Statistics and probability for Edexcel IGCSE Mathematics Specification B: Methods and Worked Examples

Statistics and probability are the topics that connect mathematics to everyday decisions. From understanding survey data to predicting outcomes, the skills tested here are practical, and the Edexcel IGCSE Mathematics Specification B exam expects you to use them accurately.

Think of statistics as the part of the exam where real-world data meets mathematical method. You are given numbers, charts or frequency tables, and your job is to extract meaning from them. Probability, on the other hand, asks you to look forward and work out how likely something is before it happens. Together, they form a significant portion of both Paper 1 and Paper 2, and the good news is that the methods are highly systematic. Once you know the steps, these questions become some of the most reliable places to earn marks.

These edexcel igcse (4MB1) mathematics specification b revision notes cover everything you need for both statistical representation and measures and probability. Every worked example below follows the format you will meet in the exam, and every self-check question at the end mirrors a real exam challenge.

Statistical representation: reading and creating charts

The specification requires you to use bar charts, pie charts and histograms. The key distinction that trips students up is between a bar chart and a histogram. A bar chart has gaps between the bars and plots frequency against discrete categories. A histogram has no gaps, plots frequency density on the vertical axis (not frequency), and is used for continuous grouped data.

For pie charts, the angle for each category is calculated as (frequency / total frequency) x 360. If 15 out of 60 people chose football, the angle is (15/60) x 360 = 90 degrees. Conversely, if you are reading a pie chart and the football sector is 90 degrees, the fraction is 90/360 = 1/4, and if the total is 60, the frequency is 15.

Measures of central tendency

You need to find the mean, median and mode for discrete data, and estimate the mean for grouped data. Here is how each one works, with the detail examiners expect.

Mean

For raw data, add all values and divide by the number of values. For a frequency table, multiply each value by its frequency, add the products, and divide by the total frequency.

Worked example: The scores of 20 students in a test are summarised below.

Sum of (score x frequency): (3 x 2) + (4 x 5) + (5 x 7) + (6 x 4) + (7 x 2) = 6 + 20 + 35 + 24 + 14 = 99

Total frequency: 2 + 5 + 7 + 4 + 2 = 20

Mean = 99 / 20 = 4.95

Estimated mean for grouped data

When data is grouped, you do not know the exact values, so you use the midpoint of each class interval. Multiply each midpoint by its frequency, add the products, and divide by the total frequency.

Worked example: The times (in minutes) taken by 40 runners to complete a course are shown below.

Total of (midpoint x frequency) = 60 + 204 + 330 + 216 = 810

Total frequency = 40

Estimated mean = 810 / 40 = 20.25 minutes

The word "estimated" matters. In grouped data the exact values are lost, so the mean is an estimate. If the question asks for the mean of grouped data, write "estimated mean" in your answer.

Median

The median is the middle value when the data is arranged in order. For n data values, the median is the ((n + 1) / 2)th value. If n = 20, the median is the 10.5th value, meaning you average the 10th and 11th values.

For grouped data, you identify the class containing the median by cumulating the frequencies. Using the runners example above: the median position is the 20.5th value. Cumulative frequencies: 5, 17, 32, 40. The 20.5th value falls in the 20-24 class (since the first two classes cover only 17 values).

Mode and modal class

The mode is the value that appears most often. For the scores table above, the mode is 5 (frequency 7). For grouped data, you identify the modal class, which is the class with the highest frequency. In the runners example, the modal class is 20-24 (frequency 15).

Probability fundamentals

Probability is measured on a scale from 0 (impossible) to 1 (certain). The probability of an event A is P(A) = number of favourable outcomes / total number of equally likely outcomes. The complement is P(not A) = 1 - P(A).

These are the edexcel igcse mathematics specification b notes you need for the probability section. Every formula here is directly testable.

The addition rule for mutually exclusive events

Two events are mutually exclusive if they cannot both happen at the same time. If A and B are mutually exclusive, P(A or B) = P(A) + P(B).

Worked example: A bag contains 3 red, 5 blue and 2 green marbles. What is the probability of drawing a red or a green marble?

P(red) = 3/10, P(green) = 2/10

P(red or green) = 3/10 + 2/10 = 5/10 = 1/2

The product rule for independent events

Two events are independent if the outcome of one does not affect the outcome of the other. If A and B are independent, P(A and B) = P(A) x P(B).

Worked example: A fair coin is tossed and a fair six-sided die is rolled. What is the probability of getting heads and a 4?

P(heads) = 1/2, P(4) = 1/6

P(heads and 4) = 1/2 x 1/6 = 1/12

Tree diagrams and conditional probability

Tree diagrams are the standard tool for combined events, especially when events are not independent (for example, drawing without replacement). Each branch represents a possible outcome, and you multiply along the branches and add between branches.

Worked example: A box contains 4 red and 6 blue counters. Two counters are drawn without replacement. Find the probability that both are red.

First draw: P(red) = 4/10

Second draw (given first was red): P(red) = 3/9

P(both red) = 4/10 x 3/9 = 12/90 = 2/15

For the probability of getting one of each colour:

P(red then blue) = 4/10 x 6/9 = 24/90

P(blue then red) = 6/10 x 4/9 = 24/90

P(one of each) = 24/90 + 24/90 = 48/90 = 8/15

Expected frequency

Expected frequency is theoretical: it tells you how many times you would expect an event to occur over a given number of trials. Expected frequency = probability x number of trials.

Worked example: A biased spinner has P(landing on red) = 0.35. If the spinner is spun 200 times, how many times would you expect it to land on red?

Expected frequency = 0.35 x 200 = 70

The actual result will rarely be exactly 70, but 70 is the expected value. Exam questions sometimes give you the observed frequency and ask you to compare it with the expected frequency, or to use relative frequency (observed frequency / number of trials) as an estimate of probability.

Relative frequency

Relative frequency = number of times the event occurs / total number of trials. As the number of trials increases, the relative frequency tends to get closer to the true probability. This is the experimental approach to probability, and the exam may ask you to explain why more trials give a better estimate.

Common mistakes in statistics and probability

Having marked many practice papers as edexcel igcse mathematics specification b practice questions, there are patterns in the errors students make.

• Forgetting to use midpoints for grouped data. Some students add the class boundaries instead. The midpoint is always (lower bound + upper bound) / 2.
• Confusing "or" with "and" in probability. "Or" means add (for mutually exclusive events). "And" means multiply (for independent events). Mixing these up inverts the calculation.
• Not adjusting for without-replacement. After one item is removed, both the numerator (if relevant) and the denominator change. Keeping the original denominator throughout is a guaranteed way to lose marks.
• Leaving probability as a percentage when a fraction or decimal is expected. Unless the question specifically asks for a percentage, give your answer as a fraction or decimal.
• Plotting frequency instead of frequency density on histograms. If the class widths are unequal, the bars will misrepresent the data.

Worked exam-style question: combined statistics and probability

The table below shows the number of goals scored by a football team in 30 matches.

(a) Find the mean number of goals per match.

Sum = (0 x 4) + (1 x 9) + (2 x 10) + (3 x 5) + (4 x 2) = 0 + 9 + 20 + 15 + 8 = 52

Mean = 52 / 30 = 1.73 (to 3 significant figures)

(b) Find the median.

With 30 values, the median is the average of the 15th and 16th values. Cumulative frequencies: 4, 13, 23, 28, 30. The 15th value falls in the "2 goals" row (cumulative reaches 23 by end of that row, and was only 13 at the end of "1 goal"). The 16th value also falls in the "2 goals" row. Median = 2.

(c) State the mode.

Mode = 2 (highest frequency, 10).

(d) A match is chosen at random. Find the probability that the team scored more than 2 goals.

Matches with more than 2 goals: 5 + 2 = 7

P(more than 2) = 7/30

Conditional probability: a careful approach

Simple conditional probability at this level typically involves two-stage experiments where the outcome of the first stage affects the second. The tree diagram is your primary tool. The edexcel igcse mathematics specification b explained approach is to label every branch with its probability, multiply along paths for "and", and add between paths for "or".

Worked example: In a class, 60% of students are girls. Of the girls, 40% wear glasses. Of the boys, 25% wear glasses. A student is chosen at random. Find the probability that the student wears glasses.

P(girl and glasses) = 0.6 x 0.4 = 0.24

P(boy and glasses) = 0.4 x 0.25 = 0.10

P(wears glasses) = 0.24 + 0.10 = 0.34

If the question then asks: "Given that the student wears glasses, find the probability that the student is a girl," you use:

P(girl | glasses) = P(girl and glasses) / P(glasses) = 0.24 / 0.34 = 12/17

Self-check questions

Test yourself on these. Cover the answers and work through each one on paper before checking.

1. A frequency table shows test marks: 1 (freq 3), 2 (freq 7), 3 (freq 5), 4 (freq 5). Find the mean mark.
   Answer: Sum = 3 + 14 + 15 + 20 = 52. Total frequency = 20. Mean = 52/20 = 2.6.
2. A bag holds 5 red, 3 blue and 2 yellow balls. Two balls are drawn without replacement. Find the probability that both are blue.
   Answer: P(blue then blue) = 3/10 x 2/9 = 6/90 = 1/15.
3. Grouped data: 0-9 (freq 8), 10-19 (freq 15), 20-29 (freq 12), 30-39 (freq 5). Find the estimated mean.
   Answer: Midpoints: 4.5, 14.5, 24.5, 34.5. Sum = 36 + 217.5 + 294 + 172.5 = 720. Total freq = 40. Estimated mean = 720/40 = 18.
4. P(A) = 0.3, P(B) = 0.5, A and B are independent. Find P(A and B).
   Answer: 0.3 x 0.5 = 0.15.
5. A spinner is spun 150 times and lands on green 42 times. What is the relative frequency of green?
   Answer: 42/150 = 0.28.

The statistics and probability edexcel igcse section rewards accuracy and method. Every step in a calculation should be visible on the page. If you can lay out your working clearly, show the formula you are using, and double-check whether the question says "with" or "without" replacement, you are covering the ground that separates a solid mark from a lost one. Use the Green Bridge CBT platform to work through statistics and probability edexcel igcse practice questions under timed conditions, and review the edexcel igcse mathematics specification b mark scheme after each attempt to see exactly where the marks are awarded.