N/4$ and $3N/4$, percentile $p$ rank $pN/100$. Read the corresponding data value from the horizontal axis.

To estimate the number/proportion below value xx, read cumulative count C(x)C(x), then use C(x)/NC(x)/N. Above xx: [N−C(x)]/N[N-C(x)]/N.

For $a<X\le b$, estimated count isC(b)-C(a),and estimated proportion is $[C(b)-C(a)]/N$.

With N=80N=80, Q3Q_3 is the x-value at cumulative frequency 6060. If C(50)=62C(50)=62 and C(30)=18C(30)=18, the estimated proportion between 3030 and 5050 is (62−18)/80=0.55(62-18)/80=0.55.

Graph readings from grouped data are estimates. Cumulative frequency is a count/rank, not itself the percentile’s data value.

Compute mean and standard deviation from totals, coding or combined sets

For $n=\sum f$:\bar x=\frac{\sum fx}{n},\qquad \sigma=\sqrt{\frac{\sum fx^2}{n}-\bar x^2}.For raw data take $f=1$; for grouped data use class midpoint $x$ and label results estimates.

Build/use n,∑x,∑x2n,\sum x,\sum x^2 (or ∑f,∑fx,∑fx2\sum f,\sum fx,\sum fx^2). Given totals can be inserted directly; do not reconstruct fictional raw values.

If $y=(x-a)/b$, then\bar x=a+b\bar y,\qquad \sigma_x=|b|\sigma_y.Coded totals $\sum(x-a)$ or $\sum(x-a)^2$ can also be expanded algebraically to recover totals.

For two datasets, add nn, ∑x\sum x and ∑x2\sum x^2, then recompute the combined mean and SD. Do not average the two means/SDs unless sample sizes and formulas justify it.

∑x2\sum x^2 means sum of squared observations, not (∑x)2(\sum x)^2. Grouped midpoint results are estimates; combined variance must be rebuilt from combined totals.