Cambridge Lower Secondary CheckpointStage 6

Statistics and Probability (6Ss)

Mathematics Stage 6 Chapter Notes

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Statistics and Probability — Statistics
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Statistics and Probability (6Ss) notes

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In this chapter

  1. 1Mean, Median, and Mode
  2. 2Range and Measuring Spread
  3. 3Frequency Tables and Bar Charts
  4. 4The Language of Probability
  5. 5Calculating Probability

1. Mean, Median, and Mode

These are three types of 'average' used to find a typical value in a set of data. The mean is what people usually mean by 'average' - you add everything up and divide. The median is the middle value when the data is in order. The mode is simply the most common value. Each is useful in different situations.

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

Median = The middle value of an ordered dataset

Mode = The most frequently occurring value

Key term

Central Tendency: A measure that represents the center or typical value of a frequency distribution.

Examiner insight

Examiners often test the median with an even number of data points, so be sure you know how to find the average of the two middle values.

Common pitfall

Forgetting to put the data in order from smallest to largest before finding the median is a very common mistake.

Worked example 15 marks

A student scores the following marks in 8 tests: 7, 9, 6, 7, 10, 8, 5, 7. Find the mean, median, and mode of their scores.

  1. 1

    Step 1: Find the Mean. Add all the scores together: 7 + 9 + 6 + 7 + 10 + 8 + 5 + 7 = 59.

  2. 2

    Divide the sum by the number of tests (8): 59 ÷ 8 = 7.375. The mean is 7.375.

  3. 3

    Step 2: Find the Median. First, put the scores in order: 5, 6, 7, 7, 7, 8, 9, 10.

  4. 4

    There is an even number of scores (8), so we find the two middle values. These are the 4th and 5th values, which are both 7.

  5. 5

    The median is the average of these two: (7 + 7) ÷ 2 = 7. The median is 7.

  6. 6

    Step 3: Find the Mode. Look for the score that appears most often in the list: 5, 6, 7, 7, 7, 8, 9, 10.

  7. 7

    The score 7 appears three times, more than any other score. The mode is 7.

Recap

  • The mean is calculated by adding all values and dividing by how many values there are.
  • The median is the middle value after you have put the data in numerical order.
  • For an even number of data points, the median is the average of the two middle values.
  • The mode is the most frequent value in the dataset.
  • Always order the data before finding the median.

Quick check

  1. Find the mode of this list of numbers: 4, 6, 2, 6, 7, 9, 6.1 mark
  2. What is the median of: 11, 5, 8, 14, 12?2 marks

2. Range and Measuring Spread

The range tells us how spread out a set of data is. It's a simple measure of spread, calculated by finding the difference between the highest and lowest values. A small range means the data points are all close to each other (they are consistent). A large range means the data is widely spread out.

Range = Highest value - Lowest value

Key term

Range: The difference between the maximum and minimum values in a dataset, indicating the spread of the data.

Examiner insight

Marks are often awarded for interpreting the range in the context of the question, for example by stating that a smaller range means more consistent results.

Common pitfall

Mistaking the range for the mean or median; it measures spread, not the central value.

Fun fact

In manufacturing, a small range in the size of a product (like a screw or a phone screen) is critical for quality control. A large range would mean many parts wouldn't fit correctly.

Worked example 14 marks

The daily temperatures in °C for a week in two cities are recorded: City A: 15, 17, 16, 15, 18, 17, 16. City B: 12, 19, 20, 11, 15, 21, 10. Calculate the range for each city and comment on what it shows.

  1. 1

    Step 1: Calculate the range for City A. Highest temperature = 18°C. Lowest temperature = 15°C.

  2. 2

    Range for City A = 18 - 15 = 3°C.

  3. 3

    Step 2: Calculate the range for City B. Highest temperature = 21°C. Lowest temperature = 10°C.

  4. 4

    Range for City B = 21 - 10 = 11°C.

  5. 5

    Step 3: Comment on the findings. City A has a much smaller range (3°C) than City B (11°C). This means the temperature in City A is more consistent and stable, while the temperature in City B varies a lot more from day to day.

Recap

  • The range is a measure of the spread or consistency of data.
  • To calculate the range, subtract the smallest value from the largest value.
  • A large range indicates that the data is widely spread.
  • A small range indicates that the data is consistent and clustered together.

Quick check

  1. Find the range of the following shoe sizes: 8, 10, 7, 7, 9, 11, 6.2 marks

3. Frequency Tables and Bar Charts

When we have a lot of data, it can be messy. We use a frequency table to organise it. This involves using a tally to count how many times each value or category appears. We can then display this organised data visually using a bar chart. A good bar chart has a title, labelled axes, bars of equal width, and equal gaps between the bars.

Key term

Frequency: The number of times a particular value or category occurs in a set of data.

Examiner insight

Full marks for drawing a graph are only given for perfect execution. This means a title, correctly labelled axes with units, an accurate scale, and correctly plotted bars are all required.

Common pitfall

Drawing bars that touch each other. In a bar chart, there should be gaps between the bars. Bars that touch are used for histograms, which show continuous data.

Worked example 15 marks

The favourite colours of 20 students are: Red, Blue, Green, Red, Blue, Yellow, Blue, Red, Green, Blue, Red, Yellow, Red, Blue, Green, Red, Blue, Blue, Yellow, Red. Create a frequency table and draw a bar chart to represent this data.

  1. 1

    Step 1: Create a frequency table. Go through the list and use tally marks to count each colour.

  2. 2
    ColourTallyFrequency
    Red
    Blue
    Green
    Yellow
    Total20
  3. 3

    Step 2: Draw the bar chart. Draw and label the axes. The horizontal axis will be 'Favourite Colour' and the vertical axis will be 'Frequency'.

  4. 4

    Step 3: Choose a suitable scale for the frequency axis. Since the highest frequency is 7, a scale going up to 8 or 10 would be appropriate.

  5. 5

    Step 4: Draw a bar for each colour. The height of the bar should match its frequency from the table. For example, the bars for Red and Blue should go up to 7. The bars for Green and Yellow should go up to 3.

  6. 6

    Step 5: Give the chart a title, such as 'Favourite Colours of 20 Students'. Ensure bars are of equal width and have gaps between them.

Recap

  • A frequency table is used to organise raw data by counting occurrences.
  • Tally marks are often used to help count frequencies accurately.
  • A bar chart is a visual representation of a frequency table.
  • Bar charts must have a title, labelled axes, a consistent scale, and gaps between the bars.

Quick check

  1. A tally chart shows '|||| |||| |' for a category. What is the frequency?1 mark
  2. On a bar chart, what does the height of each bar represent?1 mark

4. The Language of Probability

Probability is the measure of how likely an event is to happen. We can describe it using words or numbers. The numerical scale for probability goes from 0 to 1. An event with a probability of 0 is 'impossible'. An event with a probability of 1 is 'certain'. An event with a probability of 0.5 (or 1/2 or 50%) has an 'even chance' of happening.

Probability Scale: 0 ≤ P(event) ≤ 1

Key term

Probability: A measure of the likelihood that a specific event will occur, expressed as a number between 0 and 1.

Common pitfall

Confusing 'unlikely' with 'impossible'. An unlikely event has a small chance of happening, but it is not zero. An impossible event has a zero chance.

Fun fact

Insurance companies are built entirely on probability. They calculate the probability of events like car crashes or house fires to determine how much you should pay for your policy.

Worked example 13 marks

For each event below, choose the best word to describe its probability (impossible, unlikely, even chance, likely, certain) and estimate a percentage.a) The sun will rise tomorrow.b) You will roll a 7 on a standard six-sided die.c) A tossed coin will land on heads.

  1. 1

    a) The sun rising tomorrow is as close to a certainty as we can get. Word: Certain. Percentage: 100% (or very close to it, like 99.99...%).

  2. 2

    b) A standard die only has numbers 1, 2, 3, 4, 5, 6. It is not possible to roll a 7. Word: Impossible. Percentage: 0%.

  3. 3

    c) A fair coin has two sides, heads and tails. Each has an equal chance of landing face up. Word: Even chance. Percentage: 50%.

Recap

  • Probability is measured on a scale from 0 to 1.
  • 0 means impossible, 1 means certain.
  • 0.5 (or 50%) means an even chance.
  • Events with a probability between 0 and 0.5 are 'unlikely'.
  • Events with a probability between 0.5 and 1 are 'likely'.
  • Probability can be written as a fraction, decimal, or percentage.

Quick check

  1. An event has a probability of 0.9. Is it likely or unlikely?1 mark
  2. What is the probability of an event that is certain to happen?1 mark

5. Calculating Probability

To calculate the theoretical probability of an event, we use a simple formula. We count the number of ways the desired event can happen (favourable outcomes) and divide it by the total number of all possible outcomes. For experimental probability, we use the results of an experiment: we divide the number of times an event occurred by the total number of trials conducted.

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

Experimental Probability = (Frequency of the event) / (Total number of trials)

Key term

Favourable Outcome: The specific outcome or group of outcomes that you are interested in for a probability experiment.

Examiner insight

Examiners require probabilities to be given in their simplest form, so always check if your fraction can be cancelled down. For example, write 1/2 instead of 5/10.

Common pitfall

Getting the fraction the wrong way around, for example, putting total outcomes on top and favourable outcomes on the bottom.

Worked example 14 marks

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. A marble is picked at random. What is the probability that the marble is:a) Red?b) Not blue?

  1. 1

    Step 1: Find the total number of possible outcomes. This is the total number of marbles in the bag. Total marbles = 5 (red) + 3 (blue) + 2 (green) = 10.

  2. 2

    Step 2: Calculate the probability of picking a red marble. The number of favourable outcomes (red marbles) is 5. P(Red) = (Number of red marbles) / (Total marbles) = 5/10.

  3. 3

    Step 3: Simplify the fraction. 5/10 simplifies to 1/2. The probability of picking a red marble is 1/2 or 0.5 or 50%.

  4. 4

    Step 4: Calculate the probability of picking a marble that is not blue. The number of marbles that are not blue is 5 (red) + 2 (green) = 7.

  5. 5

    P(Not Blue) = (Number of non-blue marbles) / (Total marbles) = 7/10. This is 0.7 or 70%.

Worked example 23 marks

A scientist records the types of sea animals found over 10 trips. The total findings are: 25 Sea snails, 12 Clownfish, 3 Starfish, 1 Lobster. What is the experimental probability of the next specimen found being a clownfish?

  1. 1

    Step 1: Find the total number of trials (specimens found). Total = 25 + 12 + 3 + 1 = 41.

  2. 2

    Step 2: Identify the frequency of the desired event. The number of clownfish found is 12.

  3. 3

    Step 3: Calculate the experimental probability. P(Clownfish) = (Number of clownfish) / (Total specimens) = 12/41.

  4. 4

    The experimental probability of the next specimen being a clownfish is 12/41.

Recap

  • Probability is calculated by dividing favourable outcomes by total possible outcomes.
  • Always find the total number of outcomes first.
  • Probabilities are usually given as fractions in their simplest form, but decimals or percentages are also correct.
  • The probability of an event NOT happening is 1 minus the probability of it happening.
  • Experimental probability is based on data from experiments, while theoretical probability is based on ideal situations.

Quick check

  1. A fair six-sided die is rolled. What is the probability of rolling an even number?2 marks
  2. A spinner has 4 equal sections: red, blue, green, yellow. What is P(green)?1 mark

End-of-chapter exercise

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

  1. Find the mode of this dataset: 15, 12, 11, 15, 19, 11, 15.1 mark
  2. The highest daily temperature in a week was 23°C and the lowest was 14°C. Calculate the temperature range for the week.2 marks
  3. A spinner is spun 50 times. The results are recorded in the table. What is the experimental probability of landing on Blue? Give your answer as a simplified fraction. Colour | Frequency --- | --- Red | 21 Blue | 15 Green | 143 marks
  4. Find the median of this set of numbers: 23, 18, 31, 16, 28, 20.3 marks
  5. A survey asked 30 students about their favourite sport. 12 chose football, 9 chose basketball, 6 chose tennis, and 3 chose swimming. If a student is chosen at random, what is the probability their favourite sport is NOT football?3 marks
  6. The mean of five numbers is 9. Four of the numbers are 10, 7, 11, and 5. What is the fifth number?4 marks
  7. The number of goals scored by a hockey team in 10 games are: 2, 0, 3, 1, 2, 4, 1, 0, 2, 3. Create a frequency table for this data and use it to find the modal number of goals.4 marks
  8. 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
  9. The test scores for a student are 85, 90, 82, 7, 88. Calculate the mean and the median. Which average better represents the student's typical performance, and why?5 marks
  10. The data shows the number of siblings for 15 students: 1, 2, 0, 1, 3, 2, 1, 0, 1, 4, 2, 1, 3, 0, 2. Calculate the mean, median, and mode for this data. (Round the mean to one decimal place).6 marks

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