Data, Averages, Probability and Exam Technique

This lesson covers DfE content statements L1.27, L1.28, L1.29, L1.30 and L1.31 — extracting and interpreting information from tables, charts and graphs; calculating the mean and range; understanding the probability scale; calculating simple probabilities as fractions. We finish with essential exam technique for the non-calculator section.

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Reading Tables

Tables organise data in rows and columns. You need to be able to find specific values and use them in calculations.

Worked Example — Train Timetable

Scenario: Here is a train timetable:

Station	Train 1	Train 2	Train 3
London	08:00	09:15	10:30
Reading	08:25	09:40	10:55
Swindon	08:50	10:05	11:20
Bristol	09:30	10:45	12:00

Question: How long does Train 2 take from London to Bristol?

Departure: 09:15, Arrival: 10:45 Duration: 1 hour 30 minutes

Question: Which train arrives in Swindon before 11:00?

Trains 1 and 2 arrive before 11:00 (08:50 and 10:05). Train 3 arrives at 11:20, which is after 11:00.

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Charts and Graphs

At Level 1, you need to read and interpret:

Bar Charts

• Each bar represents a category.
• The height (or length) of the bar shows the value.
• Read values from the scale on the axis carefully.

Worked Example — Bar Chart

Scenario: A bar chart shows the number of coffees sold in a cafe each day:

Day	Coffees sold
Monday	45
Tuesday	38
Wednesday	52
Thursday	41
Friday	60

Question: How many more coffees were sold on Friday than Tuesday? 60 − 38 = 22 more coffees

Question: What was the total for the week? 45 + 38 + 52 + 41 + 60 = 236 coffees

Pie Charts

A pie chart is a circle divided into sections (slices). Each slice represents a proportion of the whole.

Key facts:

• The whole pie = 100% = 360°.
• To find the number from a percentage: calculate the percentage of the total.
• To find the percentage from a fraction of the pie: divide the angle by 360 and multiply by 100.

Worked Example — Pie Chart

Scenario: A pie chart shows how 200 students travel to college:

• Walk: 25%
• Bus: 40%
• Car: 20%
• Cycle: 15%

Question: How many students walk? 25% of 200 = 200 ÷ 4 = 50 students

Question: How many more students take the bus than cycle? Bus: 40% of 200 = 80 Cycle: 15% of 200 = 30 Difference: 80 − 30 = 50 more students

Line Graphs

Line graphs show how something changes over time. The x-axis usually shows time; the y-axis shows the value being measured.

Worked Example — Temperature Graph

Scenario: A line graph shows the temperature in an office throughout the day:

Time	Temperature (°C)
8 am	16
10 am	19
12 pm	22
2 pm	23
4 pm	21
6 pm	18

Question: What was the highest temperature? 23°C at 2 pm.

Question: Between which two times did the temperature rise the most? 8 am to 10 am: rose 3°C 10 am to 12 pm: rose 3°C Both had the same rise of 3°C.

Exam Tip: When reading charts and graphs, always read the axis labels and scale carefully. Check what each grid line represents — it might be 1, 2, 5, 10, 50, etc. Misreading the scale is one of the most common errors.

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Grouping Data (Tally Charts and Frequency Tables)

When you collect data, you often group it using a tally chart and then summarise it in a frequency table.

Worked Example — Shoe Sizes

Scenario: A shoe shop records the sizes sold in one day:

Size	Tally	Frequency
5	IIII	4
6	IIII II	7
7	IIII IIII	9
8	IIII III	8
9	IIII	5
10	II	2

Total pairs sold: 4 + 7 + 9 + 8 + 5 + 2 = 35

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The Mean (Average)

The mean is found by adding up all the values and dividing by how many values there are.

Mean = Total of all values ÷ Number of values

Worked Example — Mean Score

Scenario: A student scores the following marks in 5 tests: 64, 72, 58, 80, 71.

Step 1: Add up: 64 + 72 + 58 + 80 + 71 = 345 Step 2: Divide by the number of tests: 345 ÷ 5 = 69

The mean score is 69.

Worked Example — Mean in Context

Scenario: A delivery driver records the number of parcels delivered each day over 4 days: 35, 42, 28, 39.

Mean = (35 + 42 + 28 + 39) ÷ 4 = 144 ÷ 4 = 36 parcels per day

Finding a Missing Value from the Mean

Scenario: The mean of 4 numbers is 20. Three of the numbers are 18, 22 and 15. What is the fourth number?

Step 1: Total = mean × count = 20 × 4 = 80 Step 2: Sum of known numbers = 18 + 22 + 15 = 55 Step 3: Missing number = 80 − 55 = 25

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The Range

The range tells you how spread out the data is.

Range = Highest value − Lowest value

Worked Example — Comparing Ranges

Scenario: Two machines produce bolts. The lengths (in mm) of 5 bolts from each machine are:

Machine A	Machine B
50.1	49.8
50.3	50.5
49.9	49.5
50.0	50.8
50.2	49.4

Machine A range: 50.3 − 49.9 = 0.4 mm Machine B range: 50.8 − 49.4 = 1.4 mm

Machine A is more consistent (smaller range). Machine B is more variable (larger range).