8.2 Statistics: Tables, Charts, Line Graphs & Mean

Data handling and statistics evaluate a candidate's ability to extract, cross-reference, and interpret quantitative information presented in visual and tabular formats. For the published scope through Year 5, prioritise reading tables and timetables, scaled charts and line graphs, including comparisons, sums and differences. Mean calculations are Year 6 extension work, not a published Year 5 requirement.

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1. Tabular Data Representation & Categorical Diagrams

Tabular data structures require candidates to cross-reference multiple categories simultaneously.

Two-Way Frequency Tables

A two-way table organizes discrete data across two distinct criteria (for example, Year Group across the columns and Preferred After-School Club down the rows). The row totals and column totals must balance in the bottom-right grand total cell.

Worked Example: Completing a Two-Way Table A survey of 80 Year 6 pupils records their preferred school trip:

Activity	Boys	Girls	Total
Science Museum	18	$\mathbf{A}$	32
Adventure Park	$\mathbf{B}$	19	35
Theatre	5	8	$\mathbf{C}$
Total	39	$\mathbf{D}$	80

Deduce the missing values $\mathbf{A}, \mathbf{B}, \mathbf{C},$ and $\mathbf{D}$:

1. Find A (Science Museum Girls): Look across row 1: $18 + \mathbf{A} = 32 \implies \mathbf{A} = 32 - 18 = 14$.
2. Find B (Adventure Park Boys): Look across row 2: $\mathbf{B} + 19 = 35 \implies \mathbf{B} = 35 - 19 = 16$.
3. Find C (Theatre Total): Look across row 3: $\mathbf{C} = 5 + 8 = 13$.
4. Find D (Total Girls): Look down the Girls column: $14 + 19 + 8 = 41$ (or $80 - 39 = 41$).
5. Verify Row & Column Sums: $32 + 35 + 13 = 80$, and $39 + 41 = 80$. Both directions confirm accuracy.

Carroll Diagrams

A Carroll diagram sorts data into a $2 \times 2$ grid based on whether elements satisfy or do not satisfy two specific yes/no criteria:

                    │    Multiple of 3     │  NOT a Multiple of 3
────────────────────┼──────────────────────┼──────────────────────
    Even Number     │      6, 12, 18       │       2, 4, 8, 10    
────────────────────┼──────────────────────┼──────────────────────
  NOT an Even Number│       3, 9, 15       │       1, 5, 7, 11    

Venn Diagrams

A Venn diagram uses overlapping circles to illustrate logical relationships between sets:

• Intersection ($A \cap B$): The overlapping central region containing elements that possess both attributes.
• Universal Set Boundary: Elements placed outside all circles belong to the overall survey group but do not satisfy any of the tested properties.

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2. Pictograms & Fractional Key Decoding

A pictogram presents categorical frequencies using repeated visual symbols or icons. Every pictogram includes a key that defines how many units each symbol represents.

Interpreting Fractional Symbols

When frequencies are not exact multiples of the key, fractions of a symbol are used:

Key: 1 Circle (●) = 12 Books

●       = 12 books (Full symbol)
◑       = 6 books  (Half symbol: 1/2 × 12)
◔       = 3 books  (Quarter symbol: 1/4 × 12)
◕       = 9 books  (Three-quarter symbol: 3/4 × 12)

Worked Example: A school library tracks book loans across three classes:

Class	Symbols Displayed	Calculation	Total Books
Class 5A	● ● ● ◑	$3 \times 12 + 6$	42
Class 5B	● ● ◕	$2 \times 12 + 9$	33
Class 5C	● ● ● ● ◔	$4 \times 12 + 3$	51

Question: How many more books did Class 5C loan than Class 5B?

• Calculation: $51 - 33 = 18$ books.

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3. Bar Charts: Non-Unit Scales & Grouped Comparisons

Bar charts display discrete categorical data where the length or height of each rectangular bar corresponds directly to its numerical frequency.

Calibrating Non-Unit Scales

A chart scale can advance in increments of $2, 5, 10, 20,$ or $50$. You must calculate the value of one grid square before extracting data:

$$\text{Value per subdivision} = \frac{\text{Numbered Interval Difference}}{\text{Number of grid spaces between numbers}}$$

• If the axis marks $0, 20, 40$ with $4$ grid squares between $0$ and $20$: $$\text{Value per grid line} = 20 \div 4 = 5 \text{ units}$$
• If a bar terminates 3 squares above $20$, its value is $20 + (3 \times 5) = 35$.

Dual & Grouped Bar Charts

Dual bar charts position two bars side-by-side for each category, enabling direct comparisons between two sub-cohorts (e.g., Boys vs Girls, or Year 2024 vs Year 2025).

These are original chart data. A grouped bar chart should plot two clearly labelled bars at each sport, starting at the same baseline. The table keeps every value explicit.

When evaluating dual bar charts, questions typically ask for:

• The total within a single sport across both genders ($20 + 25 = 45$ for Tennis).
• The difference between boys and girls in a category ($30 - 15 = 15$ for Football).
• The category showing the greatest disparity between groups.

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4. Line Graphs: Continuous Variables & Time-Series Data

While bar charts display discrete categories (like favourite colour or mode of transport), line graphs display continuous data—measurements that can take any numerical value along a continuous scale, such as temperature, height, mass, or elapsed time.

Interpreting Line Graph Features

1. Data Points: Specific recorded observations plotted at fixed intervals.
2. Connecting Lines: Indicate trends between recorded data points.
3. Interpolation (Reading Between Points): Estimating intermediate values, assuming the drawn straight-line connection is appropriate (e.g., finding the temperature at 11:30 am when readings were logged at 11:00 am and 12:00 noon).
4. Interpreting Slopes (Gradients):
   • Steep Upward Slope: Rapid increase in the named quantity; on a distance–time graph, this indicates higher speed.
   • Gentle Upward Slope: Steady, gradual increase.
   • Horizontal (Flat) Line: No change in value (in distance-time graphs, this indicates being stationary or at rest).
   • Downward Slope: Decrease in temperature, depth, or returning to the start.

A line graph of these data rises from 0 to 2 km, stays level during a 30-minute rest, then rises by 4 km in 30 minutes. The last interval has an average speed of 8 km/h. The vertical axis is total distance travelled; reaching 6 km does not mean arriving back home. A graph of distance from home would decrease during a return journey. A flat distance-from-home graph alone need not establish rest if movement along a circular route is possible; use the stated route and quantity.

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5. Optional extension: mean and range

Key Stage 2 mathematics focuses heavily on two core statistical statistics: the arithmetic mean (an average) and the range (a measure of spread).

1. The Arithmetic Mean

The mean is calculated by summing all individual numerical values and dividing that sum by the total count of values:

$$\text{Mean} = \frac{\text{Sum of all values}}{\text{Number of values } (n)}$$

Worked Example: Find the mean of these five test scores: $14, 18, 15, 17, 16$.

• Step 1: Calculate the sum: $$\text{Sum} = 14 + 18 + 15 + 17 + 16 = 80$$
• Step 2: Count the items: $n = 5$.
• Step 3: Divide sum by count: $$\text{Mean} = \frac{80}{5} = 16$$

2. The Range

The range is not an average; it is a measure of dispersion or spread, showing how widely varied the data values are:

$$\text{Range} = \text{Maximum Value} - \text{Minimum Value}$$

• In the test scores above ($14, 18, 15, 17, 16$):
  • Maximum = $18$, Minimum = $14$.
  • $\text{Range} = 18 - 14 = 4$.
• A small range indicates consistent data; a large range indicates high variability.

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6. Optional extension: missing-value mean problems

A reverse mean problem gives a mean and asks for another quantity. Here, the mean is already given and an unknown data item must be deduced.

The Fundamental Principle of Total Sum

Whenever a mean is provided, immediately determine the grand total sum:

$$\text{Total Sum} = \text{Mean} \times \text{Number of values } (n)$$

Case 1: Finding a Single Missing Value

Worked Example: The mean score of a pupil across 5 mathematics tests is 18. Their scores on the first 4 tests are 16, 21, 14, and 19. What score did the pupil obtain on their fifth test?

• Step 1 (Find target total sum): $$\text{Target Sum} = \text{Mean} \times n = 18 \times 5 = 90$$
• Step 2 (Sum the four known tests): $$\text{Known Sum} = 16 + 21 + 14 + 19 = 70$$
• Step 3 (Calculate the missing fifth test): $$\text{Fifth Test Score} = \text{Target Sum} - \text{Known Sum} = 90 - 70 = 20$$
• The pupil scored 20 on the fifth test.

Case 2: Updating the Mean when a New Value is Added

Worked Example: A rugby team of 15 players has a mean weight of 82 kg. When a reserve player joins the squad, the new mean weight becomes 83 kg for the 16 players. What is the weight of the reserve player?

• Step 1 (Original total weight of 15 players): $$\text{Total}_1 = 15 \times 82 = 1,230\text{ kg}$$
• Step 2 (New total weight of 16 players): $$\text{Total}_2 = 16 \times 83 = 1,328\text{ kg}$$
• Step 3 (Weight of the new player): $$\text{Reserve Player Weight} = 1,328 - 1,230 = 98\text{ kg}$$

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7. Common Exam Pitfalls & Trap Avoidance