AQA GSCE: Section C - Statistics
Statistical Charts & Graphs
Pictograms
Purpose:
To represent simple categorical data using images or symbols.
How to Construct:
Choose a symbol.
Assign a key (e.g. 1 symbol = 10 items).
Draw symbols in rows to represent frequencies.
Use half-symbols or partial symbols for non-multiples of the key.
How to Interpret:
Count the number of full and partial symbols per category.
Multiply by the value of the key to get the actual frequency.
Note:
Always include a key.
Keep symbols evenly spaced.
Make sure all symbols are the same size.
Look for proportionality – the length of a row should reflect the size of the value.
Bar Charts
Purpose:
To show the frequency of different categories (discrete data).
How to Construct:
Categories go on the x-axis.
Frequencies go on the y-axis.
Draw bars for each category with equal width and equal spacing.
How to Interpret:
The height of each bar shows the frequency.
Compare heights to find most/least common categories.
Note:
Bars must not touch (unlike histograms).
Label axes clearly with units.
Use a sensible scale on the y-axis.
Pie Charts
Purpose:
To show how a total is divided between different categories proportionally.
How to Construct:
Total frequency = 360°.
For each category:
Angle=(category frequency/total frequency)×360∘
Use a protractor to draw each sector.
Label sectors or provide a key.
How to Interpret:
Larger angles = greater proportions.
You can calculate the actual frequency from the angle:
Frequency=(angle/360)×total
Note:
Add up all angles to check they total 360°.
Label sectors clearly or use a colour key.
Line Graphs
Purpose:
To show changes in data over time.
How to Construct:
Time on the x-axis, values on the y-axis.
Plot data points for each time interval.
Join points with straight lines.
How to Interpret:
Look for trends (increasing, decreasing).
Identify sharp increases/decreases or plateaus.
Useful in showing rate of change.
Note:
Points should only be connected when the variable is continuous over time.
Use gridlines to align values accurately.
Frequency Polygons
Purpose:
To represent frequency data (usually grouped) with a smooth line alternative to a bar chart.
How to Construct:
Calculate midpoint of each class interval:
Midpoint=lower limit+upper limit/2Plot midpoint against frequency.
Join points with straight lines.
Often used with or instead of a bar chart/histogram.
How to Interpret:
Compare shape of distribution: symmetric, skewed, bimodal.
Find modal group (highest peak).
Tips:
Start and finish at the x-axis by adding dummy values with 0 frequency to enclose the shape.
Time Series Graphs
Purpose:
To show how data changes over regular time intervals (e.g. months, years).
How to Construct:
Time on x-axis, values on y-axis.
Plot and connect points in order.
Label axes clearly with units and dates.
How to Interpret:
Identify trends (e.g. upward, downward).
Spot seasonal variations or cyclical patterns.
Predict future values based on patterns.
Note:
Use a ruler for connecting points cleanly.
Time intervals should be evenly spaced.
Stem-and-Leaf Diagrams
Purpose:
To show raw data in a semi-graphical format, keeping individual values visible.
How to Construct:
Split numbers into stem (e.g. tens) and leaf (e.g. units).
List all leaves in ascending order for each stem.
Include a key (e.g. 4 | 7 = 47).
How to Interpret:
Easy to find mode, median, and range.
Useful for comparing data sets (e.g. back-to-back diagrams).
Tips:
Don’t forget the key.
Group all same-stem values in the same row.
Sort the leaves for clarity.
Scatter Diagrams
Purpose:
To show the relationship between two numerical variables.
How to Construct:
Plot paired data as (x, y) coordinates.
Each point represents one data pair.
How to Interpret:
Look at overall direction:
Positive correlation: as x increases, y increases.
Negative correlation: as x increases, y decreases.
No correlation: no clear trend.
Estimate line of best fit (straight line that best follows the trend).
Use the line to make predictions (interpolation or extrapolation).
Note:
Correlation does not mean causation.
Line of best fit can be drawn by eye.
Label axes with both variable names and units.
Representing and Interpreting Data – In-Depth Guide
Grouped Frequency Tables
Purpose:
To organize large sets of continuous data into class intervals.
How to Construct:
Group raw data into intervals (e.g. 0–10, 10–20...).
Count how many values fall in each class → this is the frequency.
Interpretation:
Can't identify exact values, only how many fall within each range.
Used to estimate things like the mean, median, or mode.
Note:
Classes should be equal width (especially for histograms).
Avoid overlapping boundaries (e.g., use 10–19 not 10–20).
Estimating the Mean from Grouped Data
Why Estimate?
Because exact data values are unknown, you use midpoints of each class as estimates.
Steps:
Find midpoints of each class (e.g. for 10–20 → midpoint = 15).
Multiply midpoint × frequency for each row.
Add these up and divide by total frequency.
Note: Use a working table to stay organized.
Cumulative Frequency Curves
Purpose:
To show how the total frequency accumulates up to a certain value.
How to Construct:
Use upper class boundaries (e.g. for 10–20, use 20).
Add up frequencies cumulatively.
Plot (upper boundary, cumulative frequency).
Join points with a smooth curve (not straight lines).
Interpretation:
Use the curve to estimate:
Median (at the 50th percentile)
Quartiles (25% and 75%)
Interquartile range (IQR) = Q3 − Q1
Note: Read off values by drawing lines from the y-axis (frequency) across to the curve and down.
Box Plots (Box-and-Whisker Plots)
Purpose:
To give a quick visual summary of the distribution of data.
Key values needed:
Minimum
Lower quartile (Q1)
Median
Upper quartile (Q3)
Maximum
How to Draw:
Use a number line.
Mark the five values.
Draw a box from Q1 to Q3.
Draw a line at the median.
Add whiskers from the box to the min and max.
Interpretation:
Width of the box = Interquartile Range (spread of middle 50%)
Long whiskers = data is more spread out
Skewed if box is not symmetric
Tip: Box plots are great for comparing two sets of data!
Histograms
Purpose:
To show continuous data where class intervals are not necessarily equal.
How to Draw:
x-axis = class intervals
y-axis = frequency density
Frequency Density=Frequency/Class Width
Bars touch (since data is continuous)
Area of each bar = actual frequency
Interpretation:
Taller bars = higher density, not necessarily higher frequency.
Be careful if class widths are different!
Note: Always label axes properly and calculate frequency density clearly.
Comparing Two Sets of Data Using Summary Statistics
What to Compare:
Averages:
Mean, Median
Spread:
Range, Interquartile Range (IQR)
Standard deviation (Higher only, rarely tested)
How to Compare:
Use two clear comparisons, eg: Set A has a higher median, so the typical value is greater. However, Set B has a smaller IQR, meaning it is more consistent.
Note: Use box plots, cumulative frequency curves, or summary tables to justify comparisons.
Probability
Basic Probability
Theoretical Probability
This is the expected chance of an event happening, assuming fairness.
Probability = Number of desired outcomes/Total number of outcomes
Experimental (Relative) Probability
Based on real trials or data:
Relative frequency = Number of times event occurs/Total trials
Use this to predict future outcomes or to estimate theoretical probability.
Sample Space Diagrams
Purpose:
To list all possible outcomes of one or more events.
How to Construct:
For one event: list all options (e.g. die roll: 1–6)
For two events: create a grid or list all pairs
Use sample space to find probabilities:
P(event)=Number of favourable outcomes/Total outcomes
Relative Frequency
Already mentioned above, but to clarify:
Used when outcomes are recorded (e.g. flipping a coin 100 times).
Values may vary from theoretical probability due to random variation.
Note: As number of trials increases, relative frequency tends to approach theoretical probability.
Venn Diagrams & Set Notation
Key Symbols:
A∪B: Union (either A or B)
A∩B: Intersection (A and B)
A′: Not A (complement of A)
n(A): Number of elements in A
ξ: Universal set
Use in Probability:
Fill the Venn diagram with values.
Use set operations to calculate probabilities.
Note: Be careful not to double-count the overlap.
Tree Diagrams
Purpose:
To show all possible outcomes of two or more linked events.
How to Draw:
First branches: outcomes of first event.
Second branches: outcomes of second event (based on each first branch).
Multiply along branches for joint probability.
Add across branches for either/or situations.
Example: Tossing a coin twice
P(H then H) = 12×12=144
Independent and Dependent Events
Independent Events
One event does not affect the other.
P(A and B)=P(A)×P(B)
Example: Tossing a coin and rolling a die.
Dependent Events
One event affects the next (e.g. without replacement).
P(A then B)=P(A)×P(B ∣ A)
Example: Drawing 2 cards from a deck without replacement.
P(1st red) = 26/52
P(2nd red | 1st was red) = 25/51
So: P(2 reds)=26/52×25/51