AQA GSCE: Section D - Statistics
Pictograms
What is a Pictogram?
A pictogram (also called a pictograph) is a visual method of representing data using symbols or images. Each symbol stands for a certain number of items, and this number must be specified in a key or legend. Pictograms are useful for presenting data in a way that is easy to understand at a glance, especially for small data sets.
Key Characteristics:
Symbols represent quantities: One picture or icon typically represents multiple units. For example, one picture of a car might represent 10 actual cars.
Must include a key: A pictogram always includes a key that clearly states what each symbol represents. This allows accurate interpretation of the data.
Can use part-symbols: When a quantity cannot be represented by a whole symbol, part of a symbol is used (e.g. half a circle to show five if one circle represents ten).
Data should be discrete: Pictograms are used for discrete data (separate, countable items like number of students, number of pets, etc.).
Neat presentation is important: The symbols should be arranged in a clear, uniform way, typically in a straight line or a grid.
Example:
Suppose a pictogram shows how many ice creams were sold on different days of the week. If the key states that one picture of an ice cream represents 5 sales, and Monday has 3 whole symbols and a half-symbol, then the total number of ice creams sold on Monday is:
(3 × 5) + (0.5 × 5) = 15 + 2.5 = 17.5 (often rounded or represented as an estimate)
Advantages of Pictograms:
Easy to interpret visually
Useful for young learners or general audiences
Good for simple comparisons
Limitations:
Not suitable for large or complex datasets
Difficult to use for precise data
Drawing accurate partial symbols can be time-consuming or confusing
Bar Charts
What is a Bar Chart?
A bar chart is a diagram that uses rectangular bars to represent data. The height or length of each bar corresponds to the frequency of the category it represents. Bar charts are commonly used to compare discrete data, and they provide a clear and accurate way to display differences between groups or categories.
Key Characteristics:
Each bar represents a category: The categories being compared are listed along one axis (usually the horizontal axis).
Bar height equals frequency: The other axis (usually vertical) is labeled with a numerical scale that shows how many items are in each category.
Bars have equal width: All bars must be the same width to maintain consistency and avoid misleading interpretations.
Bars do not touch: Since the data is discrete (not continuous), there should be spaces between the bars. This distinguishes a bar chart from a histogram, where bars touch to represent continuous data.
Axes are clearly labeled: The horizontal axis should show the categories; the vertical axis should be labeled with a scale and a title, such as “Number of people” or “Frequency.”
Can be vertical or horizontal: Although vertical bar charts are more common, horizontal ones can also be used and are equally valid.
Example:
Imagine a survey where students choose their favorite subject. The results show:
10 students chose Maths
7 students chose English
12 students chose Science
5 students chose History
A bar chart would show four bars, each labeled with the subject, and the height of each bar would match the number of students. For example, the Maths bar would reach up to 10 on the vertical scale.
Advantages of Bar Charts:
They make it easy to compare data between categories
Accurate and scalable to large datasets
Visually clear for spotting the highest or lowest values
Limitations:
Cannot be used for continuous data
Overcrowding can occur if too many categories are shown
Does not show internal breakdowns of data within categories unless extended (e.g. stacked or grouped bar charts)
Pie Charts
What is a Pie Chart?
A pie chart is a circular chart divided into sectors, where each sector represents a proportion of a whole. It is used to display categorical data as parts of a total, often in percentages or fractions.
Key Characteristics:
The entire circle represents the total data set, equivalent to 360 degrees.
Each sector’s angle is calculated using the formula:
Angle = (Category frequency / Total frequency) × 360Categories with larger frequencies have larger sectors.
The chart may include labels, percentages, or a key to identify each category.
Pie charts are generally not used to show actual quantities but rather relative proportions.
When to Use:
When you want to compare parts of a whole.
When exact values are less important than visual proportions.
When data categories are mutually exclusive and the total is fixed.
Example:
Suppose a survey records favourite snacks among 100 students:
Crisps: 40
Fruit: 25
Chocolate: 20
Biscuits: 15
To draw the pie chart, calculate the angle for each category:
Crisps: (40/100) × 360 = 144°
Fruit: (25/100) × 360 = 90°
Chocolate: (20/100) × 360 = 72°
Biscuits: (15/100) × 360 = 54°
Then construct the chart using a protractor to measure and draw each sector accurately.
Advantages:
Visually displays how each category contributes to a total.
Useful for showing proportions and percentages.
Effective for small data sets with a few categories.
Limitations:
Not suitable for large numbers of categories or precise comparisons.
Difficult to interpret small differences in sector size.
Requires total data to calculate angles accurately.
Line Graphs
What is a Line Graph?
A line graph is a type of chart that displays information over time, using points plotted along a pair of axes. The points are then connected by straight lines to show trends or changes.
Key Characteristics:
The horizontal axis (x-axis) typically represents time (e.g. months, years, days).
The vertical axis (y-axis) represents the quantity being measured (e.g. temperature, sales, population).
Data points are plotted and joined by straight lines.
Can be used to show increasing or decreasing trends, fluctuations, or steady changes.
Common in time series data.
When to Use:
When data is collected at regular intervals (e.g. daily, weekly, yearly).
To observe trends over time (e.g. sales growth, weather patterns).
To identify patterns, such as seasonal variation or long-term trends.
Example:
Suppose a student records the average daily temperature at noon over five days:
Monday: 14°C
Tuesday: 16°C
Wednesday: 17°C
Thursday: 15°C
Friday: 18°C
The line graph plots these values with time (days) on the x-axis and temperature on the y-axis. Each point is plotted, then joined by straight lines.
Advantages:
Clearly shows trends and changes over time.
Good for comparing rates of change.
Easy to identify peaks, drops, and patterns.
Limitations:
Requires data to be ordered (usually over time).
Only suitable for continuous or time-based data.
Misleading if scale is uneven or if too many points are plotted without clarity.
Frequency Polygons
What is a Frequency Polygon?
A frequency polygon is a line graph used to display the distribution of grouped continuous data. It is constructed by plotting points at the midpoints of class intervals and connecting them with straight lines.
Key Characteristics:
Data must be grouped into intervals (e.g. 10–20, 20–30, etc.).
For each class interval, you calculate the class midpoint:
Midpoint = (Lower bound + Upper bound) ÷ 2The frequency is plotted on the y-axis, and the midpoint on the x-axis.
Points are connected by straight lines.
The graph often starts and ends at the horizontal axis to give it a clear shape.
When to Use:
To represent continuous data visually.
To compare two or more distributions (multiple polygons can be drawn on the same axes).
As an alternative to histograms for visual clarity.
Advantages:
Smooth representation of frequency distributions.
Easy to compare different sets of data.
Maintains the shape of distribution like a histogram but without bars.
Limitations:
Requires grouped data, which may result in loss of detail.
Less precise than a histogram for showing frequencies within intervals.
Requires accurate calculation of midpoints.
Time Series
What is a Time Series?
A time series is a set of data points that are recorded or measured at successive points in time, often at regular intervals (e.g. daily, monthly, yearly). A time series graph is used to represent this data, helping to identify trends, patterns, or seasonal effects.
Key Characteristics:
The horizontal axis (x-axis) always represents time (e.g. years, quarters, weeks).
The vertical axis (y-axis) shows the variable being measured (e.g. sales, temperature, profit).
Data points are plotted and connected with straight lines to show the progression over time.
Trends can be identified (e.g. increasing, decreasing, constant).
Seasonal variation may be visible if data follows a repeating pattern (e.g. higher sales in December).
Components of a Time Series:
Trend – General direction of the data (upward, downward, stable).
Seasonal variation – Regular fluctuations depending on time of year or cycle.
Irregular variation – Unpredictable, short-term fluctuations (e.g. spikes, drops).
Example:
If a company tracks monthly profits for a year, each month is plotted on the x-axis and the profit on the y-axis. Connecting the points will show whether profits are rising, falling, or fluctuating.
When to Use:
To examine changes over time.
To identify long-term patterns or short-term irregularities.
To make future predictions based on observed trends.
Stem-and-Leaf Diagrams
What is a Stem-and-Leaf Diagram?
A stem-and-leaf diagram is a way of organising numerical data to show its distribution while retaining the original data values. It splits each number into a "stem" (the leading digit(s)) and a "leaf" (the last digit).
Structure:
The stem represents the tens, hundreds, or other larger place values.
The leaf represents the units or the final digit.
Data is listed in ascending order along each row.
A key (or legend) must be provided to explain the format.
Advantages:
Maintains original data values (unlike bar charts or histograms).
Quickly shows distribution and spread of data.
Easier to identify mode, median, and range.
Limitations:
Only suitable for small or moderately sized data sets.
Becomes difficult to read if data contains many digits or decimal values.
Back-to-Back Stem-and-Leaf:
Used to compare two data sets side by side, with one set of leaves extending to the left and the other to the right of a shared stem.
Scatter Diagrams (Scatter Graphs)
What is a Scatter Diagram?
A scatter diagram is a graph used to explore the relationship between two quantitative variables. Each data point represents a pair of values (x, y), plotted on a Cartesian coordinate system.
Key Characteristics:
The horizontal axis (x) typically represents the independent variable.
The vertical axis (y) represents the dependent variable.
Each plotted point represents one observation (e.g. height vs weight of a person).
It is used to detect correlation or association between variables.
Types of Correlation:
Positive Correlation – As one variable increases, the other also increases.
Negative Correlation – As one variable increases, the other decreases.
No Correlation – No clear relationship between the two variables.
Line of Best Fit:
A straight line that best represents the general trend in the data.
It may not pass through any actual points, but it minimizes the overall distance from all points.
It can be used to make predictions for values not in the original data.
Example:
Suppose you plot hours studied (x-axis) against test score (y-axis). If the points show an upward trend, that suggests positive correlation — more studying tends to lead to higher scores.
Causation vs Correlation:
Correlation shows that two variables move together.
Causation implies one variable directly affects the other.
Scatter diagrams only show correlation, not causation.
Advantages:
Visually identifies relationships or patterns.
Useful for prediction if a strong correlation is found.
Can highlight outliers that do not fit the trend.
Limitations:
Not useful for categorical data.
Weak or non-linear relationships may not be obvious.
Correlation ≠ causation — other factors might be involved.