Measurement, Space & Statistics

Data Analysis and Two-Step Chance

Year 8

✓By the end of this lesson students will be able to calculate and interpret measures of central tendency (mean, median, mode) and range for a given data set.
✓By the end of this lesson students will be able to interpret and compare various data displays, including dot plots, stem-and-leaf plots, histograms, and column graphs.
✓By the end of this lesson students will be able to describe and calculate probabilities of two-step chance experiments, both with and without replacement.
✓By the end of this lesson students will be able to use lists, tables, and tree diagrams to represent and determine outcomes for two-step chance experiments.

Key concepts

Measures of Central Tendency

Measures of central tendency describe the 'centre' or typical value of a data set. These include the mean, median, and mode.

Mean

The mean (or average) is calculated by summing all the values in a data set and dividing by the number of values. It is sensitive to extreme values.

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

Median

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

Mode

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

Range

The range is a measure of spread that describes the difference between the highest and lowest values in a data set. It indicates how spread out the data is.

Range = Highest value - Lowest value

Two-Step Chance Experiments

A two-step chance experiment involves two events happening in sequence. The outcome of the first event may or may not affect the outcome of the second event. We can use lists, tables, or tree diagrams to visualise and calculate probabilities.

Probability of Two Independent Events

If two events, A and B, are independent (meaning the outcome of A does not affect the outcome of B), the probability of both events occurring is the product of their individual probabilities.

P(A and B) = P(A) × P(B)

Probability of Two Dependent Events (Without Replacement)

If two events, A and B, are dependent (meaning the outcome of A affects the outcome of B, often due to 'without replacement' scenarios), the probability of both events occurring is the probability of A multiplied by the probability of B occurring given that A has already occurred.

P(A and B) = P(A) × P(B after A)

Key facts to remember

1Always order data when finding the median or range.
2The mean is sensitive to outliers (extreme values), while the median is less affected.
3The mode is the only measure of central tendency that can be used for categorical data.
4The range provides a simple measure of the spread of data.
5For two-step chance experiments, use lists, tables, or tree diagrams to map out all possible outcomes.
6When events are independent, P(A and B) = P(A) × P(B).
7When events are dependent (e.g., 'without replacement'), the probability of the second event changes based on the outcome of the first.
8Probabilities are always between 0 and 1 (inclusive), or 0% and 100%.

Worked examples

Example 1

For the following set of test scores: 75, 82, 68, 91, 75, 88, 70. Calculate the mean, median, mode, and range.

I1. Order the data set: 68, 70, 75, 75, 82, 88, 91.

II2. Calculate the Mean: (68 + 70 + 75 + 75 + 82 + 88 + 91) / 7 = 549 / 7 = 78.43 (to 2 decimal places).

III3. Identify the Median: The middle value in the ordered set is 75.

IV4. Identify the Mode: The value that appears most frequently is 75.

V5. Calculate the Range: Highest value - Lowest value = 91 - 68 = 23.

Answer

Mean = 78.43, Median = 75, Mode = 75, Range = 23

Always order the data first when finding the median or range.

Example 2

A fair coin is tossed twice. What is the probability of getting two heads?

I1. List all possible outcomes using a tree diagram or a list: HH, HT, TH, TT.

II2. Determine the total number of possible outcomes: There are 4 possible outcomes.

III3. Determine the number of favourable outcomes (two heads): There is 1 outcome (HH).

IV4. Calculate the probability: P(HH) = (Number of favourable outcomes) / (Total number of possible outcomes) = 1/4.

VAlternatively, using the multiplication rule for independent events:

VIP(Head on 1st toss) = 1/2

VIIP(Head on 2nd toss) = 1/2

VIIIP(HH) = P(Head on 1st) × P(Head on 2nd) = 1/2 × 1/2 = 1/4.

Answer

P(two heads) = 1/4

Each toss of a fair coin is an independent event.

Example 3

A bag contains 3 red marbles and 2 blue marbles. Two marbles are drawn at random without replacement. What is the probability that both marbles drawn are red?

I1. Determine the probability of drawing a red marble first: P(1st Red) = (Number of red marbles) / (Total number of marbles) = 3 / 5.

II2. After drawing one red marble without replacement, there are now 2 red marbles left and a total of 4 marbles remaining in the bag.

III3. Determine the probability of drawing a second red marble, given the first was red: P(2nd Red | 1st Red) = 2 / 4 = 1/2.

IV4. Calculate the probability of both events occurring: P(Both Red) = P(1st Red) × P(2nd Red | 1st Red) = (3/5) × (1/2) = 3/10.

Answer

P(both red) = 3/10

The phrase 'without replacement' means the events are dependent, and the probabilities change for the second draw.

Common mistakes

✗Not ordering the data set before finding the median.
✗Forgetting to adjust the total number of outcomes and favourable outcomes when calculating probabilities 'without replacement'.
✗Confusing mean, median, and mode, or incorrectly calculating them.
✗Incorrectly adding probabilities for 'and' events instead of multiplying them.
✗Not listing all possible outcomes for two-step chance experiments, leading to incorrect probability calculations.

Exam tips

★Read the question carefully to determine if events are 'with replacement' (independent) or 'without replacement' (dependent).
★For data analysis questions, clearly show your working for ordering data and each calculation (mean, median, mode, range).
★When dealing with two-step chance, draw a tree diagram or create a table to visualise all possible outcomes and their probabilities.
★Always simplify fractions when stating probabilities, unless otherwise specified.

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