Graphical Representation of Data

When statisticians collect raw data — exam scores, rainfall totals, survey responses — the numbers alone rarely tell a clear story. Graphical representations transform lists of values into visual patterns that reveal shape, spread, centre, and outliers almost instantly. Among the most important tools for displaying one-variable (univariate) data are bar graphs, histograms, and frequency polygons, each suited to a different type of data and a different analytical question. A bar graph is used for categorical (qualitative) data, such as favourite sports or car colours, where each bar represents a distinct category with no implied numerical order. A histogram, by contrast, is designed for continuous or grouped numerical (quantitative) data: the horizontal axis is a number line, bars share edges (no gaps), and the area of each bar is proportional to the frequency of that class interval. A frequency polygon connects the midpoints of histogram bars with straight line segments, making it easy to compare multiple distributions on the same axes. Choosing the correct graph is not merely an aesthetic decision — it determines whether the display is mathematically honest. Putting categorical data into a histogram implies a number-line relationship that does not exist, while using a bar graph for continuous measurements obscures the density structure of the data. Mastering these three representations is the first step toward sophisticated statistical literacy.

Key takeaways

Bar Graphs — Categorical Data
Choosing Class Intervals for a Histogram
Drawing a Histogram
Frequency Polygons
Reading Shape, Centre, and Spread

Bar Graphs — Categorical Data

List every category on one axis (usually the horizontal). Draw a bar whose height equals the frequency (count) or relative frequency (proportion) of that category. Bars must NOT touch, emphasising that categories are discrete and unordered. The vertical axis should start at zero; truncating it exaggerates differences and misleads readers. Example categories: {Red, Blue, Green, Yellow} with counts {14, 22, 9, 5}.

Choosing Class Intervals for a Histogram

For n data values, Sturges' Rule suggests k = 1 + 3.322 log₁₀(n) classes as a starting point, rounded to a whole number. Once k is chosen, compute the class width: w = (maximum − minimum) / k, rounding up to a convenient value. All classes must have equal width (unless specifically justified) so that bar area correctly represents frequency. Classes are written as [lower, upper) — closed on the left, open on the right — to avoid ambiguity about where boundary values belong.

Drawing a Histogram

Place the class boundaries on the horizontal axis (a continuous number line). Draw adjacent bars — NO gaps — with height equal to frequency (for a frequency histogram) or relative frequency = frequency ÷ n (for a relative-frequency histogram). If class widths are equal, height is proportional to frequency; if widths differ, use frequency density = frequency ÷ class width on the vertical axis so that area = frequency still holds. Label both axes fully, including units.

Frequency Polygons

Find the midpoint of each class interval: midpoint = (lower boundary + upper boundary) / 2. Plot the point (midpoint, frequency) for every class. Connect consecutive points with straight line segments. To 'close' the polygon and anchor it to the baseline, add a phantom class of zero frequency immediately before the first class and immediately after the last class, then draw segments down to the horizontal axis. Frequency polygons are especially powerful when overlaying two distributions (e.g., boys' vs. girls' test scores) because overlapping lines are far easier to read than overlapping histograms.

Reading Shape, Centre, and Spread

Once a histogram is drawn, describe its shape: symmetric (roughly mirror-image), skewed right (long tail toward larger values), or skewed left (long tail toward smaller values). A unimodal distribution has one peak; bimodal has two. The modal class is the class with the greatest frequency. Visually estimate centre near the 'balance point' and spread from the range covered. These qualitative descriptions guide every subsequent numerical calculation — mean, median, standard deviation.

The Key Equation

w = \frac{x_{\max} - x_{\min}}{k}, \qquad k \approx 1 + 3.322\,\log_{10}(n)

w — Class width (same as data units): The span of each equal-width interval on the histogram's horizontal axis.
x_{\max} - x_{\min} — Range (same as data units): The difference between the largest and smallest observed values in the dataset.
k — Number of classes (dimensionless): The number of bars in the histogram; estimated by Sturges' Rule and adjusted for readability.
n — Sample size (dimensionless): The total count of data values collected.

Worked Example

Problem. A teacher records the following 20 exam scores (out of 100): 45, 52, 58, 61, 63, 65, 67, 68, 70, 72, 73, 75, 76, 78, 80, 83, 85, 88, 91, 97. Construct a frequency table with appropriate class intervals and describe the shape of the resulting histogram.

k=6,\; w=10,\; \text{modal class } [70,80),\; \text{shape: left-skewed}

Answer. 6 class intervals of width 10 starting at 40. Modal class [70, 80) with frequency 6. Distribution is mildly left-skewed.

Figure

Diagram

Origins · 1 minWilliam Playfair and how this was discovered

William Playfair

1759–1823 · Scottish

William Playfair, a Scottish engineer and political economist, invented the bar chart (1786), the line graph, and the pie chart (1801) — virtually creating statistical graphics as a discipline. His 'Commercial and Political Atlas' (1786) presented trade data as bar graphs for the first time in history, demonstrating that a visual representation could communicate trends far more powerfully than tables of numbers. His frequency histogram concepts laid groundwork later formalised by Karl Pearson. Playfair was not an academic mathematician but a businessman and pamphleteer who also worked as a spy for the British government. He once attempted to destabilise the French economy by flooding France with counterfeit currency.

Measures of Central Tendency — After constructing a histogram, the next step is to calculate the mean, median, and mode — numerical summaries of the centre that complement what the graph shows visually.
Measures of Spread (Range, Variance, Standard Deviation) — The width and shape of a histogram give a visual sense of variability; range and standard deviation quantify that spread precisely.
Frequency Tables and Tally Charts — A frequency table is the organisational step that directly precedes drawing a histogram or frequency polygon; all bar heights come from the table's frequency column.
Cumulative Frequency and Ogives — An ogive is a natural extension of a frequency polygon: instead of plotting interval frequency, you plot running totals, enabling estimation of medians and percentiles graphically.