Mathematics · Topic 1.4
Statistics
Statistics is the collection, organisation, and interpretation of data. In this topic you will learn to summarise data with averages and display it using graphs and charts.
What You'll Learn
- Calculate the mean, median, mode, and range of a data set
- Understand when to use each type of average
- Read and draw bar graphs, pie charts, and pictograms
- Construct and interpret stem-and-leaf plots
- Describe what data tells us in context
- Understand the difference between discrete and continuous data
IB Assessment Focus
Criterion A: Calculate averages accurately and select the most appropriate type.
Criterion B: Look for patterns in data and describe trends.
Criterion C: Present data clearly with correct labels, titles, and scales.
Criterion D: Apply statistics to real-world contexts and comment on whether results make sense.
Key Vocabulary
| Term | Definition |
|---|---|
| Data | Information collected for analysis (e.g., test scores, heights) |
| Mean | The average: sum of all values ÷ number of values |
| Median | The middle value when data is arranged in order |
| Mode | The value that occurs most frequently |
| Range | The difference between the highest and lowest values |
| Frequency | How many times a value appears in a data set |
| Discrete data | Data that can only take certain values (e.g., number of pets) |
| Continuous data | Data that can take any value in a range (e.g., height, weight) |
Mathematics · Topic 1.4
Mean, Median, and Mode
These three "averages" each summarise a data set in a different way. Knowing which one to use and how to calculate each is essential.
The Mean (Average)
Formula
Mean = Sum of all valuesNumber of values
Example: Find the mean of 5, 8, 3, 10, 4.
Sum = 5 + 8 + 3 + 10 + 4 = 30
Number of values = 5
Mean = 30 ÷ 5 = 6
The Median (Middle Value)
- Arrange all values in ascending order
- If there is an odd number of values: the median is the middle value
- If there is an even number of values: the median is the mean of the two middle values
Odd example: 3, 5, 7, 9, 11 (5 values)
Middle value = 3rd value = 7
Even example: 2, 4, 6, 8 (4 values)
Two middle values = 4 and 6
Median = (4 + 6) ÷ 2 = 5
The Mode (Most Common)
The mode is the value that appears most often. A data set can have:
- One mode: 2, 3, 4, 4, 4, 5, 6 → mode = 4
- Two modes (bimodal): 1, 2, 2, 3, 5, 5, 7 → modes = 2 and 5
- No mode: 1, 2, 3, 4, 5 → every value appears once, so there is no mode
Which Average Should You Use?
| Average | Best Used When | Weakness |
|---|---|---|
| Mean | Data is evenly spread with no extreme values | Affected by outliers (very high/low values) |
| Median | Data has outliers or is skewed | Ignores most of the data values |
| Mode | Data is categorical (e.g., favourite colour) | May not exist or may not be unique |
Critical Rule: Always arrange data in order before finding the median. Finding the median from unsorted data is one of the most common errors in statistics.
Mathematics · Topic 1.4
Range & Data Types
The range tells us how spread out data is, and understanding data types helps us choose the right graphs.
The Range
Formula
Range = Highest value − Lowest value
Example: Data: 3, 7, 2, 9, 5
Range = 9 − 2 = 7
Range = 9 − 2 = 7
The range is a measure of spread, not an average. A large range means the data is very spread out; a small range means the values are close together.
Discrete vs Continuous Data
Discrete data can only take specific values (usually whole numbers). You can count it.
- Number of students in a class
- Number of pets owned
- Goals scored in a match
Continuous data can take any value in a range. You measure it.
- Height of students (e.g., 152.3 cm)
- Weight of parcels (e.g., 2.7 kg)
- Time to run 100m (e.g., 13.45 seconds)
Frequency Tables
A frequency table organises data by counting how often each value (or group of values) appears.
Example: Shoe sizes of 10 students: 5, 6, 5, 7, 6, 5, 8, 6, 5, 7
| Shoe Size | Tally | Frequency |
|---|---|---|
| 5 | IIII | 4 |
| 6 | III | 3 |
| 7 | II | 2 |
| 8 | I | 1 |
Mathematics · Topic 1.4
Bar Graphs & Pie Charts
Graphs and charts help us visualise data so patterns and comparisons are easier to see.
Bar Graphs
A bar graph uses rectangular bars to represent data. The height (or length) of each bar shows the frequency or value.
Rules for drawing bar graphs:
- Give the graph a clear title
- Label both axes (x-axis = categories, y-axis = frequency)
- Use a consistent scale on the y-axis
- Leave equal gaps between bars
- All bars should be the same width
Pie Charts
A pie chart is a circle divided into sectors (slices). Each sector represents a proportion of the whole.
Drawing a pie chart:
- Find the total of all values
- For each category: angle = (value ÷ total) × 360°
- Draw each sector using a protractor
- Label each sector with its category and percentage (or value)
Example: 30 students choose their favourite sport: Football = 12, Basketball = 8, Tennis = 6, Swimming = 4.
Football angle = (12/30) × 360 = 144°
Basketball angle = (8/30) × 360 = 96°
Tennis angle = (6/30) × 360 = 72°
Swimming angle = (4/30) × 360 = 48°
Check: 144 + 96 + 72 + 48 = 360° ✓
Choosing the Right Graph
| Graph Type | Best For |
|---|---|
| Bar graph | Comparing quantities across categories |
| Pie chart | Showing proportions (parts of a whole) |
| Pictogram | Showing frequencies using symbols (easy to read) |
| Line graph | Showing changes over time |
Common Mistake: When drawing pie charts, always check that your angles add up to 360°. If they don't, you have made a calculation error.
Mathematics · Topic 1.4
Stem-and-Leaf Plots
A stem-and-leaf plot is a way to organise numerical data that preserves all the individual values while showing the shape of the distribution.
How It Works
- The stem is the tens digit (or higher place value)
- The leaf is the units digit
- Each leaf represents one data value
- Leaves must be in ascending order
Reading a Stem-and-Leaf Plot
Example:
| Stem | Leaf |
|---|---|
| 2 | 1 3 5 8 |
| 3 | 0 2 5 5 7 |
| 4 | 1 4 6 |
Key: 2 | 3 means 23
This represents the data: 21, 23, 25, 28, 30, 32, 35, 35, 37, 41, 44, 46
There are 12 data values in total.
Constructing a Stem-and-Leaf Plot
Data: 15, 22, 18, 31, 25, 14, 29, 33, 27, 16
- Identify the stems: 1, 2, 3
- Sort the data: 14, 15, 16, 18, 22, 25, 27, 29, 31, 33
- Write leaves in order:
| Stem | Leaf |
|---|---|
| 1 | 4 5 6 8 |
| 2 | 2 5 7 9 |
| 3 | 1 3 |
Key: 1 | 5 means 15
Finding Averages from a Stem-and-Leaf Plot
Because all individual values are preserved, you can find the mean, median, mode, and range directly from the plot:
- Median: Count total leaves, find the middle value(s)
- Mode: Look for repeated leaves in the same stem
- Range: Last value − first value
Always include a key! Without a key, the reader does not know what the stem and leaf represent. For example, Key: 3 | 5 means 35.
Mathematics · Topic 1.4
Worked Examples
Step-by-step solutions showing the working expected in assessments.
EXAMPLE 1Find the mean, median, mode, and range of: 4, 7, 3, 7, 9, 2, 7.
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Full Solution
Step 1: Arrange in order: 2, 3, 4, 7, 7, 7, 9.
Mean = (2+3+4+7+7+7+9) ÷ 7 = 39 ÷ 7 ≈ 5.6.
Median = 4th value = 7 (middle of 7 values).
Mode = 7 (appears 3 times, more than any other).
Range = 9 − 2 = 7.
Mean = (2+3+4+7+7+7+9) ÷ 7 = 39 ÷ 7 ≈ 5.6.
Median = 4th value = 7 (middle of 7 values).
Mode = 7 (appears 3 times, more than any other).
Range = 9 − 2 = 7.
EXAMPLE 2The mean of 5 numbers is 8. Four of the numbers are 6, 10, 9, and 7. Find the fifth number.
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Full Solution
Step 1: Total sum = mean × count = 8 × 5 = 40.
Step 2: Sum of known values = 6 + 10 + 9 + 7 = 32.
Step 3: Fifth number = 40 − 32 = 8.
Step 2: Sum of known values = 6 + 10 + 9 + 7 = 32.
Step 3: Fifth number = 40 − 32 = 8.
EXAMPLE 3Find the median of: 12, 5, 8, 3, 15, 9.
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Full Solution
Step 1: Arrange in order: 3, 5, 8, 9, 12, 15.
Step 2: Even number of values (6), so median = mean of 3rd and 4th values.
Step 3: Median = (8 + 9) ÷ 2 = 8.5.
Step 2: Even number of values (6), so median = mean of 3rd and 4th values.
Step 3: Median = (8 + 9) ÷ 2 = 8.5.
EXAMPLE 4Calculate the angle for a pie chart sector if 15 out of 60 students chose Science as their favourite subject.
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Full Solution
Angle = (value ÷ total) × 360°
= (15 ÷ 60) × 360
= 0.25 × 360
= 90°
= (15 ÷ 60) × 360
= 0.25 × 360
= 90°
EXAMPLE 5From a stem-and-leaf plot: Stem 1 | Leaves 2, 5, 8; Stem 2 | Leaves 0, 3, 3, 7; Stem 3 | Leaves 1, 4. Find the median and mode.
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Full Solution
Values: 12, 15, 18, 20, 23, 23, 27, 31, 34 (9 values).
Median = 5th value = 23.
Mode = 23 (appears twice, all others appear once).
Median = 5th value = 23.
Mode = 23 (appears twice, all others appear once).
EXAMPLE 6A class recorded temperatures over 5 days: 18°C, 22°C, 19°C, 25°C, 21°C. Is this discrete or continuous data? Find the mean and range.
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Full Solution
Temperature is continuous data (it can take any value, not just whole numbers).
Mean = (18+22+19+25+21) ÷ 5 = 105 ÷ 5 = 21°C.
Range = 25 − 18 = 7°C.
Mean = (18+22+19+25+21) ÷ 5 = 105 ÷ 5 = 21°C.
Range = 25 − 18 = 7°C.
Mathematics · Topic 1.4
Practice Q&A
Attempt each question before revealing the model answer.
CALCULATEFind the mean of 12, 15, 18, 20, 25.
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Model Answer
Sum = 12 + 15 + 18 + 20 + 25 = 90. Mean = 90 ÷ 5 = 18.
CALCULATEFind the median of 14, 8, 22, 5, 11, 19, 3.
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Model Answer
In order: 3, 5, 8, 11, 14, 19, 22. Middle (4th) value = 11.
IDENTIFYFind the mode and range of: 6, 3, 8, 3, 5, 9, 3, 7.
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Model Answer
Mode = 3 (appears 3 times). Range = 9 − 3 = 6.
EXPLAINA data set has values 2, 3, 4, 5, 100. Why is the median a better average than the mean for this data?
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Model Answer
The mean = (2+3+4+5+100) ÷ 5 = 22.8, which does not represent the typical values. The median = 4, which is much closer to most of the data. The value 100 is an outlier that pulls the mean up. The median is not affected by outliers, making it a better measure of the "centre" here.
CALCULATEIn a pie chart, a sector represents 25% of the data. What angle should it be?
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Model Answer
25% of 360° = 0.25 × 360 = 90°.
DESCRIBEGive an example of discrete data and an example of continuous data.
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Model Answer
Discrete: Number of siblings (0, 1, 2, 3…) — you can count them and they can only be whole numbers.
Continuous: Height in cm (e.g., 145.2 cm) — it is measured and can take any value.
Continuous: Height in cm (e.g., 145.2 cm) — it is measured and can take any value.
CALCULATEThe mean of 4 numbers is 10. Three of the numbers are 8, 12, and 6. Find the fourth number.
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Model Answer
Total = 10 × 4 = 40. Sum of three known = 8 + 12 + 6 = 26. Fourth number = 40 − 26 = 14.
CONSTRUCTConstruct a stem-and-leaf plot for: 23, 15, 31, 28, 17, 34, 22, 19, 25, 30.
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Model Answer
Sorted: 15, 17, 19, 22, 23, 25, 28, 30, 31, 34.
Stem 1 | 5 7 9
Stem 2 | 2 3 5 8
Stem 3 | 0 1 4
Key: 1 | 5 means 15
Stem 1 | 5 7 9
Stem 2 | 2 3 5 8
Stem 3 | 0 1 4
Key: 1 | 5 means 15
Mathematics · Review
Flashcard Review
Tap each card to reveal the answer. Try to answer from memory first.
How do you calculate the mean?
Sum of all values ÷ number of values.
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How do you find the median?
Arrange data in order, then find the middle value. If even number, take the mean of the two middle values.
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What is the mode?
The most frequently occurring value in a data set.
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How do you find the range?
Range = highest value − lowest value.
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What is an outlier?
A value that is much higher or lower than the rest of the data. It can distort the mean.
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Discrete data vs continuous data?
Discrete = counted, specific values (e.g., number of pets). Continuous = measured, any value in a range (e.g., height).
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How do you calculate a pie chart angle?
Angle = (value ÷ total) × 360°
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What must all pie chart angles add up to?
360° (a full circle).
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In a stem-and-leaf plot, what is the stem?
The tens digit (or higher place value).
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In a stem-and-leaf plot, what is the leaf?
The units digit. Each leaf represents one data value.
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What does 3 | 7 mean in a stem-and-leaf plot?
The value 37 (stem = 3 tens, leaf = 7 units).
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When is the median better than the mean?
When there are outliers (extreme values) that would pull the mean away from the typical value.
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What is frequency?
The number of times a particular value appears in a data set.
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What graph is best for showing proportions?
A pie chart — it shows parts of a whole.
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What must you always do before finding the median?
Arrange the data in ascending order!
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Mathematics · Practice Test
Practice Test — 20 Questions
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