Comparing Data Sets (AQA GCSE Maths): Revision Notes

Comparing data sets

When studying statistics, you'll often need to compare different groups of data to understand patterns and draw conclusions. This involves looking at the shape, average values, and spread of different data sets using various statistical tools.

What does comparing data sets involve?

Comparing data sets means examining how different groups of data are similar or different. You might compare the distributions shown in graphs and charts, or look at specific measures like averages (mean, median, mode) and measures of spread (range, interquartile range). The key is to always consider what your findings mean in the real-world context of the data.

Comparing data sets using box plots

Box plots are particularly useful for comparing data sets because they clearly show the median, quartiles, and spread of each group. When you have two or more box plots side by side, you can quickly spot differences in the typical values and how spread out the data is.

Reading and comparing box plots

From a box plot, you can easily identify the median (the line in the middle of the box) and calculate the range and interquartile range (IQR). When comparing groups, look at both the position of the median and the size of the box and whiskers.

A larger spread in the data means the values are less consistent, showing more variation. You can compare spreads by looking at both the range (total spread from minimum to maximum) and the IQR (the spread of the middle 50% of data).

Practical example of box plot comparison

Important limitations of box plots

Using scatter plots for comparison

Scatter plots help you compare two variables for the same group of people or items. They show relationships between different measurements and can reveal patterns that other graphs might miss.

Adding box plots to scatter plots

You can create box plots from scatter plot data to compare the distributions of each variable. This involves finding the quartiles for each set of scores and drawing the corresponding box plots.

• Minimum score: $3$
• Maximum score: $20$

There are $11$ values in the data set, so:

• Lower quartile ($Q\_1$): position $\frac{11 + 1}{4} = 3^\text{rd}$ value $\Rightarrow Q\_1 = 4$
• Median ($Q\_2$): position $\frac{11 + 1}{2} = 6^\text{th}$ value $\Rightarrow Q\_2 = 12$
• Upper quartile ($Q\_3$): position $3 \times \frac{11 + 1}{4} = 9^\text{th}$ value $\Rightarrow Q\_3 = 17$

These five key values — minimum, $Q\_1$, median ($Q\_2$), $Q\_3$, and maximum — form the basis of your box plot.

Use them to compare:

• Range: $20 - 3 = 17$
• Interquartile range (IQR): $Q\_3 - Q\_1 = 17 - 4 = 13$
• Median values: $Q\_2 = 12$ for comparison of central tendency
• Skewness or distribution shape: check if the box plot is symmetrical or skewed

Comparing data sets using histograms

Histograms are excellent for comparing the shape and frequency of distributions, especially when dealing with grouped data. They show you not just the typical values, but also how the data is distributed across different ranges.

Example:

A histogram shows the time (in seconds) taken by a group of children to solve a puzzle

Step 1: Complete a Table

Fill in the missing information using the formula: Frequency = Frequency Density × Class Width

Time interval	Frequency Density	Frequency ($f$)	Midpoint ($x$)	$f \times x$
$0 \< t \leq 20$	$0.25$	$0.25 \times 20 = 5$	$10$	$5 \times 10 = 50$
$20 \< t \leq 30$	$0.8$	$0.8 \times 10 = 8$	$25$	$8 \times 25 = 200$
$30 \< t \leq 40$	$1.5$	$1.5 \times 10 = 15$	$35$	$15 \times 35 = 525$
$40 \< t \leq 50$	$0.9$	$0.9 \times 10 = 9$	$45$	$9 \times 45 = 405$
$50 \< t \leq 80$	$0.1$	$0.1 \times 30 = 3$	$65$	$3 \times 65 = 195$

• Total frequency: $5 + 8 + 15 + 9 + 3 = 40$
• Total of $f \times x$ column: $50 + 200 + 525 + 405 + 195 = 1375

Step 2: Estimate the Mean

Use the formula: $\text{Mean} = \dfrac{\text{Total of } f \times x}{\text{Total frequency}}$ $\text{Mean} = \dfrac{1375}{40} = 34.4$ seconds (to 1 decimal place

Step 3: Identify the Modal Class

The modal class is the class with the highest frequency density, not the highest frequency.

In this case: Modal class = $30 \< t \leq 40$ (since frequency density = $1.5$

Step 4: Estimate the Range

The range is calculated using the class boundaries: $\text{Range} = 80 - 0 = 80$ seconds

Understanding histograms for comparison

Calculating key statistics from histograms

From a histogram, you can estimate the mean by using the mid-point of each class interval and the frequency of each class. You can also identify the modal class, which is the class interval with the highest frequency density (not necessarily the highest frequency).

Comparing distributions with histograms

When comparing two histograms, look at the shape of each distribution, where the peaks occur, and how spread out the data is. You can also compare calculated statistics like the mean and modal class between different groups.

Making valid comparisons

Interpreting comparisons in context

The most important skill in comparing data sets is explaining what your findings mean in the real world. Statistical measures only become useful when you can explain what they tell you about the actual situation being studied.

When you find differences between groups, consider whether these differences are meaningful and what might cause them. Also think about what factors might affect your results and whether your data is representative of the populations you're trying to understand.