Statistics: Averages to Histograms
Statistics for GCSE Success
Lesson duration is: 60 minItems in this lesson
Statistics for GCSE Success
Central lesson thread: how to choose, calculate and interpret the right statistical measure or graph, then use it accurately in Edexcel-style exam questions.
Learning goal
I can choose, calculate and interpret averages and statistical diagrams, and use them to solve GCSE exam-style problems accurately.
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Averages: what each one tells us
Mean: total of the data ÷ number of values
Median: middle value when data is in order
Mode: most frequent value
The best average depends on the data set
In exams, always check for extreme values and grouped data
Stress that students must not treat all averages as interchangeable. Link the choice of average to the shape and context of the data.
Discrete data and averages
Discrete data takes separate values, often whole numbers.
Example data: 3, 4, 4, 6, 8
Mean = (3 + 4 + 4 + 6 + 8) ÷ 5 = 25 ÷ 5 = 5
Median = 4
Mode = 4
For small lists, write the values in order first.
Point out that many errors come from finding the median before ordering the data.
Continuous data and grouped tables
Continuous data can take any value in an interval, such as height or time.
For grouped frequency tables:
Use the class midpoint for each group
Estimate mean using
estimated mean = Σ(f × x) ÷ Σf
The result is an estimate because the exact values are not known
Use x for midpoint and f for frequency consistently. Remind the student that grouped continuous data usually leads to an estimated mean, not an exact one.
Estimated mean example
Table
Class | f | x | f × x
0 < t ≤ 10 | 3 | 5 | 15
10 < t ≤ 20 | 5 | 15 | 75
20 < t ≤ 30 | 2 | 25 | 50
Σf = 10
Σ(f × x) = 140
Estimated mean
= Σ(f × x) ÷ Σf
= 140 ÷ 10
= 14
Calculation
Emphasise the midpoint values 5, 15 and 25. Ask the student to justify why 15 is the midpoint of 10 < t ≤ 20.
Exam-style checks with averages
If the question says estimate, use midpoints
If one very large or very small value appears, the mean may be misleading
The median is usually better for skewed data
The mode is useful for the most common category or score
Always give units if the context has units
This slide is about exam technique rather than new content. Keep the student alert to command words such as calculate, estimate, compare and justify.
For grouped continuous data, which average is usually an estimate?
Median
Mode
Estimated mean
Range
Checks the key distinction from slides 4 to 6: grouped continuous data uses class midpoints, so the mean is estimated rather than exact.
Complete the method
For grouped continuous data, use the class midpoint for each group. Then calculate Σ(f × x) and divide by Σf. The answer is an estimate because the exact values are not known.
Echoes the notation on the content slides and reinforces exam wording: if the question says estimate, use midpoints.
Comparing data sets
When comparing two data sets, look at:
Average for typical value
Spread for variation
A fair comparison usually needs both
A set can have a higher mean but also a larger spread.
Make clear that 'better' depends on the context: consistency often matters as much as size.
Measures of spread
Range = highest − lowest
Interquartile range = upper quartile − lower quartile
A smaller spread means the data is more consistent
Range uses only two values
Interquartile range uses the middle 50% and is less affected by extremes
This prepares for box plots later. Mention that GCSE comparison questions often reward one comment on average and one on spread.
Exam-style comparison
Mean score = 58
Range = 12
Comparison:
School A has the higher mean, but School B is more consistent because it has the smaller range.
School B
School A
Mean score = 62
Range = 24
Model the wording of a 2-mark comparison: one statement about average, one about spread, both linked to values.
Which measure is less affected by extreme values?
Range
Interquartile range
Highest value
Class width
Targets the slide 8 idea that the interquartile range uses the middle 50%, so it is less influenced by extremes than the range.
Scatter diagrams: spotting correlation
Each point represents a pair of values
Positive correlation: as one increases, the other tends to increase
Negative correlation: as one increases, the other tends to decrease
No correlation: no clear pattern
Keep the focus on relationship, not causation. Students often overstate what a scatter graph proves.
Correlation does not mean cause
A scatter diagram can show an association between two variables.
It does not prove that one variable causes the other. In exam answers, describe the correlation first, then only predict if the pattern supports it.
Useful misconception check. Give a quick example such as shoe size and reading score having no obvious causal link even if some pattern appears.
Line of best fit
Draw a straight line that follows the overall trend
Aim for a balance of points above and below the line
Use the line to estimate unknown values
Do not join point to point
Be careful when estimating outside the data range
Introduce the word extrapolation for estimates outside the data range, and explain why they are less reliable than interpolation.
Exam-style scatter question
A scatter diagram of revision time and test score shows positive correlation.
A suitable exam answer might be:
As revision time increases, test score tends to increase. A line of best fit could be used to estimate a score for a given revision time, but any estimate outside the data range is less reliable.
This slide models the standard wording expected in Edexcel responses.
A scatter diagram can show that two variables are linked, but not that one variable definitely causes the other. What is this idea called?
No correlation
Class boundary
Causation limit
Correlation only
Built from slides 10 to 13. The student should recognise that scatter diagrams show association/correlation, not proof of causation.
Complete the scatter diagram summary
A scatter diagram shows pairs of values. If one variable increases and the other tends to increase, this is positive correlation. A line of best fit can be used to estimate values, but estimating outside the data range is called extrapolation and is less reliable.
Uses the exact Edexcel-style phrasing taught. Distinguishes line of best fit from joining points and interpolation from extrapolation.
Pie charts and angles
A full circle is 360°
Sector angle = category frequency ÷ total frequency × 360°
Larger frequency means larger angle
Always find the total first
Remind the student that a pie chart shows proportion, not just raw numbers.
Pie chart example
Favourite sport survey:
Football 18
Tennis 9
Netball 3
Total = 30
Football: 18 ÷ 30 × 360° = 216°
Tennis: 9 ÷ 30 × 360° = 108°
Netball: 3 ÷ 30 × 360° = 36°
Data
Angles
If needed, simplify the fraction first: 18 ÷ 30 = 3/5, then 3/5 of 360° = 216°.
Reading a pie chart
If the angle is known and the total is known:
Frequency = angle ÷ 360° × total
Proportion = angle ÷ 360
Percentage = angle ÷ 360 × 100
This is a common reverse question in GCSE exams.
Students often manage drawing pie charts but struggle to work backwards from a sector angle.
In a pie chart, what do you find first before working out sector angles?
The total
The mode
The range
The median
Direct check on slide 14 exam technique. Students often rush into angle calculations without finding the total frequency.
Complete the pie chart facts
A full circle is 360°. To find a sector angle, use category frequency ÷ total frequency × 360°. To work backwards, frequency = angle ÷ 360° × total.
Checks both forward and reverse pie chart methods from slides 14 to 16.
Cumulative frequency
Cumulative frequency is a running total
It is used to find medians and quartiles
Plot cumulative frequency against the upper class boundary
Join with a smooth curve
Use upper class boundaries carefully, especially if the student confuses class limits and boundaries.
Finding median and quartiles
For a data set of size n:
Median is at n ÷ 2
Lower quartile is at n ÷ 4
Upper quartile is at 3n ÷ 4
Read these values across from the cumulative frequency axis, then down to the data axis.
For GCSE graph reading, students are usually locating positions on the graph rather than using a strict separate-item formula for exact lists.
Box plots
A box plot shows the five-number summary:
minimum
lower quartile
median
upper quartile
maximum
It helps compare centre and spread quickly.
Link this directly to cumulative frequency: quartiles found from the graph can be transferred into a box plot.
Comparing box plots
Set A has the higher median, so its typical value is greater. Set B has the smaller interquartile range, so it is more consistent.
Median for typical value
Interquartile range for spread of middle 50%
Range for overall spread
What to look for
Exam wording
Encourage precise comparative language: 'higher median' and 'smaller interquartile range' rather than vague comments.
On a cumulative frequency graph, cumulative frequency is plotted against the...
Lower class boundary
Upper class boundary
Class midpoint
Frequency density
Common GCSE misconception: students may use class midpoints from estimated mean work, but cumulative frequency uses upper class boundaries.
Complete the box plot summary
A box plot shows the five-number summary: minimum, lower quartile, median, upper quartile and maximum. To compare two box plots, use the median for typical value and the interquartile range for spread.
Connects cumulative frequency reading to box plot interpretation and comparison language.
Histograms
Used for continuous grouped data
Bar width = class width
Bar height = frequency density
Area of each bar represents frequency
Students often confuse histograms with bar charts. The key idea is that area, not just height, represents frequency.
Frequency density
Frequency density = frequency ÷ class width
So:
if class widths are equal, taller bars usually mean greater frequency
if class widths are different, compare areas, not heights alone
This is why frequency density is needed.
This is the key conceptual slide for histograms. Pause here and check the student understands why unequal class widths change the graph.
Histogram example
Table
Class | Frequency | Width
0 < m ≤ 5 | 10 | 5
5 < m ≤ 15 | 24 | 10
15 < m ≤ 20 | 8 | 5
Density
0 < m ≤ 5: 10 ÷ 5 = 2
5 < m ≤ 15: 24 ÷ 10 = 2.4
15 < m ≤ 20: 8 ÷ 5 = 1.6
These are the bar heights.
Highlight that the middle class has the greatest frequency and also the greatest density here, but those are not always the same in other examples.
Percentages in statistical problems
Percentages often appear when interpreting data.
Useful links:
percentage = part ÷ whole × 100
part = percentage ÷ 100 × whole
percentage change = change ÷ original × 100
Connect this to pie charts and comparisons: percentages describe proportion clearly across different totals.
Using percentages to compare data
Percentages are useful when totals are different.
Example:
Team A won 18 out of 24 games = 75%
Team B won 21 out of 30 games = 70%
Team A has the better win rate, even though Team B won more games.
This is an important exam comparison idea: raw totals can mislead when group sizes differ.
Statistics exam technique
Choose the diagram or average that matches the data type
For grouped data, check whether the answer is exact or estimated
When comparing, use one measure of average and one of spread
In histograms, use frequency density
In scatter diagrams, describe correlation carefully
This slide draws the lesson together before the glossary. It reflects common Edexcel exam habits and pitfalls.
In a histogram, what represents the frequency?
Bar height
Bar width
Bar area
Bar label
Checks the crucial histogram idea from slides 21 to 23: frequency is shown by area, not height alone.
Complete the statistics links
In a histogram, bar height is frequency density, found by frequency ÷ class width. When comparing groups with different totals, using a percentage is often fairer than using a raw total.
Brings together the final block: histogram method and percentage comparisons when totals differ.
Which topic needs the most more practice?
Averages
Scatter diagrams
Pie charts
Cumulative frequency
Histograms
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What is one method from today that now makes more sense?
Reflection only. Invite the student to name a method, phrase or exam habit that now feels secure.
Which part of this lesson felt hardest, and why?
Reflection only. Encourage a specific answer such as histograms, choosing average and spread, or reverse pie chart questions.
Glossary
mean - the total of the values divided by the number of values
median - the middle value when the data is written in order
mode - the value that appears most often
range - the difference between the highest and lowest values
interquartile range - the difference between the upper quartile and lower quartile
discrete data - data that takes separate values
continuous data - data that can take any value in an interval
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Glossary
scatter diagram - a graph showing pairs of values to look for correlation
correlation - a relationship between two variables
cumulative frequency - the running total of frequencies
box plot - a diagram showing the five-number summary of a data set
histogram - a graph for continuous grouped data where area represents frequency
frequency density - frequency divided by class width
estimated mean - an approximate mean found from grouped data using class midpoints
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