Cambridge Lower Secondary CheckpointStage 7

Statistics and Probability (7Ss)

Mathematics Stage 7 Chapter Notes

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Statistics and Probability - Statistics
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Statistics and Probability (7Ss) notes

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In this chapter

  1. 1Averages and Range
  2. 2The Probability Scale
  3. 3Calculating Theoretical Probability
  4. 4Experimental Probability (Relative Frequency)
  5. 5Sampling and Bias

1. Averages and Range

To understand and compare datasets, we use statistical measures. Averages (or measures of central tendency) tell us a typical value, while the range (a measure of spread) tells us how varied the data is. There are three main types of average: the Mean is the sum of all values divided by the number of values; the Median is the middle value when the data is in order; the Mode is the most frequent value. A dataset can have one mode, more than one mode (bimodal), or no mode at all.

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

Range = Largest value – Smallest value

Key term

Median: The middle value in a set of data that has been arranged in order of size.

Examiner insight

Examiners often ask you to choose the 'best' average for a dataset. The median is best if there are extreme outliers, while the mode is useful for non-numerical data.

Common pitfall

Forgetting to order the data from smallest to largest before finding the median is the most common mistake.

Worked example 14 marks

Find the mean, median, mode, and range of this set of numbers: 7, 4, 5, 10, 4.

  1. 1
    1. Find the Mean: Add the numbers together: 7 + 4 + 5 + 10 + 4 = 30. Divide by the number of values (5): 30 ÷ 5 = 6. The mean is 6.
  2. 2
    1. Find the Median: First, order the numbers: 4, 4, 5, 7, 10. The middle value is the 3rd number, which is 5. The median is 5.
  3. 3
    1. Find the Mode: Look for the most frequent number. The number 4 appears twice, more than any other number. The mode is 4.
  4. 4
    1. Find the Range: Subtract the smallest value from the largest: 10 – 4 = 6. The range is 6.

Worked example 23 marks

The ages of 50 children at a school are recorded in this frequency table. Find the mean age.

  1. 1
    1. To find the mean from a frequency table, we first multiply each age by its frequency: (6×12), (7×9), (8×8), (9×10), (10×8), (11×3).
  2. 2
    1. Calculate these values: 72, 63, 64, 90, 80, 33.
  3. 3
    1. Find the total sum of ages by adding these products: 72 + 63 + 64 + 90 + 80 + 33 = 402.
  4. 4
    1. Find the total number of children (the total frequency): 12 + 9 + 8 + 10 + 8 + 3 = 50.
  5. 5
    1. Calculate the mean: Total sum of ages ÷ Total number of children = 402 ÷ 50 = 8.04. The mean age is 8.04 years.

Recap

  • The mean is the sum of values divided by the count of values.
  • The median is the middle value of an ordered dataset.
  • The mode is the most frequently occurring value.
  • The range measures the spread of data (Largest - Smallest).
  • Always order the data before finding the median.

Quick check

  1. What is the median of these scores: 9, 2, 5, 11, 7?2 marks

2. The Probability Scale

Probability measures how likely an event is to happen. It's represented on a scale from 0 to 1. An event with probability 0 is impossible. An event with probability 1 is certain. An event with probability 0.5 (or 1/2 or 50%) has an even chance of happening. Probabilities can be written as fractions, decimals, or percentages.

0 ≤ P(event) ≤ 1

Key term

Probability: A measure of the likelihood that an event will occur, expressed as a number between 0 (impossible) and 1 (certain).

Fun fact

The mathematical theory of probability started when a gambler asked the famous mathematician Blaise Pascal for help on how to win at dice games in the 17th century.

Worked example 12 marks

Put these probabilities in order, from smallest to largest: 60%, 5/8, 0.55

  1. 1
    1. To compare, convert all probabilities to the same format, for example, decimals.
  2. 2
    1. 60% = 60/100 = 0.60
  3. 3
    1. 5/8 = 5 ÷ 8 = 0.625
  4. 4
    1. 0.55 is already a decimal.
  5. 5
    1. Now order the decimals: 0.55, 0.60, 0.625.
  6. 6
    1. Write the final answer in the original format: 0.55, 60%, 5/8.

Recap

  • Probability is measured on a scale from 0 to 1.
  • A probability of 0 means the event is impossible.
  • A probability of 1 means the event is certain.
  • A probability of 0.5 means an even chance.
  • Probabilities can be written as fractions, decimals, or percentages.

Quick check

  1. An event is 'very unlikely'. Which of these could be its probability: 0.9, 0.5, 0.1?1 mark

3. Calculating Theoretical Probability

Theoretical probability is what we expect to happen in an ideal situation. It's calculated when all possible outcomes are equally likely. For example, with a fair coin, a Head or a Tail are equally likely. We use a formula to find the probability of a specific event happening.

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

Key term

Outcome: An outcome is a single possible result of a probability experiment.

Examiner insight

Examiners award marks for showing your method. Always write down the fraction with the number of favourable outcomes over the total outcomes, even if you simplify it later.

Common pitfall

Miscounting the total number of outcomes is a frequent error, especially in more complex scenarios like rolling two dice (total is 36, not 12).

Worked example 12 marks

A fair 6-sided dice is rolled. What is the probability of rolling a prime number?

  1. 1
    1. List the total possible outcomes: {1, 2, 3, 4, 5, 6}. There are 6 total outcomes.
  2. 2
    1. Identify the favourable outcomes (prime numbers): {2, 3, 5}. There are 3 favourable outcomes. (Note: 1 is not a prime number).
  3. 3
    1. Use the formula: P(prime) = (Number of prime numbers) / (Total numbers) = 3/6.
  4. 4
    1. Simplify the fraction: 3/6 = 1/2. The probability is 1/2 or 0.5.

Worked example 22 marks

A bag contains 5 red, 3 blue and 2 green counters. A counter is chosen at random. What is the probability it is NOT blue?

  1. 1
    1. Find the total number of counters: 5 + 3 + 2 = 10. This is the total number of possible outcomes.
  2. 2
    1. Find the number of counters that are NOT blue. These are the red and green ones: 5 + 2 = 7. This is the number of favourable outcomes.
  3. 3
    1. Use the formula: P(not blue) = (Number of non-blue counters) / (Total number of counters) = 7/10.
  4. 4
    1. Alternatively, find P(blue) first: P(blue) = 3/10. The sum of probabilities is 1, so P(not blue) = 1 - P(blue) = 1 - 3/10 = 7/10.

Recap

  • Theoretical probability applies when all outcomes are equally likely.
  • The formula is: Favourable Outcomes / Total Outcomes.
  • The probability of an event NOT happening is 1 minus the probability of it happening.
  • Always simplify your probability fraction unless told not to.

Quick check

  1. What is the probability of picking a vowel from the letters of the word 'PROBABILITY'?2 marks

4. Experimental Probability (Relative Frequency)

Experimental probability, also called relative frequency, is based on the results of an actual experiment. It's a calculation of what actually happened, not what was expected to happen. The more times you repeat an experiment (more trials), the more reliable your experimental probability becomes, and it will generally get closer to the theoretical probability.

Experimental P(event) = (Number of times event occurs) / (Total number of trials)

Key term

Relative Frequency: An estimate of probability based on the results of an experiment or observation, calculated by dividing the number of times an event occurred by the total number of trials.

Examiner insight

When asked to compare experimental and theoretical probabilities, a good answer will calculate both and then comment on whether the experimental result is higher or lower than expected, often linking it to the number of trials.

Common pitfall

Confusing experimental probability (what did happen) with theoretical probability (what should happen). For a biased coin, they will be different.

Worked example 13 marks

A computer simulates spinning three coins 160 times. The results are shown in the table. What is the experimental probability of getting at least two heads?

  1. 1
    1. Identify the total number of trials. The simulation was run 160 times, so the total number of trials is 160.
  2. 2
    1. Identify the successful outcomes. 'At least two heads' means getting 2 heads OR 3 heads.
  3. 3
    1. Find the frequencies for these outcomes from the table: Frequency of 2 heads = 68. Frequency of 3 heads = 14.
  4. 4
    1. Add these frequencies together: 68 + 14 = 82. This is the number of times the event occurred.
  5. 5
    1. Calculate the experimental probability: P(at least 2 heads) = 82 / 160.
  6. 6
    1. Simplify the fraction: 82/160 = 41/80.

Worked example 24 marks

A student spins a biased spinner 80 times. It lands on red 24 times.a) What is the experimental probability of landing on red?b) If the spinner is spun 400 times, estimate the number of times it will land on red.

  1. 1

    a) 1. Use the formula for experimental probability: P(red) = (Number of times it landed on red) / (Total spins).

  2. 2

    a) 2. P(red) = 24 / 80.

  3. 3

    a) 3. Simplify the fraction: 24/80 = 3/10.

  4. 4

    b) 1. Use the experimental probability from part(a) as the best estimate: P(red) = 3/10.

  5. 5

    b) 2. Multiply this probability by the new number of spins: (3/10) * 400.

  6. 6

    b) 3. (3 * 400) / 10 = 1200 / 10 = 120. The estimated number of times is 120.

Recap

  • Experimental probability is based on results from an experiment.
  • The formula is: Frequency of event / Total number of trials.
  • More trials generally lead to a more reliable estimate of probability.
  • You can use experimental probability to predict future outcomes.

Quick check

  1. A drawing pin is dropped 100 times. It lands 'point up' 35 times. What is the experimental probability of it landing 'point up'?1 mark

5. Sampling and Bias

When we want to find out information about a large group (a 'population'), it's often impractical to ask everyone. Instead, we survey a smaller group, called a 'sample'. For the results to be meaningful, the sample must be 'representative' of the whole population. If the sample is not representative, it is 'biased'. For example, asking only your friends about their favourite music will not represent the tastes of your whole school, as your friends likely share similar tastes to you.

Key term

Bias: A systematic error in a sampling method that results in a sample that does not accurately represent the population.

Examiner insight

When asked to criticise a sampling method, always use the word 'bias' and explain which part of the population is being over-represented or under-represented.

Fun fact

Political polls that get elections wrong are often criticised for biased sampling, such as only calling landlines and therefore missing younger voters who only use mobile phones.

Worked example 12 marks

A school wants to investigate what its 150 students think about the quality of teaching. A student decides to give a questionnaire to 30 of their friends. Explain why this is not a good sampling method.

  1. 1
    1. Identify the population (150 students) and the sample (30 friends).
  2. 2
    1. State the problem: The sample is not representative of the entire population of the school.
  3. 3
    1. Explain the bias: The student's friends are likely to share similar opinions, ages, or be in the same classes. Their views might not reflect the views of students in other year groups or friendship circles.
  4. 4
    1. Conclude: This method is biased because it over-represents one particular social group and does not give every student in the school an equal chance of being selected.

Worked example 22 marks

How could the student in the previous question choose a more representative sample of 30 learners?

  1. 1
    1. The goal is to give every student in the population an equal chance of being picked. This is called random sampling.
  2. 2
    1. A good method would be to obtain a list of all 150 students in the school.
  3. 3
    1. Assign each student a number from 1 to 150.
  4. 4
    1. Use a random number generator (on a calculator or computer) to select 30 unique numbers between 1 and 150. The students corresponding to these numbers would form the sample.

Recap

  • A sample is a small group taken from a larger population.
  • A good sample must be representative of the whole population.
  • A sample is biased if it does not represent the population accurately.
  • Random sampling, where everyone has an equal chance of being chosen, helps to avoid bias.

Quick check

  1. A TV channel asks viewers to call in to vote on an issue. Why is this a biased sample of the general public's opinion?1 mark

End-of-chapter exercise

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

  1. The number of goals scored in 7 football matches were: 3, 0, 2, 5, 2, 1, 4. Calculate the mean, median, mode and range of the number of goals.4 marks
  2. A bag contains 12 counters. 5 are red, 4 are blue, and the rest are yellow. A counter is picked at random. What is the probability that the counter is: a) yellow? b) not red?3 marks
  3. A biased coin is flipped 200 times. It lands on heads 70 times. a) What is the experimental probability of getting tails? b) If the coin is flipped 50 more times, estimate how many times it will land on heads.4 marks
  4. A supermarket manager wants to find out the average time customers spend in her store. She surveys customers who are leaving the store between 9am and 10am on a Monday morning. Explain why this sample might be biased.2 marks
  5. The mean weight of 4 boys is 65 kg. A fifth boy weighing 80 kg joins the group. What is the new mean weight of the group?3 marks
  6. The two-way table shows the transport methods of 100 students. What is the probability that a randomly selected student is a girl who travels by bus?2 marks
  7. The probability that it will rain on Saturday is 0.8. The probability that it will rain on Sunday is 0.4. What is the probability that it does not rain on Saturday?1 mark
  8. A fair 8-sided dice, numbered 1 to 8, is rolled. What is the probability of rolling a number that is a factor of 12?3 marks
  9. A spinner is marked with numbers 1, 2, 3, 4. The probability of it landing on each number is shown in the table: P(1)=0.4, P(2)=0.1, P(3)=x, P(4)=0.3. Find the value of x.2 marks
  10. A survey of 234 men found that 46 were over 180 cm tall. a) Calculate the experimental probability that a man is over 180 cm tall. b) In a group of 1000 men, how many would you expect to be over 180 cm tall?3 marks

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