Measurement, Space & Statistics

Understanding Data and Chance: Summary Statistics and Probability

Year 7

✓Calculate the mean, median, mode, and range for a given set of data.
✓Interpret and describe data using measures of centre and spread.
✓Identify the sample space and specific events for simple chance experiments.
✓Determine theoretical probabilities for simple events and express them as fractions, decimals, or percentages.
✓Distinguish between theoretical and experimental probability.

Key concepts

Data Set

A collection of numerical or categorical information gathered for a particular purpose.

Mean (Average)

The arithmetic average of a data set. It is calculated by adding all the values and then dividing by the count of values.

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

Median

When a data set is arranged in ascending or descending order, the median is the middle value. If there is an even number of values, the median is the average of the two middle values.

Mode

The value that appears most frequently in a data set. A data set can have one mode (unimodal), multiple modes (multimodal), or no mode if all values appear with the same frequency.

Range

The range is a measure of spread that indicates the difference between the maximum and minimum values in a data set.

Range = Highest value - Lowest value

Experiment

In probability, an experiment is any activity or process that involves an element of chance and results in one or more possible outcomes. Examples include rolling a die or flipping a coin.

Outcome

An outcome is one of the individual results that can occur when an experiment is performed.

Sample Space

The sample space is a list or set of all the distinct outcomes that could possibly occur in a probability experiment. It is often denoted by 'S'.

Event

An event is a particular result or a group of results that we are interested in from a probability experiment. It is a subset of the sample space.

Likelihood Scale

A scale used to describe how likely an event is to happen, ranging from impossible (0%) to certain (100%). Common terms include impossible, unlikely, even chance, likely, and certain.

Theoretical Probability

The probability of an event occurring based on mathematical theory, without actually performing the experiment. It is calculated by dividing the number of favourable outcomes by the total number of possible outcomes.

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

Experimental Probability

The probability of an event occurring based on actual trials or observations from an experiment. It is calculated by dividing the number of times an event occurred by the total number of trials.

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

Key facts to remember

1Mean: Sum of values / Number of values.
2Median: The middle value of an *ordered* data set.
3Mode: The most frequent value in a data set.
4Range: Highest value - Lowest value.
5Sample Space: The set of all possible outcomes of an experiment.
6Theoretical Probability: (Number of favourable outcomes) / (Total number of possible outcomes).
7Experimental Probability: (Number of times event occurred) / (Total number of trials).
8Probabilities are always between 0 (impossible) and 1 (certain), inclusive. They can be expressed as fractions, decimals, or percentages.

Worked examples

Example 1

A student recorded the number of goals scored by their favourite soccer team in their last 7 matches: 2, 0, 3, 1, 2, 4, 2. Calculate the mean, median, mode, and range for this data set.

IOrder the data: 0, 1, 2, 2, 2, 3, 4

IICalculate the Mean:

IIISum of values = 0 + 1 + 2 + 2 + 2 + 3 + 4 = 14

IVNumber of values = 7

VMean = 14 / 7 = 2

VICalculate the Median:

VIIThe middle value in the ordered set (0, 1, 2, 2, 2, 3, 4) is 2.

VIIICalculate the Mode:

9The value that appears most frequently is 2 (it appears 3 times).

10Calculate the Range:

11Highest value = 4

12Lowest value = 0

13Range = 4 - 0 = 4

Answer

Mean = 2, Median = 2, Mode = 2, Range = 4

Example 2

A standard six-sided die is rolled.\na) List the sample space.\nb) What is the theoretical probability of rolling an even number?\nc) What is the theoretical probability of rolling a number greater than 4?

Ia) The possible outcomes when rolling a standard six-sided die are 1, 2, 3, 4, 5, 6.

IISample Space, S = {1, 2, 3, 4, 5, 6}

IIITotal number of possible outcomes = 6

IVb) Event: Rolling an even number.

VFavourable outcomes = {2, 4, 6}

VINumber of favourable outcomes = 3

VIIP(Even number) = (Number of favourable outcomes) / (Total number of possible outcomes) = 3 / 6 = 1/2

VIIIAs a decimal: 0.5

9As a percentage: 50%

10c) Event: Rolling a number greater than 4.

11Favourable outcomes = {5, 6}

12Number of favourable outcomes = 2

13P(Number > 4) = (Number of favourable outcomes) / (Total number of possible outcomes) = 2 / 6 = 1/3

14As a decimal (approx): 0.333

15As a percentage (approx): 33.3%

Answer

a) S = {1, 2, 3, 4, 5, 6}\nb) P(Even) = 1/2 (or 0.5 or 50%)\nc) P(Number > 4) = 1/3 (or approx. 0.333 or 33.3%)

Probabilities can be expressed as fractions, decimals, or percentages. Fractions should be simplified.

Example 3

A coin is flipped 20 times. It lands on heads 12 times and tails 8 times.\na) What is the experimental probability of landing on heads?\nb) What is the theoretical probability of landing on heads?

Ia) Total number of trials = 20

IINumber of times heads occurred = 12

IIIExperimental P(Heads) = (Number of times heads occurred) / (Total number of trials) = 12 / 20 = 3/5

IVAs a decimal: 0.6

VAs a percentage: 60%

VIb) For a fair coin, the possible outcomes are Heads, Tails.

VIITotal number of possible outcomes = 2

VIIINumber of favourable outcomes (Heads) = 1

9Theoretical P(Heads) = (Number of favourable outcomes) / (Total number of possible outcomes) = 1 / 2

10As a decimal: 0.5

11As a percentage: 50%

Answer

a) Experimental P(Heads) = 3/5 (or 0.6 or 60%)\nb) Theoretical P(Heads) = 1/2 (or 0.5 or 50%)

Experimental probability often differs from theoretical probability, especially with a small number of trials. As the number of trials increases, experimental probability usually gets closer to theoretical probability.

Common mistakes

✗Not ordering data for the median: Always arrange the data in ascending or descending order before finding the median.
✗Confusing mean, median, and mode: Remember their distinct definitions and calculation methods.
✗Incorrectly identifying the sample space: Ensure all possible outcomes are listed, and no outcome is missed or duplicated.
✗Not simplifying fractions for probability: Always simplify probability fractions to their lowest terms.
✗Confusing theoretical and experimental probability: Understand that theoretical probability is what *should* happen, while experimental probability is what *did* happen.

Exam tips

★Show all working: Clearly show the steps for calculating mean, median, mode, and range, and for probability calculations.
★Order data carefully: For median, write out the ordered data set to avoid errors.
★List the sample space: For probability questions, listing the sample space can help identify all possible outcomes and favourable outcomes accurately.
★Check your answers: Double-check calculations, especially for the mean, and ensure probabilities are reasonable (between 0 and 1).

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