Mathematics · Topic 1.4
Statistics and Probability
Statistics describes and interprets data. Probability measures how likely events are to occur. Together, these tools help us make sense of the world using numbers and evidence.
What You'll Learn
- Classify and display data using appropriate graphs and charts
- Calculate mean, median, mode, and range from a data set
- Understand and apply theoretical probability
- Calculate experimental probability from trials and compare it to theoretical probability
- Use the complement rule: P(A') = 1 − P(A)
- Read and construct two-way tables
IB Assessment Focus
Criterion A: Calculate averages and probabilities correctly in familiar and unfamiliar situations.
Criterion B: Discover patterns by comparing theoretical and experimental results across multiple trials.
Criterion C: Communicate statistical reasoning clearly, including correct notation (e.g., P(A) = 3/10).
Criterion D: Apply probability and statistics to real-world contexts (surveys, games, scientific experiments).
Key Vocabulary at a Glance
| Term | Definition |
|---|---|
| Experimental probability | P = (times event occurred) ÷ (total trials) — based on observed results |
| Theoretical probability | P = (favourable outcomes) ÷ (total outcomes) — based on equally likely outcomes |
| Sample space | The set of all possible outcomes |
| Complementary event | The event that A does NOT happen: P(A') = 1 − P(A) |
| Two-way table | A table showing the frequency of two categorical variables simultaneously |
| Relative frequency | Frequency expressed as a fraction or percentage of the total |
Mathematics · Topic 1.4
Data Types & Display
Before analysing data, you must understand what type of data you have and choose the best way to display it.
Types of Data
| Type | Description | Examples |
|---|---|---|
| Discrete | Countable, separate values (whole numbers only) | Number of goals, number of students |
| Continuous | Any value within a range (measurable) | Height, temperature, time |
| Categorical | Non-numerical, placed in groups | Favourite colour, genre of book |
Choosing the Right Graph
| Graph Type | Best Used For |
|---|---|
| Bar chart | Comparing discrete or categorical data |
| Pie chart | Showing proportions of a whole |
| Line graph | Showing trends over time (continuous data) |
| Histogram | Showing the distribution of continuous data in intervals |
| Stem-and-leaf plot | Displaying all individual values while showing shape |
| Two-way table | Comparing two categorical variables simultaneously |
Two-Way Tables
A two-way table organises data by two variables at once. Each cell shows the frequency (count) for a combination.
Example: 30 students chose a sport and were categorised by gender.
| Football | Basketball | Swimming | Total | |
|---|---|---|---|---|
| Boys | 8 | 5 | 2 | 15 |
| Girls | 3 | 4 | 8 | 15 |
| Total | 11 | 9 | 10 | 30 |
P(student chose swimming) = 10/30 = 1/3 ≈ 33.3%
Key Point: Always check that the row and column totals add up correctly. The grand total in the bottom-right corner should equal both the sum of all row totals and the sum of all column totals.
Mathematics · Topic 1.4
Measures of Average
Measures of average (central tendency) summarise a data set with a single value. The range measures the spread of the data.
Mean
Mean = Sum of all valuesNumber of values
Range
Range = Largest value − Smallest value
Mean, Median, Mode, Range
| Measure | How to Find It | Best Used When |
|---|---|---|
| Mean | Add all values, divide by number of values | Data has no extreme outliers |
| Median | Order the data; find the middle value (or average of two middle values) | Data has outliers that would skew the mean |
| Mode | The value that appears most often (can be more than one) | Categorical data or finding most popular item |
| Range | Largest value minus smallest value | Measuring how spread out the data is |
Example: Data set: 3, 7, 7, 8, 12, 14, 15
- Mean = (3 + 7 + 7 + 8 + 12 + 14 + 15) ÷ 7 = 66 ÷ 7 ≈ 9.4
- Median = 8 (the 4th value when ordered)
- Mode = 7 (appears twice, more than any other value)
- Range = 15 − 3 = 12
Finding the Median with an Even Number of Values
Example: Data set: 4, 6, 9, 11, 14, 20
Data is already ordered; 6 values (even) → median = average of 3rd and 4th
3rd value = 9, 4th value = 11
Median = (9 + 11) ÷ 2 = 10
Common Mistake: You MUST order the data first before finding the median. The median is not simply the middle-positioned number in the original list — it must be the middle value after ordering from smallest to largest.
Mathematics · Topic 1.4
Probability Basics
Theoretical probability is calculated by reasoning about equally likely outcomes, without carrying out an experiment.
Theoretical Probability
P(event) = number of favourable outcomestotal number of equally likely outcomes
- Probability is always between 0 and 1: 0 ≤ P(A) ≤ 1
- P(certain event) = 1 (will definitely happen)
- P(impossible event) = 0 (cannot happen)
- P(A) + P(A') = 1 (an event and its complement sum to 1)
The Complement Rule
Complement Rule
P(A') = 1 − P(A) where A' means "A does NOT happen"
Example: A bag contains 4 red and 6 green balls. Find P(not red).
P(red) = 4/10 = 2/5
P(not red) = 1 − 2/5 = 3/5 = 0.6 = 60%
P(red) = 4/10 = 2/5
P(not red) = 1 − 2/5 = 3/5 = 0.6 = 60%
Sample Space and Listing Outcomes
The sample space is the complete list of all possible outcomes. For a fair six-sided die:
Sample space = {1, 2, 3, 4, 5, 6}
- P(rolling a 4) = 1/6
- P(rolling an even number) = 3/6 = 1/2 (outcomes: 2, 4, 6)
- P(rolling a number greater than 4) = 2/6 = 1/3 (outcomes: 5, 6)
Probability on a Scale
| Probability | Description | Example |
|---|---|---|
| 0 | Impossible | Rolling a 7 on a standard die |
| 0.1 – 0.3 | Unlikely | Picking a heart from a deck of cards (P = 1/4) |
| 0.5 | Even chance | Flipping heads on a fair coin |
| 0.7 – 0.9 | Likely | Not rolling a 6 on a die (P = 5/6) |
| 1 | Certain | Rolling a number between 1 and 6 on a standard die |
Mathematics · Topic 1.4
Experimental Probability
Experimental probability is based on the results of actual trials. It is also called relative frequency.
Experimental Probability
P(event) = number of times event occurredtotal number of trials
Comparing Experimental and Theoretical Probability
Key Principle:
- Experimental and theoretical probabilities are not always equal
- As the number of trials increases, experimental probability gets closer to theoretical probability
- With a small number of trials, results can differ significantly from theoretical predictions
Example: A coin is flipped 100 times. It lands heads 47 times.
Experimental P(heads) = 47/100 = 0.47
Theoretical P(heads) = 1/2 = 0.50
The results are close but not exactly equal. The difference (0.03) is due to chance variation. If the experiment were repeated with 10,000 flips, the experimental probability would be expected to be much closer to 0.50.
Experimental P(heads) = 47/100 = 0.47
Theoretical P(heads) = 1/2 = 0.50
The results are close but not exactly equal. The difference (0.03) is due to chance variation. If the experiment were repeated with 10,000 flips, the experimental probability would be expected to be much closer to 0.50.
Relative Frequency Tables
Example: A spinner is spun 40 times. Results:
| Colour | Frequency | Relative Frequency | Theoretical P |
|---|---|---|---|
| Red | 12 | 12/40 = 0.30 | 1/4 = 0.25 |
| Blue | 11 | 11/40 = 0.275 | 1/4 = 0.25 |
| Green | 9 | 9/40 = 0.225 | 1/4 = 0.25 |
| Yellow | 8 | 8/40 = 0.20 | 1/4 = 0.25 |
| Total | 40 | 1.00 | 1.00 |
The experimental results are close to the theoretical 0.25, but not identical after only 40 trials.
Critical Rule: Experimental probability and theoretical probability are not always equal — they converge as the number of trials increases. With a small number of trials, experimental results can differ significantly from theoretical predictions. This is called chance variation.
Mathematics · Topic 1.4
Worked Examples
These examples show the reasoning expected at Grade 7. Lay out your work clearly and state your method.
EXAMPLE 1A bag contains 3 red, 5 blue, and 2 green marbles. What is the probability of picking a blue marble?
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Full Solution
Step 1: Find the total number of marbles = 3 + 5 + 2 = 10
Step 2: Identify the number of favourable outcomes (blue) = 5
Step 3: Apply the formula: P(blue) = 5/10 = 1/2 = 0.5 = 50%
Step 2: Identify the number of favourable outcomes (blue) = 5
Step 3: Apply the formula: P(blue) = 5/10 = 1/2 = 0.5 = 50%
EXAMPLE 2Find the mean, median, mode, and range of: 8, 3, 12, 8, 5, 10, 8
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Full Solution
Ordered data: 3, 5, 8, 8, 8, 10, 12
Mean = (3+5+8+8+8+10+12) ÷ 7 = 54 ÷ 7 ≈ 7.7
Median = 4th value = 8 (middle of 7 values)
Mode = 8 (appears 3 times, more than any other)
Range = 12 − 3 = 9
Mean = (3+5+8+8+8+10+12) ÷ 7 = 54 ÷ 7 ≈ 7.7
Median = 4th value = 8 (middle of 7 values)
Mode = 8 (appears 3 times, more than any other)
Range = 12 − 3 = 9
EXAMPLE 3A fair six-sided die is rolled. Find: (a) P(even) (b) P(greater than 4) (c) P(not a 3)
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Full Solution
Sample space = {1, 2, 3, 4, 5, 6}; total outcomes = 6
(a) Even numbers: {2, 4, 6} → P(even) = 3/6 = 1/2
(b) Greater than 4: {5, 6} → P(>4) = 2/6 = 1/3
(c) P(3) = 1/6, so P(not 3) = 1 − 1/6 = 5/6
(a) Even numbers: {2, 4, 6} → P(even) = 3/6 = 1/2
(b) Greater than 4: {5, 6} → P(>4) = 2/6 = 1/3
(c) P(3) = 1/6, so P(not 3) = 1 − 1/6 = 5/6
EXAMPLE 4A coin is tossed 200 times and lands heads 94 times. Calculate the experimental probability and compare to theoretical probability.
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Full Solution
Experimental P(heads) = 94/200 = 0.47
Theoretical P(heads) = 1/2 = 0.50
The experimental probability (0.47) is close to but not exactly equal to the theoretical probability (0.50). The difference of 0.03 is due to chance variation. With more trials, the experimental probability would be expected to get closer to 0.50.
Theoretical P(heads) = 1/2 = 0.50
The experimental probability (0.47) is close to but not exactly equal to the theoretical probability (0.50). The difference of 0.03 is due to chance variation. With more trials, the experimental probability would be expected to get closer to 0.50.
EXAMPLE 5Use the two-way table to find (a) P(girl who swims) and (b) P(football player given they are a boy).
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Full Solution
Using the earlier table (30 students: Boys: Football=8, Basketball=5, Swimming=2; Girls: Football=3, Basketball=4, Swimming=8):
(a) Girls who swim = 8 out of 30 total students
P(girl who swims) = 8/30 = 4/15 ≈ 0.267
(b) Boys who play football = 8; total boys = 15
P(football | boy) = 8/15 ≈ 0.533
(a) Girls who swim = 8 out of 30 total students
P(girl who swims) = 8/30 = 4/15 ≈ 0.267
(b) Boys who play football = 8; total boys = 15
P(football | boy) = 8/15 ≈ 0.533
EXAMPLE 6Scores: 14, 22, 19, 22, 31, 18, 22, 27. Find the mean and median. Which is the better measure of average and why?
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Full Solution
Ordered: 14, 18, 19, 22, 22, 22, 27, 31
Mean = (14+18+19+22+22+22+27+31) ÷ 8 = 175 ÷ 8 = 21.875
Median = average of 4th and 5th values = (22+22)/2 = 22
The data has no extreme outliers, so the mean (21.875) is a suitable measure, but the median (22) is also very close. Both represent the data well. If there were an extreme value like 100, the median would be preferred as it is more resistant to outliers.
Mean = (14+18+19+22+22+22+27+31) ÷ 8 = 175 ÷ 8 = 21.875
Median = average of 4th and 5th values = (22+22)/2 = 22
The data has no extreme outliers, so the mean (21.875) is a suitable measure, but the median (22) is also very close. Both represent the data well. If there were an extreme value like 100, the median would be preferred as it is more resistant to outliers.
EXAMPLE 7A pack of 52 cards has 4 suits (13 cards each). A card is drawn at random. Find P(Ace) and P(not a heart).
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Full Solution
P(Ace) = 4 aces out of 52 cards = 4/52 = 1/13 ≈ 0.077
P(heart) = 13/52 = 1/4
P(not a heart) = 1 − 1/4 = 3/4 = 0.75
P(heart) = 13/52 = 1/4
P(not a heart) = 1 − 1/4 = 3/4 = 0.75
Mathematics · Topic 1.4
Practice Q&A
Attempt each question before revealing the model answer. Show your working clearly.
CALCULATEFind the mean of: 15, 22, 18, 30, 10, 25
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Model Answer
Mean = (15+22+18+30+10+25) ÷ 6 = 120 ÷ 6 = 20
CALCULATEFind the median of: 7, 14, 3, 9, 21, 6
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Model Answer
Ordered: 3, 6, 7, 9, 14, 21 (6 values)
Median = (7 + 9) ÷ 2 = 16 ÷ 2 = 8
Median = (7 + 9) ÷ 2 = 16 ÷ 2 = 8
CALCULATEA bag has 4 red, 3 blue, and 5 yellow sweets. If one is picked at random, find P(yellow).
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Model Answer
Total = 4 + 3 + 5 = 12 sweets.
P(yellow) = 5/12 ≈ 0.417 (41.7%)
P(yellow) = 5/12 ≈ 0.417 (41.7%)
CALCULATEUsing the bag above, find P(not blue).
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Model Answer
P(blue) = 3/12 = 1/4
P(not blue) = 1 − 1/4 = 3/4 = 0.75 (75%)
P(not blue) = 1 − 1/4 = 3/4 = 0.75 (75%)
DESCRIBEA die is rolled 60 times and a 6 appears 8 times. Describe the experimental probability and compare to theoretical probability.
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Model Answer
Experimental P(6) = 8/60 = 2/15 ≈ 0.133
Theoretical P(6) = 1/6 ≈ 0.167
The experimental probability is lower than the theoretical. This difference (0.033) is due to chance variation. With more trials, experimental probability would converge towards 1/6.
Theoretical P(6) = 1/6 ≈ 0.167
The experimental probability is lower than the theoretical. This difference (0.033) is due to chance variation. With more trials, experimental probability would converge towards 1/6.
IDENTIFYFind the mode of: 5, 3, 8, 5, 9, 2, 5, 3, 7, 3
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Model Answer
Tally: 5 appears 3 times, 3 appears 3 times. Both 5 and 3 are modes. This dataset is bimodal with modes 3 and 5.
APPLYThe probability that it rains on any day in April is 0.35. What is the probability that it does NOT rain on a randomly chosen day in April?
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Model Answer
P(rain) = 0.35
P(no rain) = 1 − 0.35 = 0.65
P(no rain) = 1 − 0.35 = 0.65
APPLYA student scored: 68, 75, 80, 68, 72 on five tests. The teacher says the typical score is 68. Is the teacher using the mean, median, or mode? Is this a fair summary?
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Model Answer
The teacher is using the mode (68 appears twice).
Mean = (68+75+80+68+72) ÷ 5 = 363 ÷ 5 = 72.6
Median (ordered: 68, 68, 72, 75, 80) = 72
The mode of 68 is not the fairest summary — both the mean (72.6) and median (72) are higher and more representative of the student's actual performance. The teacher's use of the mode understates the student's ability.
Mean = (68+75+80+68+72) ÷ 5 = 363 ÷ 5 = 72.6
Median (ordered: 68, 68, 72, 75, 80) = 72
The mode of 68 is not the fairest summary — both the mean (72.6) and median (72) are higher and more representative of the student's actual performance. The teacher's use of the mode understates the student's ability.
Mathematics · Review
Flashcard Review
Tap each card to reveal the answer. Try to answer from memory first.
What is theoretical probability?
P(event) = favourable outcomes ÷ total equally likely outcomes. Calculated by reasoning, not by experiment.
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What is experimental probability?
P(event) = times event occurred ÷ total trials. Based on actual results from an experiment.
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What is the complement rule?
P(A') = 1 − P(A). The probability that an event does NOT happen equals 1 minus the probability it does happen.
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What is the sample space?
The complete set of all possible outcomes. For a fair die: {1, 2, 3, 4, 5, 6}.
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How do you calculate the mean?
Add all values together, then divide by the number of values. Mean = sum ÷ count.
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How do you find the median?
Order the data. The median is the middle value. For even number of values, average the two middle values.
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What is the mode?
The value that appears most often in a data set. A data set can have no mode, one mode, or multiple modes (bimodal, multimodal).
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What is the range?
The largest value minus the smallest value. It measures the spread of the data.
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What is the probability range?
0 ≤ P(A) ≤ 1. Impossible events have P = 0. Certain events have P = 1. All probabilities are between these.
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As the number of trials increases, what happens to experimental probability?
It gets closer to (converges toward) the theoretical probability. With very many trials, the two should be nearly equal.
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What is a two-way table?
A table that organises data by two categorical variables simultaneously. Each cell shows the frequency for that combination.
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What is relative frequency?
Frequency divided by total; expressed as a fraction, decimal, or percentage. Same as experimental probability.
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When should you use the median instead of the mean?
When the data contains outliers (extreme values) that would distort the mean. The median is more resistant to outliers.
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A bag has 3 red and 7 green balls. Find P(red).
Total = 10. P(red) = 3/10 = 0.3 = 30%.
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What does it mean for a data set to be bimodal?
It has two modes — two values that both appear more often than any others with the same frequency.
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Mathematics · Practice Test
Practice Test — 20 Questions
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