Comparing Data

Description of Levels

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Level 1 - Summary Statistics

Level 2 - Raw Data

Level 3 - Frequency Charts

Level 4 - Stem-and-Leaf

Level 5 - Box Plots

Level 6 - Cumulative Frequency

Level 7 - Histograms

Level 8 - Mixed Representations

Exam Style Questions - A collection of problems in the style of GCSE or IB/A-level exam paper questions (worked solutions are available for Transum subscribers).

More on this topic including lesson Starters, visual aids, investigations and self-marking exercises.

Comparing Two Distributions

This exercise asks you to choose which of the given statements are true, false, or cannot be determined. In exams you are often asked to come up with your own statements comparing two distributions.

Whatever diagram the data is presented in, a good comparison does two things:

1. Compares a measure of average (usually the median) — which set of values is typically higher?
2. Compares a measure of spread (usually the interquartile range, or the range) — which set is more consistent?

Always write your comparison in context and comparatively. For example: “The median journey time on the west route was longer than on the east route, but the east route times were more spread out.” A single sentence that only describes one distribution on its own is not a comparison.

It is just as important to know what a diagram cannot tell you. Summary diagrams often hide the individual data values, the exact number of items, or how the data is distributed within a class. If a question asks for something the diagram does not show, the honest answer is “cannot be determined”.

A note on method: the counting method described above is for discrete data where you have every value. For grouped or continuous data (cumulative frequency graphs and histograms) we instead estimate the quartiles at the \(\frac{n}{2}\), \(\frac{n}{4}\) and \(\frac{3n}{4}\) positions.

Raw data

For discrete distributions, there is no universal agreement on selecting the quartile values, but for the purpose of this exercise, we'll use the most popular method:

1. The median is the value in the middle of the data set when it is arranged in order of size.
2. Use the median to divide the ordered data set into two halves. The median becomes the second quartile.
3. If there are an odd number of data points in the original ordered data set, do not include the median (the central value in the ordered list) in either half.
4. If there are an even number of data points in the original ordered data set, split this data set exactly in half.
5. The lower quartile value is the median of the lower half of the data. The upper quartile value is the median of the upper half of the data.

Read more about the methods for finding quartiles of discrete data in the December 2024 Transum Newsletter.

The interquartile range is the difference between the upper and lower quartiles: \( IQR = Q_3 - Q_1 \).

Outliers are the points lying beyond the upper boundary of \(Q_3 + 1.5 \times IQR\) and the lower boundary of \(Q_1 - 1.5 \times IQR\).

Frequency charts

A frequency chart or table records how many times each value (or category) occurs. Find the total \(n\) by adding the frequencies. The mode is the value with the highest frequency. To find the median, count along the frequencies to the middle position; the quartiles are found the same way using the method above.

You can read: mode, median, quartiles, range and the exact value of \(n\). You cannot read: anything not recorded in the chart.

Stem and leaf diagrams

A stem and leaf diagram keeps every original data value, already arranged in order, so it is one of the most informative displays. Count the leaves to find \(n\), then count in from the ends to locate the median and quartiles exactly. Remember to read the key so you interpret each value correctly.

You can read: every individual value, the exact median, quartiles, range, mode and any outliers. You cannot read: nothing is hidden — this diagram shows the full data set. A back-to-back stem and leaf diagram lets you compare two sets directly.

Box plots (box and whisker diagrams)

A box plot displays the five-number summary: minimum, lower quartile, median, upper quartile and maximum. The length of the box is the interquartile range and the distance between the whisker ends is the range. The position of the median line inside the box, and the relative whisker lengths, indicate the skew of the data.

You can read: median, quartiles, IQR, range, and the fact that roughly a quarter of the data lies in each section. You cannot read: the number of data values \(n\), the mean, or any individual value. You cannot tell how many items lie in a particular interval — only that 25% fall between each pair of marks.

Cumulative frequency graphs

A cumulative frequency graph (ogive) plots a running total against the upper boundary of each class. Read the median across from \(\frac{n}{2}\), the lower quartile from \(\frac{n}{4}\) and the upper quartile from \(\frac{3n}{4}\); you can also estimate how many values fall below (or above) any chosen amount. When two curves share the same axes, the curve further to the right represents larger values, and a steeper curve means the data is more tightly concentrated (smaller spread). Where two curves cross, the same number of items from each set lies below that value.

You can read: estimates of the median, quartiles, percentiles, IQR and the number of values below a given amount. You cannot read: individual values, the exact mean, or how the data is distributed within a single class.

Histograms

When the class widths are unequal, the vertical axis must be frequency density, not frequency:

\[ \text{frequency density} = \frac{\text{frequency}}{\text{class width}} \qquad \text{so} \qquad \text{frequency} = \text{frequency density} \times \text{class width} \]

This means it is the area of each bar, not its height, that represents the frequency. The tallest bar is not necessarily the one containing the most data — always multiply by the class width. The modal class is the bar with the greatest frequency density. Quartiles can be estimated by finding the values that divide the total area into halves and quarters.

You can read: the frequency in each class (frequency density × width), the total \(n\), the modal class, and estimates of the median and quartiles. You cannot read: individual values, or how the data is spread within a class.

Scatter graphs

A scatter graph plots paired data, with each point showing two measurements for the same item (for example, one student’s statistics mark and physics mark). To compare the two distributions, treat each axis separately: read all the horizontal values as one data set and all the vertical values as another, then compare their medians and quartiles in the usual way.

Be careful to keep two ideas apart. Comparing the distributions asks which subject’s marks were higher and more spread out. Correlation and a line of best fit describe the relationship between the two variables — a different question. A strong upward trend does not by itself tell you which set of marks was higher; for that you must compare the two sets of values directly.

You can read: every individual pair of values, so the median, quartiles and range of each variable can be found exactly. You cannot read: nothing is hidden, but remember that the line of best fit answers a question about relationship, not about comparing the two distributions.

Some questions have a hint that can help you find a way to solve the problem. If you see a?  button, click it to reveal the hint.

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