Maths Skills (Statistics) Revision Notes for AQA A-Level Business

Maths Skills (Statistics)

Introduction to statistics in business

Companies handle vast amounts of numerical data every day. This includes sales figures, cost information, revenue amounts, profit calculations, and market research findings. Understanding what these numbers actually tell us about business performance is crucial for making informed decisions and forecasting future success.

To make sense of all this data, businesses present information visually using diagrams and charts. This makes complex numbers easier to interpret and helps identify patterns and trends at a glance. However, it's important to develop the skills to both create and critically analyse these visual representations.

infoNote

Visual data presentation isn't just about making information look attractive — it's about transforming raw numbers into insights that can drive business decisions. The ability to interpret these visuals critically is just as important as creating them.

Types of diagrams used in business

Pie charts

Pie charts show how a total is divided into different parts, making them ideal for displaying market share — the percentage of total market sales that belong to each competitor.

How pie charts work:

The complete circle represents 100% of the data
Each segment shows a proportion of the whole
Every 1% share equals a 3.6° section (since $360° \div 100 = 3.6$ )

Benefits:

Simple to create and interpret
Can be generated quickly using spreadsheet software
Provide an instant visual impression of relative proportions

lightbulbExample

Example: Cat Food Market Share

A pie chart showing the cat food market might display segments for brands like "Whiskas", "Felix", "Kitty Treats", and "Purr", with each segment sized according to their market share. If Whiskas has 40% market share, its segment would occupy $40 \times 3.6° = 144°$ of the circle.

infoNote

Exam tip: Remember that pie charts work best when you're comparing parts of a whole. If the data doesn't add up to 100%, a pie chart probably isn't the right choice.

Bar charts

Bar charts use rectangular bars to display values for a single variable across different categories. The height (or length) of each bar represents the value being measured.

Key features:

Easy to construct and interpret
Provide high visual impact — differences between values are immediately obvious
Can show data for multiple time periods or categories side by side
Bars can vary in both width and height

lightbulbExample

Example: Monthly Revenue Analysis

A bar chart might show monthly revenue for a sun lotion company across a year, with each bar representing one month's sales figures (e.g., January through December). This would clearly reveal seasonal patterns, with peak sales during summer months and lower sales in winter.

UK business example: A supermarket chain like Tesco might use bar charts to compare sales performance across different store locations, making it easy to identify which branches are performing above or below average.

Histograms

Histograms look similar to bar charts but serve a different purpose. In a histogram, the area of each block represents the value being measured, not just the height.

chatImportant

Key Differences from Bar Charts:

Use the variable being measured (such as height) on the horizontal axis
No gaps between bars
Bars can vary in both width and height
The area of each bar is proportional to the frequency it represents

These differences are crucial — using the wrong type of chart can lead to misinterpretation of data.

When to use histograms: Histograms are particularly suitable for displaying variables with large ranges of values. For instance, comparing customer spending patterns where amounts range from £5 to £500 would work well in a histogram.

Pictograms

Pictograms are bar charts or histograms where bars are replaced with pictures or images relevant to the data being shown. These are often used in corporate brochures and marketing materials to make data more engaging and memorable.

Example: A chocolate manufacturer like Cadbury might use pictures of their chocolate bars in their sales charts, with each image representing a certain number of units sold.

Line graphs

Line graphs plot one variable against another, typically showing how something changes over time. They're excellent for identifying trends and making comparisons.

Key characteristics:

Usually show change over a time period
More than one line can be displayed on the same graph for comparison
Lines should be in different colours to keep the graph easy to read
Best for showing continuous data and trends

lightbulbExample

Example: Year-on-Year Sales Comparison

A line graph could display sales of "Benny's Bubbles" gum over the months of 2013 and 2014, with two different coloured lines showing each year's performance. This makes it easy to compare whether sales are improving or declining year-on-year. You could quickly spot if the 2014 line sits consistently above the 2013 line, indicating growth.

When diagrams can be misleading

While visual representations are useful, they can sometimes create a distorted picture of reality. It's essential to examine graphs and charts critically before drawing conclusions.

False impressions from graphs

Graphs and charts can give a false impression of what's actually happening in the data. This might be intentional (to make results look better) or unintentional (poor design choices).

The problem with non-zero scales

If the scale on a graph doesn't start at zero, it becomes difficult to accurately interpret the data. The visual representation can be distorted, making changes appear more dramatic than they actually are.

chatImportant

Critical Warning: Non-Zero Scales

A bar chart showing annual profits for a hair salon might appear to show that profit has tripled between 2010 and 2013. However, if the vertical axis starts at £20,000 rather than zero, the actual increase might only be 10% (from £20,500 to £22,500). The visual impression is misleading because we don't see the full scale.

Always check where the axes start on any graph you're analysing. If they don't start at zero, be cautious about the visual impression and look at the actual numbers instead.

Analysing data and graphs critically

Being able to read graphs is just the starting point. You need to develop the ability to analyse what the data is actually telling you.

infoNote

What to Look For in Data Analysis:

Identify the important bit of the chart — what's the key message? This might be an upward trend in sales or a large market share
Consider what might be causing the pattern you observe
Think about potential effects — what are the implications of this data?

lightbulbExample

Example: Market Share Analysis

If you notice a decrease in market share, consider what might have caused this. Perhaps a new competitor entered the market, or the marketing budget was reduced, forcing the company to increase spending to regain lost ground. This type of critical thinking goes beyond just reading the numbers — it connects data to real business decisions and outcomes.

Measures of average and spread

When working with datasets, we need ways to summarise the information. Averages help us understand what's typical, while measures of spread show how varied the data is.

Mean

The mean is what most people think of as the "average". It's calculated by adding together all the values in a dataset and dividing by the number of values.

Formula:

$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values}}$

lightbulbExample

Worked Example: Calculating Mean Customer Spend

Five customers spend £5.90, £27.97, £13.62, £24.95, and £78.81.

Step 1: Add all values together $5.90 + 27.97 + 13.62 + 24.95 + 78.81 = 151.25$

Step 2: Divide by the number of values $\text{Mean spend} = \frac{151.25}{5} = £30.25$

Business application: Shops could use mean spend per customer to set targets or evaluate pricing strategies. For example, if the mean spend is £30.25, they might aim to increase this to £35 through upselling techniques.

Median

The median is the middle value in a dataset once all values have been arranged in ascending order (from smallest to largest).

How to find the median:

Arrange all values from lowest to highest
Identify the value in the middle position

infoNote

Business application: A company might rank all salespeople by the revenue they've generated over the past month, then identify the median. Everyone above this position could receive a bonus for good performance. The median is particularly useful because it isn't affected by extreme values (very high or very low figures).

Mode

The mode is the most common number in a dataset — the value that appears most frequently.

Business application: A clothing retailer like Marks & Spencer might check the modal dress size when planning shop displays. This ensures their mannequins reflect the most common body size among British women, making displays more relevant to most customers.

infoNote

Important note: A dataset can have more than one mode if multiple values appear with equal frequency.

Range

The range shows the spread of data by measuring the difference between the largest and smallest values in the dataset.

Formula:

$\text{Range} = \text{Largest value} - \text{Smallest value}$

What it tells us: The range isn't an average, but it's often used alongside averages to give a fuller picture of the data. A large range indicates high variability, while a small range suggests values are clustered closely together.

Confidence intervals

A confidence interval is a range of values used to show the uncertainty of an estimate. It acknowledges that predictions and estimates are unlikely to be perfectly accurate.

How it works: If a business estimates sales of 2,200 units, they might say they are 95% confident that actual sales will be between 2,000 and 2,400 units. This range is the confidence interval.

infoNote

Exam tip: You don't need to calculate confidence intervals yourself — just understand what they mean. They show the likely range within which the true value will fall.

Index numbers

Index numbers provide a simple way to show percentage changes in data over time. They make it easy to track trends without getting lost in complex calculations.

How index numbers work

Businesses take a set of data showing revenue or profits over several years and create an index by:

Choosing the earliest year as the base year
Setting the value for the base year as 100
Expressing following years as a percentage of the base year figure

lightbulbExample

Example: Revenue Index for an Italian Restaurant

Year	Total Revenue	Revenue Index (2010 = 100)
2010	£17,000	100
2011	£19,550	115
2012	£21,250	125
2013	£22,440	132
2014	£24,650	145

How to calculate an index number:

To work out the revenue index for any year:

Take the total revenue from that year
Divide it by the total revenue in the base year
Multiply by 100

For 2013: $\text{Index} = \frac{£22,440}{£17,000} \times 100 = 132$

This shows that revenue in 2013 was 132% of the 2010 base year value — representing 32% growth over three years.

Benefits of using index numbers

The main advantage of indexing is that it makes trends easy to see within the business. Rather than comparing large numbers (like £17,000 versus £24,650), you can simply see that the index has increased from 100 to 145, showing 45% growth over the period.

Rearranging formulas

Sometimes you'll need to rearrange a formula before substituting numbers. The goal is to get the value you're trying to find on one side of the formula, with everything else on the other side.

General principle:

Identify which variable you need to find
Rearrange the formula so that variable is alone on one side
Then substitute the known values

lightbulbExample

Worked Example: Finding Break-Even Selling Price

Scenario: A company selling novelty doorbells has:

Fixed costs of £6,000
Variable cost per unit of £15
They need to sell 1,500 units to break even
What should the selling price per unit be?

Step 1: Start with the contribution formula: $\text{Contribution per unit} = \text{Selling price per unit} - \text{Variable costs per unit}$

Step 2: But we don't know contribution per unit yet, so we need another formula: $\text{Break-even output} = \frac{\text{Fixed costs}}{\text{Contribution per unit}}$

Step 3: Rearrange to find contribution per unit: $\text{Contribution per unit} = \frac{\text{Fixed costs}}{\text{Break-even output}}$

$\text{Contribution per unit} = \frac{£6,000}{1,500} = £4$

Step 4: Now rearrange the first formula: $\text{Selling price per unit} = \text{Contribution per unit} + \text{Variable costs per unit}$

$\text{Selling price per unit} = £4 + £15 = £19$

Answer: Each novelty doorbell must sell for £19 to break even.

Percentage changes

Businesses regularly calculate percentage increases or decreases in figures like sales volume, revenue, profit, and market share. This helps them assess how performance is progressing over time and identify trends.

Calculating percentage change

Formula:

$\text{Percentage change} = \frac{\text{New figure} - \text{Previous figure}}{\text{Previous figure}} \times 100$

lightbulbExample

Example: Sales Growth Calculation

Sales of hats increased from 9,000 to 11,000 units.

$\text{Percentage increase} = \frac{11,000 - 9,000}{9,000} \times 100$

$\text{Percentage increase} = \frac{2,000}{9,000} \times 100 = 22.2\%$

Increasing a figure by a percentage

You can rearrange the percentage change formula to calculate what a new figure will be after a percentage increase.

Formula:

$\text{New figure} = \frac{\text{Percentage change} \times \text{Previous figure}}{100} + \text{Previous figure}$

lightbulbExample

Example: Projecting Future Profit

A business's profit was £40,000 in 2013. It increased by 20% in 2014.

$\text{2014 profit} = \frac{20 \times £40,000}{100} + £40,000$

$\text{2014 profit} = £8,000 + £40,000 = £48,000$

infoNote

Exam tip: For percentage increases, you can also multiply the original figure by 1.2 (for a 20% increase), 1.15 (for 15%), etc. For decreases, multiply by 0.8 (for a 20% decrease), 0.85 (for 15%), etc.

Converting between percentages, fractions, and ratios

You might encounter data about a company or product expressed in different formats. Being able to convert between these formats is an essential skill.

Fractions to percentages

To convert from fractions to percentages, multiply by 100.

Example: If $\frac{1}{4}$ of a company's total revenue is profit, then $\left(\frac{1}{4} \times 100\right) = 25\%$ of its total revenue is profit.

Percentages to fractions

To convert from percentages to fractions, divide by 100 and simplify.

Example: $25\% = \frac{25}{100} = \frac{1}{4}$ (simplified)

Fractions to ratios

A ratio compares one amount to another. You can convert fractions to ratios by considering the relationship between parts.

Example: If profit is $\frac{1}{4}$ of revenue, this means 1 part profit to 4 parts revenue, so the ratio of profit to revenue is 1:4. This means for every £4 of revenue, the company makes £1 of profit.

lightbulbExample

Worked Example: Labour Turnover in Multiple Formats

Scenario: During one year, the average number of employees at a company is 100, and 20 employees leave. Calculate labour turnover as a percentage, as a fraction, and find the ratio.

As a percentage: $\text{Labour Turnover (\%)} = \frac{\text{Number of staff leaving}}{\text{Average number of staff employed}} \times 100$

$\text{Labour Turnover} = \frac{20}{100} \times 100 = 20\%$

As a fraction: $\text{Labour turnover} = \frac{20}{100} = \frac{1}{5} \text{ (simplified)}$

As a ratio: The ratio of employees leaving to average number of employees is 20:100, which simplifies to 1:5.

Interpretation: For every 5 employees, 1 leaves during the year. This could indicate issues with employee satisfaction or working conditions that management should address.

bookmarkSummary

Key Points to Remember:

Visual representations matter: Pie charts show market share, bar charts compare single variables, line graphs reveal trends over time, and histograms work best for large data ranges.
Check scales carefully: Graphs that don't start at zero can create misleading impressions. Always examine the actual numbers alongside the visual representation.
Three types of average: Mean (add all values and divide), median (middle value in order), and mode (most common value). Each tells you something different about the data.
Index numbers simplify trends: Setting a base year as 100 makes percentage changes over time much easier to spot and understand.
Percentage change formula is essential: $\frac{\text{New figure} - \text{Previous figure}}{\text{Previous figure}} \times 100$ . Master this formula and its rearranged versions for exam success.

Maths Skills (Statistics) (AQA A-Level Business): Revision Notes