Representing and Comparing Data Distributions

What a data distribution tells us

A data set is more than a list of numbers. Its distribution describes where the values lie, how spread out they are and whether there are any unusual features. Here we are dealing with univariate empirical data: values for one observed or measured variable, such as the journey time of each pupil.

To analyse a distribution, ask three broad questions:

1. Where is its centre, or typical value?
2. How widely are its values spread?
3. What does its graphical shape reveal, such as clusters, gaps or possible outliers?

When comparing two groups, the purpose is not merely to list facts about each one. You must connect corresponding features: compare centre with centre and spread with spread, then interpret those differences in the context of the question.

Recognising the type of data

Discrete data takes separate permitted values. The number of books read might be 3 or 4, but not 3.6. Ungrouped discrete data can be displayed using a frequency table or a vertical line chart, where each possible value has its own frequency.

Continuous data can take any value in an interval, even though measurements are rounded when recorded. Height, mass and time are continuous. When there are many values, they are often grouped into class intervals such as $150 < h \leq 160$.

Grouping makes a large data set easier to see, but it loses information: you know how many values lie in each class, not their exact values. Conclusions calculated from grouped data are therefore generally estimates.

Choosing a useful representation

Different diagrams reveal different features.

• A vertical line chart is suitable for ungrouped discrete numerical data because the possible values are separate.
• A histogram represents grouped continuous data. Its bars touch because the scale is continuous. The horizontal width shows the class interval, while the area of a bar represents frequency.
• A cumulative frequency graph shows a running total at the upper boundary of each class. It can be used to estimate the median and quartiles.
• A box plot gives a compact summary of centre and spread and is particularly useful for comparing distributions. Box plots are Higher-tier material in the supporting GCSE notes.

The most suitable diagram depends on the data and the purpose. A box plot makes comparisons quick, but it does not show every frequency, gap or mode. A histogram reveals the shape of grouped data more clearly, but reading quartiles from it is less direct.

Reading histograms correctly

A histogram can have equal or unequal class widths. When widths are unequal, bar height cannot represent frequency directly. Instead,

$$\text{frequency density}=\frac{\text{frequency}}{\text{class width}}$$

and therefore

$$\text{frequency}=\text{class width}\times\text{frequency density}.$$

This makes the area of each bar proportional to its frequency. A tall narrow bar can therefore represent fewer observations than a shorter, wider bar. When analysing a histogram, use bar areas to judge how many observations occupy different intervals, and look for concentration, gaps and a longer tail on one side.

Cumulative frequency and grouped distributions

Cumulative frequency is a running total. For each class, add its frequency to all previous frequencies and plot the total against that class's upper boundary. The resulting curve never decreases.

If there are $n$ observations, the median is read at cumulative frequency $n/2$. The lower and upper quartiles are read at $n/4$ and $3n/4$. Move horizontally from the required cumulative frequency to the curve, then vertically down to the variable axis. Because the original values were grouped and the curve is used for reading, the answers are estimates.

Understanding a box plot

A box plot is built from the five-number summary:

• minimum;
• lower quartile, $Q_1$;
• median, $Q_2$;
• upper quartile, $Q_3$;

The box extends from $Q_1$ to $Q_3$, with a line at the median. Lines often called whiskers extend to the minimum and maximum in the GCSE convention shown here. The box contains the middle 50% of the ordered data.

Its widths encode measures of spread:

$$\text{range}=\text{maximum}-\text{minimum}$$ $$\text{interquartile range}=Q_3-Q_1.$$

The range uses the two extreme values and can be strongly affected by an outlier. The interquartile range, or IQR, describes the middle 50%, so it is less affected by extreme values.

To construct a box plot, draw and label a numerical scale, mark all five summary values, draw the box from $Q_1$ to $Q_3$, place the median line inside it, and connect the box to the minimum and maximum. Box plots intended for comparison must be drawn on the same scale.

Comparing two distributions

A strong comparison normally addresses both centre and spread.

If group A has a higher median than group B, a typical value in A is higher. This does not prove that every value in A is higher than every value in B: the two distributions may overlap.

If group A has a smaller IQR, its middle 50% is less spread out. In context, its results may be described as more consistent. If A has a larger range, however, its full set of values is more spread out from minimum to maximum. These two conclusions can both be true because IQR and range describe different parts of the data.

Comparisons should quote relevant values where they can be read. For example: “The median journey time for route A is higher, but its IQR is smaller, so route A is typically slower but its middle half of journey times is more consistent.” This is more informative than saying only that one box plot is further to the right.

Interpreting with care

A graph summarises the available observations; it does not automatically explain why the groups differ. A difference between two distributions may be real in the recorded data, but the graph alone does not establish a cause.

Also notice the information a representation hides. Grouping removes exact individual values, a box plot hides detailed frequencies and modes, and a cumulative frequency graph gives estimates. Good statistical analysis uses the strengths of the chosen representation without claiming more than it shows.