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GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 2–5 · Statistics

Sampling, Frequency Polygons and Time Series

Before you can draw conclusions from data, you need a fair sample. Then you need the right diagram to show it. This page covers sampling and bias, frequency polygons and time-series graphs, three GCSE topics that IGCSE textbooks often skip.

1Populations and samples

The population is everyone (or everything) you want to know about. A sample is the smaller group you actually ask or measure.

Advantages of a sampleDisadvantages of a sample
Quicker and cheaperMay not represent the population
Practical when the population is hugeResults are estimates, not exact
Key ideaA bigger random sample gives more reliable results. In a random sample, every member of the population has an equal chance of being chosen.

2Bias and how to avoid it

A sample is biased if some groups are more likely to be chosen than others.

Worked example
A council asks people at a train station at 8 am whether the town needs more parking. Give two reasons the sample may be biased.
  1. Most people at a train station at 8 am are commuters using the train, so they may not need parking.
  2. One place and one time misses people who drive, work from home or shop later in the day.
Not representative: one place, one time, one type of person
Exam tipTo improve a sample: choose people at random from the whole population, use a larger sample, and sample at different times and places.

3Frequency polygons

  1. Find the midpoint of each class.
  2. Plot (midpoint, frequency) for each class.
  3. Join the points with straight lines. Don’t join the first point back to the last.
Height h (cm)0 < h ≤ 1010 < h ≤ 2020 < h ≤ 3030 < h ≤ 4040 < h ≤ 50
Midpoint515253545
Class A491485
Class B2610157
048121601020304050Height (cm)Frequency
Class A (solid) and Class B (dashed)

Class B’s polygon is further to the right, so on average Class B’s plants are taller. The estimated means are 25.25 cm and 29.75 cm.

Common errors
  • Plotting at the end of the class (e.g. 10) instead of the midpoint (5).
  • Joining the last point back to the first, or drawing a bar chart instead.

4Time-series graphs

A time-series graph shows how something changes over time. Look for the overall trend and any repeating seasonal pattern.

0102030Q1Q2Q3Q4Q1Q2Q3Q4Q1Q2Q3Q4QuarterSales (£000)
Quarterly sales of ice cream over three years
Worked example
Describe the time series above.
  1. Trend: sales increase slightly each year.
  2. Seasonal pattern: sales peak every Q3 (summer) and are lowest in Q1.
Upward trend with a peak each Q3

5Comparing distributions

When comparing two sets of data, always compare:

  • an average (mean or median): which is bigger, in context;
  • a measure of spread (range or IQR): which is more consistent.
Exam tipWrite each comparison in context: “On average the Class B plants were taller”, not “B has a higher mean”.
Common mistakes
  • Saying a sample is biased without saying why it doesn’t represent the population.
  • Plotting frequency polygons at class boundaries instead of midpoints.
  • Comparing only averages and forgetting spread, or making comparisons without context.
  • Describing individual values on a time series instead of the trend.

★Key Vocabulary Recap

Population
The whole group being studied.
Sample
Part of the population that is surveyed.
Bias
When a sample does not fairly represent the population.
Random sample
Every member has an equal chance of selection.
Frequency polygon
Points at class midpoints joined by straight lines.
Time series
Data recorded at regular intervals over time.
Specification reference (Pearson Edexcel GCSE (9–1) Mathematics 1MA1: S1, S2, S4, S5)
  • S1: infer properties of populations from a sample, knowing the limitations of sampling.
  • S2: interpret and construct tables, charts and diagrams, including frequency polygons and time-series graphs.
  • S4: interpret, analyse and compare distributions through appropriate measures of central tendency and spread.
  • S5: apply statistics to describe a population.
Next stepReady to practise? Try the 8 exam-style questions, then check your working against the worked solutions.
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Practice questions

GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 2–5 · Statistics

Sampling, Frequency Polygons and Time Series — Practice Questions

8 exam-style questions (17 marks), building from warm-up to grade 8–9. Show full working: Edexcel gives method marks even when the final answer is wrong.

Foundation · Grades 2–3
  1. (1 mark)
    Grade 2
    Give one advantage of using a sample instead of the whole population.
  2. (2 marks)
    Grade 3
    Sam wants to know how often students in his school use the library. He asks 10 students who are in the library at lunchtime. Give two reasons why his sample is biased.
  3. (2 marks)
    Grade 3
    Describe how Sam could choose a random sample of 50 students from the 1200 students in his school.
Foundation and Higher · Grades 4–5
  1. (3 marks)
    Grade 4
    The table shows the times, t minutes, taken by 40 people to finish a puzzle. Draw a frequency polygon.
    Time t (min)0 < t ≤ 2020 < t ≤ 4040 < t ≤ 6060 < t ≤ 80
    Frequency3111610
  2. (3 marks)
    Grade 4
    Use the table in Question 4 to estimate the mean time.
  3. (2 marks)
    Grade 4
    Look at the time-series graph of quarterly ice-cream sales in the notes. Describe (a) the trend (b) the seasonal pattern.
  4. (2 marks)
    Grade 5
    Using the frequency polygons for Class A and Class B in the notes, compare the heights of the plants in the two classes.
  5. (2 marks)
    Grade 5
    In a random sample of 80 students from a school of 1200, 12 said they cycle to school. Estimate how many students in the whole school cycle, and state an assumption you have made.
When you’re doneRevise the method in the notes, then mark your answers with the worked solutions.
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Worked solutions

Show worked solutions
GCSE Maths · Edexcel 1MA1 · Foundation & Higher · Grades 2–5 · Statistics

Sampling, Frequency Polygons and Time Series — Worked Solutions

Full worked solutions to all 8 questions (17 marks), set out the way an Edexcel examiner wants to see them.

Foundation · Grades 2–3
  1. (1 mark)
    Grade 2
    Give one advantage of using a sample instead of the whole population.
    Solution
    1. A sample is quicker and cheaper to collect.
    Quicker / cheaper
  2. (2 marks)
    Grade 3
    Sam wants to know how often students in his school use the library. He asks 10 students who are in the library at lunchtime. Give two reasons why his sample is biased.
    Solution
    1. Students in the library use it more than average.
    2. 10 students is a very small sample from one time of day.
    Only library users; sample too small
  3. (2 marks)
    Grade 3
    Describe how Sam could choose a random sample of 50 students from the 1200 students in his school.
    Solution
    1. Number every student from 1 to 1200 using the school list.
    2. Use a random number generator to pick 50 different numbers, and survey those students.
    Number the list; pick 50 random numbers
Foundation and Higher · Grades 4–5
  1. (3 marks)
    Grade 4
    The table shows the times, t minutes, taken by 40 people to finish a puzzle. Draw a frequency polygon.
    Time t (min)0 < t ≤ 2020 < t ≤ 4040 < t ≤ 6060 < t ≤ 80
    Frequency3111610
    Solution
    1. Midpoints: 10, 30, 50, 70.
    2. Plot (10, 3), (30, 11), (50, 16), (70, 10) and join with straight lines.
    0481216020406080Time (min)Frequency
    Points (10,3), (30,11), (50,16), (70,10)
  2. (3 marks)
    Grade 4
    Use the table in Question 4 to estimate the mean time.
    Solution
    1. Σ(midpoint × frequency) = 10×3 + 30×11 + 50×16 + 70×10 = 30 + 330 + 800 + 700 = 1860.
    2. Mean ≈ 1860 ÷ 40 = 46.5.
    46.5 minutes
  3. (2 marks)
    Grade 4
    Look at the time-series graph of quarterly ice-cream sales in the notes. Describe (a) the trend (b) the seasonal pattern.
    Solution
    1. (a) Sales increase slightly from year to year.
    2. (b) Sales are highest every Q3 (summer) and lowest in Q1.
    Rising trend; Q3 peak
  4. (2 marks)
    Grade 5
    Using the frequency polygons for Class A and Class B in the notes, compare the heights of the plants in the two classes.
    Solution
    1. Average: Class B’s polygon is further right (estimated mean 29.75 cm vs 25.25 cm), so on average Class B’s plants were taller.
    2. Spread: both cover the same range of classes, 0–50 cm.
    Class B taller on average
  5. (2 marks)
    Grade 5
    In a random sample of 80 students from a school of 1200, 12 said they cycle to school. Estimate how many students in the whole school cycle, and state an assumption you have made.
    Solution
    1. Proportion = 12 ÷ 80 = 0.15.
    2. 0.15 × 1200 = 180.
    3. Assumption: the sample is representative of the whole school.
    ≈ 180 students
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