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GRADE 7 · MATHEMATICS

Statistics, Probability and Data Interpretation

Statistics helps us collect, organise and interpret information. Probability helps us describe how likely an event is. In this lesson, you will work with samples, tables, averages, graphs, outcomes and simple experiments.

Learning time: approximately 35–45 minutes · Developing level

What You Will Learn

Distinguish a population from a sample and recognise possible bias.
Organise data with frequency tables and interpret graphs.
Calculate and compare mean, median, mode and range.
Describe probability on a scale from 0 to 1.
List outcomes, calculate simple theoretical probability and interpret experimental results.

1. Population, Sample and Fair Data

A population is the whole group you want to learn about. A sample is a smaller group selected from that population to provide information.

Example: A school has 800 students. Asking every student about their preferred school lunch would involve the population. Asking 80 carefully selected students would use a sample.

A sample should reflect the population as fairly as possible. If you ask only members of the school football team which sport is most popular, the results may be biased because the group is not representative of all students.

Representative sample: reflects important features of the population.
Biased sample: is selected in a way that may unfairly favour some answers.
Random selection: gives members of the population a fair chance of being chosen.

Think critically: A large sample is not automatically fair. How the sample is selected matters too.

2. Types of Data

Categorical data describes groups or labels, such as favourite subjects or transport type. Numerical data is expressed as numbers, such as age, distance or test scores.

Numerical data may be:

Discrete: counted values, often whole numbers, such as the number of books borrowed.
Continuous: measured values that can take any value within a range, such as height or time.

Identifying the data type helps you choose a suitable table or graph.

3. Frequency Tables

Frequency tells how many times a value or category occurs. A frequency table organises these counts so that patterns are easier to see.

Suppose 20 students were asked how they travel to school:

Transport	Frequency
Walk	7
Bus	8
Car	3
Bicycle	2
Total	20

Check the total: 7 + 8 + 3 + 2 = 20. The frequencies should add up to the number of responses when every person gives exactly one answer.

4. Mean, Median, Mode and Range

Mean

Add all values and divide by the number of values.

Mean = sum of values ÷ number of values

Example: Scores are 6, 7, 8, 9 and 10. Their total is 40. There are 5 scores, so the mean is 40 ÷ 5 = 8.

Median

Arrange values from smallest to largest. The median is the middle value. If there are two middle values, calculate their mean.

Example: 3, 4, 6, 8, 9 has median 6. For 2, 4, 7, 9, the median is (4 + 7) ÷ 2 = 5.5.

Mode

The mode is the value that occurs most often. A set may have one mode, more than one mode, or no mode if no value repeats.

Range

The range measures the distance between the greatest and smallest values.

Range = greatest value − smallest value

Example: For 4, 6, 6, 7, 12: mode = 6 and range = 12 − 4 = 8.

5. Compare Data Sets and Notice Outliers

An outlier is a value that is unusually far from most of the other values. It can strongly affect the mean, while the median is often less affected.

Compare: Set A = 4, 5, 5, 6, 7. Set B = 4, 5, 5, 6, 30. Both have median 5, but the mean of Set B is much larger because of 30.

When comparing two groups, do not look at an average alone. Consider the spread, sample size, how the data was collected and whether an unusual value changes the picture.

6. Read and Question Graphs

Bar chart: compares separate categories; bars are usually separated by gaps.
Line graph: often shows how a measurement changes over time.
Pie chart: shows how a whole is divided into categories. All sectors together represent 100% or 360°.

Example bar chart: ways students travel

Walk

Bus

Car

Bicycle

The bus is the most common response in this sample. The chart represents these 20 responses only.

Before interpreting a graph, read its title, axis labels, units, scale and source. A vertical axis that starts above zero can make small differences look much larger than they are. A graph should not hide relevant categories or use a misleading scale.

Pie chart example

If 5 out of 20 students choose cycling, the fraction is 5/20 = 1/4. The sector angle is (5 ÷ 20) × 360° = 90°.

7. Probability: How Likely Is an Event?

Probability describes the chance that an event will happen. For an event that is impossible, the probability is 0. For an event that is certain, the probability is 1.

Probability scale

0	0.25	0.5	0.75	1
Impossible	Unlikely	Even chance	Likely	Certain

Values between 0 and 1 describe different degrees of likelihood.

Probability can be written as a fraction, decimal or percentage. For equally likely outcomes:

P(event) = favourable outcomes ÷ total possible outcomes

Example: A fair six-sided die has six equally likely outcomes: 1, 2, 3, 4, 5 and 6. The probability of rolling a 4 is 1/6. The probability of rolling an even number is 3/6 = 1/2.

The probability of an event and its complement (the event not happening) add to 1:

P(not A) = 1 − P(A)

Example: If the probability of rain is 0.3, the probability of no rain is 1 − 0.3 = 0.7, assuming these are the only two outcomes considered.

8. Sample Spaces and Simple Outcomes

A sample space is a list of all possible outcomes of an experiment. Listing outcomes helps make sure none are missed.

Example: flipping a coin and rolling a four-sided spinner numbered 1 to 4.
Possible pairs are H1, H2, H3, H4, T1, T2, T3 and T4. There are 8 equally likely combinations if the coin and spinner are fair and independent. The probability of getting heads and a 3 is 1/8.

For a simple event, count the outcomes that match the event, then divide by the total number of equally likely outcomes. Do not use this shortcut when outcomes are not equally likely.

9. Theoretical and Experimental Probability

Theoretical probability is calculated from a model of equally likely outcomes. Experimental probability is estimated from what actually happens in repeated trials.

Experimental probability = times event occurs ÷ number of trials

Example: A coin is flipped 40 times and lands on heads 23 times. The experimental probability of heads is 23/40 = 0.575 = 57.5%.

A fair coin has theoretical probability 1/2 = 0.5 = 50% of heads. The experimental result does not have to match 50% exactly. With more independent trials, the relative frequency often gets closer to the theoretical probability, but it is not guaranteed to equal it.

Remember: A short run of results does not change the probability of the next independent fair coin flip. If a coin is fair, the chance of heads on the next flip remains 1/2.

10. Worked Mixed Examples

Example A — Find the median and range: Data: 11, 5, 8, 8, 13, 7. Ordered data: 5, 7, 8, 8, 11, 13. Median = (8 + 8) ÷ 2 = 8. Range = 13 − 5 = 8.

Example B — Probability from a bag: A bag contains 3 red, 2 blue and 5 yellow counters. Total = 10. P(blue) = 2/10 = 1/5. P(not blue) = 8/10 = 4/5.

Example C — Interpret a sample: If 18 of 30 surveyed students prefer reading, the sample proportion is 18/30 = 0.6 = 60%. This describes the surveyed sample; it does not prove that exactly 60% of the whole school prefers reading.

11. Common Mistakes to Avoid

Assuming a biased sample represents everyone.
Finding the median before sorting the data.
Confusing the mode (most frequent) with the median (middle).
Forgetting to divide the mean by the number of values.
Using favourable outcomes ÷ total outcomes when outcomes are not equally likely.
Assuming experimental probability must equal theoretical probability in a small number of trials.
Interpreting a graph without checking its scale, labels or source.

12. Practice Exercises

Try these questions before opening the solutions.

A school has 600 students. A survey asks 60 students about lunch choices. Identify the population and sample.
Find the mean of 5, 7, 8 and 12.
Find the median of 3, 9, 5, 7 and 11.
Find the mode and range of 2, 4, 4, 6, 9.
A bag contains 4 green and 6 orange counters. Find the probability of choosing green and the probability of not choosing green.
A fair six-sided die is rolled. Find the probability of rolling a number greater than 4.
A coin lands on heads 14 times in 25 flips. Find the experimental probability of heads as a fraction and percentage.
Explain why asking only basketball players about the school's favourite sport may produce a biased sample.

Show practice answers and explanations

Population = all 600 students; sample = the 60 students surveyed.
(5 + 7 + 8 + 12) ÷ 4 = 32 ÷ 4 = 8.
Ordered: 3, 5, 7, 9, 11. Median = 7.
Mode = 4; range = 9 − 2 = 7.
P(green) = 4/10 = 2/5. P(not green) = 6/10 = 3/5.
Outcomes greater than 4 are 5 and 6, so probability = 2/6 = 1/3.
Experimental probability = 14/25 = 0.56 = 56%.
Basketball players may prefer basketball, so their answers may not reflect the views of all students.

14. Printable Worksheet

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Learnora — Grade 7 Mathematics

Statistics and Probability Worksheet

Name: ______________________________ Date: _______________

Explain the difference between a population and a sample.
Find the mean of 4, 6, 8, 10 and 12.
Find the median of 3, 7, 5, 9, 11 and 13.
Find the mode and range of 2, 5, 5, 6, 8, 10.
A bag contains 2 red, 3 blue and 5 green counters. Find P(blue).
A fair die is rolled. Find P(rolling a number less than 3).
A coin lands on heads 18 times in 30 flips. Find the experimental probability as a fraction and percentage.
Give one reason why a survey sample might be biased.

Show your working and include units where appropriate.

Learnora — Learn Better. Practice Smarter.

Teacher / Parent Answer Key

Population is the whole group being studied; sample is the smaller selected group.
(4 + 6 + 8 + 10 + 12) ÷ 5 = 40 ÷ 5 = 8.
Ordered: 3, 5, 7, 9, 11, 13. Median = (7 + 9) ÷ 2 = 8.
Mode = 5; range = 10 − 2 = 8.
Total = 10; P(blue) = 3/10.
Outcomes are 1 and 2; probability = 2/6 = 1/3.
18/30 = 3/5 = 0.6 = 60%.
For example, asking only one sports team may not represent the wider population.

15. Lesson Summary

A population is the whole group; a sample is a selected part of it.
Samples should be representative and collected fairly.
Mean, median, mode and range describe different features of numerical data.
Graphs must be read with attention to labels, scales and sources.
Probability ranges from 0 (impossible) to 1 (certain).
Theoretical probability uses a model; experimental probability uses observed trials.

Educational note: This is curriculum-neutral learning support. Check your school's course sequence and teacher guidance.

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