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Statistical RepresentationsEdexcel GCSE Maths: Revision notes

Section 1

How do you construct and interpret bar charts, pie charts and pictograms?

Bar charts display categorical data using rectangular bars, where the height (or length) represents frequency. The bars must be of equal width, and there should be a gap between each bar. The axes must be clearly labelled with a scale.

Pie charts represent data as sectors of a circle, where each sector's angle is proportional to the frequency. To find the angle for a category, use the formula:

Angle = (Frequency ÷ Total Frequency) × 360°

Pictograms use repeated symbols or pictures to represent data, where each symbol represents a fixed frequency. A key must always be provided (e.g. one symbol = 10 items). Fractional parts of symbols can be used to represent smaller frequencies.

When interpreting these charts:

  • Read values carefully from the axes or scale
  • Identify the highest and lowest categories
  • Calculate totals where necessary
  • Compare frequencies between categories
Key termsbar chartpie chartpictogramfrequencycategorical data
Exam tip

For pie charts, examiners expect you to show the calculation of angles clearly. Always write out: 'Angle = (frequency ÷ total) × 360°' and show your working.

Common mistake

Forgetting to include a key on pictograms, or using inconsistent symbol sizes. Every pictogram must state what each symbol represents.

Section 2

What are frequency diagrams and histograms with equal class widths?

Frequency diagrams are bar charts used for numerical (grouped) data, where the height of each bar represents the frequency for that class interval. Unlike bar charts, the bars touch each other because the data is continuous.

Histograms with equal class widths use the same principle: frequency is represented by the height of the bar. The key difference from bar charts is that:

  • The bars represent continuous data intervals
  • The bars are touching (no gaps)
  • The width of each bar represents the class width
  • The height represents frequency (for equal widths only)

To construct a histogram with equal class widths:

  1. Draw the horizontal axis with class intervals (e.g. 0–10, 10–20, 20–30)
  2. Draw the vertical axis labelled 'Frequency'
  3. Draw bars touching each other with heights corresponding to the frequency
  4. Include a title and axis labels

When interpreting: read the frequency from the height of each bar and identify which class has the highest frequency.

Key termsfrequency diagramhistogramclass widthclass intervalcontinuous data
Exam tip

For equal class width histograms, the height of the bar directly equals the frequency. Examiners want to see bars touching and clearly labelled axes.

Think of it like this

Think of a histogram like a row of adjoining houses on a street—there are no gaps between them because the data flows continuously from one class to the next.

Section 3

How do you construct histograms with unequal class widths using frequency density? (HT)

When class intervals have unequal widths, you cannot use frequency as the height directly. Instead, you must use frequency density.

Frequency density = Frequency ÷ Class Width

The height of each bar in the histogram is the frequency density, not the frequency. This ensures that the area of each bar represents the frequency.

Key relationship: Area of bar = Frequency

To construct:

  1. Calculate the class width for each interval
  2. Calculate frequency density for each class: Frequency ÷ Class Width
  3. Draw the histogram with frequency density on the vertical axis
  4. Draw bars touching each other with height equal to frequency density

To read a histogram with unequal class widths:

  1. Identify the frequency density from the height of the bar
  2. Read the class width from the horizontal axis
  3. Calculate frequency: Frequency = Frequency Density × Class Width
ClassFrequencyWidthFrequency Density
0–51553
5–1530103
15–201052
Key termsfrequency densityunequal class widthsclass widtharea of bar
Exam tip

Always remember: in a histogram with unequal class widths, the AREA of the bar equals the frequency, not the height. Show your frequency density calculation explicitly.

Example

A histogram bar has height (frequency density) 2.5 and width 4. The frequency = 2.5 × 4 = 10. Conversely, if frequency is 20 and width is 8, then frequency density = 20 ÷ 8 = 2.5.

Common mistake

Using frequency as the height when class widths are unequal. This creates a misleading histogram where wider classes appear to have higher frequencies even if they don't.

Section 4

How do you construct and interpret cumulative frequency graphs?

Cumulative frequency is the running total of frequencies. It increases (or stays the same) as you move through the data from lowest to highest value.

To construct a cumulative frequency table:

  1. List the class intervals in order
  2. Calculate the frequency for each class
  3. Add each frequency to the sum of all previous frequencies
  4. Record the cumulative frequency at the upper class boundary of each interval

To draw a cumulative frequency graph:

  1. Plot points at (upper class boundary, cumulative frequency)
  2. Join the points with a smooth curve (not straight lines)
  3. The curve should start at the lower boundary of the first class with cumulative frequency 0
  4. Label axes clearly: horizontal axis is the variable, vertical axis is cumulative frequency

Reading from a cumulative frequency graph:

  • Find the median by locating ½ of the total cumulative frequency on the vertical axis, then reading across to the curve and down
  • Find the lower quartile (LQ) at ¼ of total cumulative frequency
  • Find the upper quartile (UQ) at ¾ of total cumulative frequency
  • Find the interquartile range (IQR) = UQ − LQ

These quartile values are used to construct box plots.

Key termscumulative frequencycumulative frequency graphmedianquartileinterquartile range
Exam tip

Always plot cumulative frequency at the upper class boundary, not the midpoint. Draw a smooth curve through your points; examiners mark this carefully.

Example

If a dataset has 100 values, the median is at cumulative frequency 50, LQ at 25, and UQ at 75. Read these from the graph to find the exact values.

Section 5

How do you construct and use box plots to compare distributions? (HT)

A box plot (also called a box-and-whisker diagram) displays the distribution of data using five key values: minimum, lower quartile (Q1), median (Q2), upper quartile (Q3), and maximum.

To construct a box plot from cumulative frequency data:

  1. Find Q1 (lower quartile) at ¼ of cumulative frequency
  2. Find Q2 (median) at ½ of cumulative frequency
  3. Find Q3 (upper quartile) at ¾ of cumulative frequency
  4. Identify the minimum and maximum values from the dataset
  5. Draw a horizontal line for the minimum value
  6. Draw a vertical line at Q1 and a box extending to Q3
  7. Draw a line inside the box at Q2 (the median)
  8. Draw a horizontal line extending to the maximum value

Comparing distributions using box plots:

  • Median: Compare the position of the median line in each box. A higher median indicates higher central values
  • Spread: Compare the length of the whiskers and the box. Longer whiskers indicate greater range; larger boxes indicate larger interquartile range
  • Skewness: If the median line is not centred in the box, the distribution is skewed. A median closer to Q1 suggests right skew; closer to Q3 suggests left skew
  • Outliers: Values lying beyond 1.5 × IQR from the quartiles can be identified as outliers

Box plots allow rapid visual comparison of multiple datasets on the same scale.

Key termsbox plotquartilemedianinterquartile rangerangeskewness
Exam tip

When comparing box plots, examiners expect you to mention median, spread (IQR or range), and any apparent skewness. Use precise language: 'Dataset A has a higher median' or 'Dataset B has a larger IQR'.

Think of it like this

A box plot is like a visual summary card for data—the box shows where most values cluster, and the whiskers show the extremes.

Section 6

How do you interpret scatter diagrams, correlation and lines of best fit?

A scatter diagram plots two variables on a graph (one on each axis) to show the relationship between them. Each point represents a pair of values.

Correlation describes the strength and direction of the relationship:

  • Positive correlation: As one variable increases, the other tends to increase (points slope upward left to right)
  • Negative correlation: As one variable increases, the other tends to decrease (points slope downward left to right)
  • No correlation: No clear pattern; points are scattered randomly
  • Strong correlation: Points lie close to a clear trend line
  • Weak correlation: Points are more scattered but still show a general trend

Line of best fit (regression line):

  • A straight line that passes through (or near) the data points, minimising the distance between the line and all points
  • Used to estimate values and identify the trend
  • Should have roughly equal numbers of points above and below the line
  • Can be drawn by eye or calculated using statistical methods

To draw a line of best fit by eye:

  1. Identify the general trend of the data
  2. Draw a straight line that passes through the middle of the data
  3. Ensure approximately equal scatter above and below the line
  4. Extend the line slightly beyond the data range if appropriate

Using the line of best fit for prediction:

  • Read values from the line for interpolation (within the data range) or extrapolation (beyond the range)
  • Be cautious with extrapolation; trends may not continue beyond the data range

Interpreting correlation:

  • Correlation does not imply causation: A strong correlation between two variables does not mean one causes the other
Key termsscatter diagrampositive correlationnegative correlationline of best fitinterpolationextrapolation
Exam tip

Examiners expect precise language about correlation: describe both strength ('strong/weak') and direction ('positive/negative'). Always state whether you're interpolating or extrapolating.

Common mistake

Drawing a line that passes through the first and last points. The line of best fit should balance the scatter around it, not necessarily touch any specific points.

Example

A scatter plot of temperature vs ice cream sales shows strong positive correlation. A line of best fit helps predict sales at a given temperature, but correlation alone doesn't prove hot weather causes higher sales—other factors (day of week, marketing) may be involved.

Section 7

How do you interpret time series graphs?

A time series graph shows how a variable changes over time. The horizontal axis represents time (in consistent intervals: days, months, years, etc.) and the vertical axis represents the quantity being measured.

Key features of time series graphs:

  • Trend: The general direction (upward, downward, or stable) over the time period
  • Seasonality: Regular, repeating patterns that occur at fixed intervals (e.g. higher sales each December)
  • Fluctuations: Short-term variations in the data
  • Anomalies: Unusual spikes or drops that deviate from the overall pattern

Reading a time series graph:

  1. Identify the time period and variable being measured
  2. Read values from the graph for specific time points
  3. Calculate change: Final Value − Initial Value
  4. Identify the trend over the entire period
  5. Spot any seasonal patterns by looking for regular repetition

Trend analysis:

  • A moving average can be calculated to smooth out fluctuations and reveal the underlying trend
  • Moving average of order n = Mean of n consecutive data points
  • Plot moving average points at the centre of their time period
  • This removes seasonal variation and makes long-term trends clearer

Making predictions from time series data:

  • Extend the trend line or moving average line to estimate future values
  • Be cautious: assume the trend and seasonal patterns will continue as before
  • This method is less reliable for long-term predictions
Key termstime seriestrendseasonalitymoving averagefluctuationanomaly
Exam tip

When describing a time series, identify and describe the trend explicitly ('increasing', 'decreasing', 'stable'), and note any seasonal patterns. Examiners want you to distinguish between trend and seasonal variation.

Example

A time series of monthly sales may show an upward trend (increasing sales over the year) with seasonality (peaks in December, troughs in January). A 3-month moving average would smooth out monthly fluctuations to reveal this underlying trend.

Must Know

  • Bar charts, pie charts and pictograms represent categorical data; pie chart angles are calculated as (Frequency ÷ Total) × 360°; pictograms require a key
  • Histograms with equal class widths: height = frequency; bars touch each other (continuous data); for unequal widths, use frequency density = Frequency ÷ Class Width and remember Area of Bar = Frequency
  • Cumulative frequency graphs: plot points at upper class boundaries with smooth curves; use to find median (at ½ total CF), lower quartile (¼), and upper quartile (¾)
  • Box plots show minimum, Q1, median, Q3, and maximum; use to compare distributions by examining median position, spread (IQR), and skewness
  • Scatter diagrams: identify positive/negative correlation (direction and strength); draw line of best fit balancing points above and below; use for interpolation (within range) and extrapolation (beyond range, with caution)
  • Time series graphs: identify trend (overall direction), seasonality (regular patterns), and fluctuations; use moving averages to smooth data and reveal trends

That's the notes covered.

Carry on to the next subtopic.