Cambridge Lower Secondary CheckpointStage 6

Statistics and Probability (6Sp)

Mathematics Stage 6 Chapter Notes

What this chapter covers

Statistics and Probability — Probability
ShareWhatsAppPost
Statistics and Probability (6Sp) notes

Unable to load PDF

The notes viewer could not load. Please refresh the page.

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Statistics and Probability (6Sp) notes as text: skim, search, and jump between subtopics.

~12 min read

In this chapter

  1. 1Measures of Central Tendency
  2. 2Measures of Spread: Range
  3. 3Organising Data with Frequency Tables
  4. 4Averages from Frequency Tables
  5. 5The Basics of Probability
  6. 6Relative and Expected Frequency

1. Measures of Central Tendency

Measures of central tendency are single values that represent the 'average' or 'middle' of a dataset. There are three main types: the Mean, the Median, and the Mode. The mean is what people usually mean by 'average' - you add everything up and divide. The median is the literal middle value when the data is in order. The mode is simply the most common value in the list.

Mean = (Sum of all values) / (Number of values)

Median position in an ordered list = (n + 1) / 2

Key term

Central Tendency: A single value that attempts to describe a set of data by identifying the central position within that set.

Examiner insight

Examiners often use data sets with an even number of values to test if students can correctly find the median by averaging the two middle numbers.

Common pitfall

Forgetting to put the data in ascending order before finding the median is the most common mistake.

Worked example 14 marks

The shoe sizes sold for men at a shop were: 40, 47, 40, 46, 43, 40, 40, 44, 40, 38. Find the mean, median and mode of this data.

  1. 1

    Mode: The mode is the most frequent value. The size '40' appears 5 times, which is more than any other size. So, the mode is 40.

  2. 2

    Median: First, put the data in order: 38, 40, 40, 40, 40, 40, 43, 44, 46, 47. There are 10 values (an even number). The middle values are the 5th and 6th values.

  3. 3

    The 5th value is 40 and the 6th value is 40. The median is the average of these two: (40 + 40) / 2 = 40.

  4. 4

    Mean: Add up all the values: 38 + 40 + 40 + 40 + 40 + 40 + 43 + 44 + 46 + 47 = 418.

  5. 5

    Divide the sum by the number of values (10): 418 / 10 = 41.8. The mean is 41.8.

Recap

  • The mean is calculated by adding all values and dividing by the count of values.
  • The median is the middle value of an ordered dataset.
  • If there is an even number of data points, the median is the average of the two middle values.
  • The mode is the most frequently occurring value in the dataset.
  • Always order the data before finding the median.

Quick check

  1. Find the mode of this list of numbers: 5, 2, 8, 2, 9, 10, 2.1 mark
  2. What is the median of this ordered list: 3, 5, 8, 11, 12?1 mark

2. Measures of Spread: Range

While averages tell you about the center of the data, the 'range' tells you how spread out the data is. It is a 'measure of spread' or 'dispersion'. The range is simply the difference between the highest and lowest values in a dataset. A small range indicates that the data points are all quite similar (consistent), while a large range indicates they are very different (variable).

Range = Highest value – Lowest value

Key term

Range: A measure of spread calculated as the difference between the maximum and minimum values in a dataset.

Common pitfall

Calculating the range from a frequency table by subtracting the smallest frequency from the largest frequency, instead of using the actual data values.

Fun fact

In quality control for manufacturing, a small range in product measurements (like the weight of a bag of crisps) is a sign of a high-quality production process.

Worked example 13 marks

The shoe sizes sold for men were 40, 47, 40, 46, 43, 40, 40, 44, 40, 38. The sizes for women were 35, 39, 30, 33, 37, 35, 36, 35, 34, 35. Calculate the range for both and comment on what it shows.

  1. 1

    Men's data: Highest value = 47, Lowest value = 38.

  2. 2

    Men's Range = 47 - 38 = 9.

  3. 3

    Women's data: Highest value = 39, Lowest value = 30.

  4. 4

    Women's Range = 39 - 30 = 9.

  5. 5

    Comment: Both sets of data have the same range (9). This suggests that the spread of shoe sizes sold is similar for both men and women in this sample.

Recap

  • The range measures how spread out your data is.
  • To calculate the range, subtract the smallest value from the largest value.
  • A larger range means the data is more variable.
  • A smaller range means the data is more consistent.

Quick check

  1. The daily temperatures in a week were 15°C, 17°C, 14°C, 18°C, 19°C, 13°C, 16°C. What is the range?2 marks

3. Organising Data with Frequency Tables

When you have a large amount of data, listing it all out is messy. A frequency table is a neat way to organise data by showing each unique data value and how many times it appears (its frequency). To create one, you list the data values in one column and use a tally column to count occurrences before writing the final frequency as a number.

Key term

Frequency: The number of times a particular data value or group of values occurs in a dataset.

Examiner insight

Clear tallying and correct totals in frequency tables are fundamental skills. Careless errors here can lose easy marks and affect all subsequent calculations, so double-check your total frequency.

Worked example 13 marks

A student records the number of siblings for 20 classmates. The results are: 1, 2, 1, 0, 3, 1, 2, 1, 4, 0, 1, 2, 2, 1, 3, 0, 1, 2, 1, 1. Create a frequency table to organise this data.

  1. 1

    First, set up a table with three columns: 'Number of Siblings', 'Tally', and 'Frequency'.

  2. 2

    Go through the list and make a tally mark for each data point. For '1', make a mark next to 1. For '2', make a mark next to 2, and so on.

  3. 3

    For the value '0', the tallies are 'III'. The frequency is 3.

  4. 4

    For the value '1', the tallies are '|||| ||||'. The frequency is 9.

  5. 5

    For the value '2', the tallies are '||||'. The frequency is 5.

  6. 6

    For the value '3', the tallies are 'II'. The frequency is 2.

  7. 7

    For the value '4', the tally is 'I'. The frequency is 1.

  8. 8

    Final check: Add up the frequency column: 3 + 9 + 5 + 2 + 1 = 20. This matches the number of classmates, so the table is correct.

Recap

  • Frequency tables help organise large datasets.
  • A tally column is used to count the data points accurately.
  • The 'frequency' is the final count for each data value.
  • The sum of the frequencies should always equal the total number of data points.

Quick check

  1. In a frequency table, what should the sum of the 'Frequency' column be equal to?1 mark

4. Averages from Frequency Tables

Once data is in a frequency table, you can find the averages without seeing the original long list. The mode is the value with the highest frequency. To find the mean, you need a new column, 'fx', where you multiply each data value(x) by its frequency (f). The mean is the total of the 'fx' column divided by the total frequency.

Mean from a frequency table = Σ(fx) / Σf

Key term

Estimated Mean: An approximation of the mean calculated from a grouped frequency table by using the midpoints of the class intervals.

Examiner insight

Examiners look for a clearly shown 'fx' column when asking for the mean from a frequency table. Simply writing the answer, even if correct, will not earn full marks for method.

Common pitfall

When calculating the mean, a common error is to divide the sum of the 'fx' column by the number of rows in the table, instead of by the total frequency (Σf).

Worked example 13 marks

The table shows the number of goals scored by a football team in 40 matches. Calculate the mean number of goals scored per match. Goals (x): 0, 1, 2, 3, 4 Frequency (f): 8, 15, 10, 5, 2

  1. 1

    To calculate the mean, we first need to add an 'fx' column to the table. This is 'frequency × goals'.

  2. 2

    For 0 goals: fx = 0 × 8 = 0

  3. 3

    For 1 goal: fx = 1 × 15 = 15

  4. 4

    For 2 goals: fx = 2 × 10 = 20

  5. 5

    For 3 goals: fx = 3 × 5 = 15

  6. 6

    For 4 goals: fx = 4 × 2 = 8

  7. 7

    Now find the totals (Σ). Total frequency Σf = 8 + 15 + 10 + 5 + 2 = 40. Total fx Σ(fx) = 0 + 15 + 20 + 15 + 8 = 58.

  8. 8

    Use the formula: Mean = Σ(fx) / Σf = 58 / 40 = 1.45.

  9. 9

    The mean number of goals scored is 1.45.

Recap

  • The mode from a frequency table is the value with the highest frequency.
  • To find the mean, create a new column by multiplying each value by its frequency (fx).
  • The mean is the total of the 'fx' column divided by the total of the 'f' column.
  • The median is found by locating the position of the middle value using cumulative frequency.

Quick check

  1. If a data value of 5 has a frequency of 10, what is the value of 'fx' for that row?1 mark

5. The Basics of Probability

Probability is a way of measuring how likely an event is to happen. It is always a value between 0 and 1. A probability of 0 means the event is impossible. A probability of 1 means the event is certain. To calculate the probability of an event, you divide the number of ways the event can happen by the total number of possible outcomes. Probabilities can be written as fractions, decimals, or percentages.

P(event) = (Number of favourable outcomes) / (Total number of possible outcomes)

Key term

Sample Space: The set of all possible outcomes of an experiment.

Common pitfall

Writing probability as a ratio (e.g., 1:4) instead of a fraction (1/5), decimal (0.2), or percentage (20%).

Fun fact

If you shuffle a standard deck of 52 cards properly, it's virtually certain that the exact order of cards you get has never existed before in the history of the universe.

Worked example 13 marks

A bag contains 5 red balls, 3 blue balls, and 2 green balls. A ball is picked at random. What is the probability that it is:a) red,b) not blue?

  1. 1

    First, find the total number of possible outcomes. Total balls = 5 + 3 + 2 = 10.

  2. 2

    a) Probability of picking a red ball. Number of red balls (favourable outcomes) = 5.

  3. 3

    P(Red) = (Number of red balls) / (Total number of balls) = 5/10 = 1/2.

  4. 4

    b) Probability of picking a ball that is not blue. The number of balls that are not blue is 5 (red) + 2 (green) = 7.

  5. 5

    P(Not Blue) = (Number of non-blue balls) / (Total number of balls) = 7/10.

Recap

  • Probability measures the chance of an event happening.
  • The probability scale is from 0 (impossible) to 1 (certain).
  • The formula is P(event) = Favourable Outcomes / Total Outcomes.
  • The sum of probabilities of all possible outcomes is always 1.
  • Probabilities can be expressed as fractions, decimals, or percentages.

Quick check

  1. You roll a standard six-sided die. What is the probability of rolling a number greater than 4?2 marks
  2. An event has a probability of 0. What word describes this event?1 mark

6. Relative and Expected Frequency

Sometimes we can't calculate theoretical probability. Instead, we can perform an experiment and find the 'relative frequency', which is an estimate of probability based on results. Expected frequency uses probability to predict how many times an event should happen over a number of trials. For example, if you know the probability of an event, you can estimate how many times it will occur in 100 trials.

Relative Frequency = (Number of times an event occurs) / (Total number of trials)

Expected Frequency = P(event) × Number of trials

Key term

Relative Frequency: An estimate of probability based on the results of an experiment or observation.

Examiner insight

Examiners expect students to understand that relative frequency is an estimate of probability that becomes more reliable as the number of trials increases.

Common pitfall

Confusing relative frequency (which is calculated from an experiment's results) with theoretical probability (which is calculated from the sample space).

Worked example 13 marks

The probability that a biased coin lands on heads is 0.4. If the coin is flipped 300 times, how many times would you expect it to land on tails?

  1. 1

    First, find the probability of landing on tails. The probabilities must add up to 1.

  2. 2

    P(Tails) = 1 - P(Heads) = 1 - 0.4 = 0.6.

  3. 3

    Now, calculate the expected number of tails in 300 flips.

  4. 4

    Expected Frequency = P(Tails) × Number of trials.

  5. 5

    Expected Tails = 0.6 × 300 = 180.

  6. 6

    You would expect it to land on tails 180 times.

Recap

  • Relative frequency is an experimental probability.
  • Expected frequency is a prediction of how many times an event will occur.
  • The formula is: Expected Frequency = Probability × Number of Trials.
  • The more trials you do, the more reliable the relative frequency becomes as an estimate of theoretical probability.

Quick check

  1. A spinner has a 1/4 chance of landing on blue. If it's spun 80 times, what is the expected number of times it will land on blue?1 mark

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. The heights of 10 plants in cm are: 15, 23, 18, 31, 17, 23, 20, 19, 23, 21. Find the mode, median, mean, and range of the heights.5 marks
  2. A survey asked 50 people how many pets they own. The results are in the frequency table below. Calculate the mean number of pets per person. Pets (x): 0, 1, 2, 3, 4 Frequency (f): 18, 20, 7, 4, 13 marks
  3. A bag contains only red, blue, and yellow counters. The probability of picking a red counter is 0.2 and the probability of picking a blue counter is 0.5. What is the probability of picking a yellow counter?2 marks
  4. A biased spinner is spun. The probability of it landing on 'Win' is 0.15. If Sarah spins the spinner 400 times, how many times should she expect to win?2 marks
  5. A dice is rolled 120 times. The results are recorded: Score: 1, 2, 3, 4, 5, 6 Frequency: 15, 25, 20, 18, 22, 20 a) What is the relative frequency of rolling a 6? b) Is the dice fair? Explain your answer.4 marks
  6. Two friends, Alex and Ben, record their test scores (out of 50) for 5 tests. Alex: 45, 42, 38, 49, 41 Ben: 50, 35, 42, 48, 30 Calculate the mean and range for both students and use these statistics to comment on who is the more consistent student.5 marks
  7. A school has 800 students. The probability that a randomly selected student is left-handed is 1/10. How many left-handed students are there in the school?2 marks
  8. Find the median of these numbers: 10, 4, 8, 15, 9, 2, 11.2 marks
  9. A machine produces light bulbs. In a sample of 200 bulbs, 8 were found to be defective. a) What is the relative frequency of a bulb being defective? b) If the machine produces 10,000 bulbs in a day, estimate how many will be defective.3 marks
  10. The mean of five numbers is 12. The numbers are 7, 15, 10, 11 and x. Find the value of x.3 marks

Go deeper

Practise and revise with member-only material for this chapter.

Workbook

Exam-style practice questions with full worked solutions.

Unlock with a plan

Chapter at a Glance

A single-page summary of every key idea for fast revision.

Unlock with a plan

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Number (6Ni)Number (6Np)Number (6Nf)Algebra (6Ae)Algebra (6As)Geometry and Measure (6Gg)