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D6.1 Stage 6 Data

Averages, spread & pie charts

Pie charts; the mean (median/mode/range at KS3); distributions (centre, spread, outliers); dot plots, histograms, box plots; scatter graphs / bivariate data.

Data is only useful once you can summarise it. This concept is the toolkit for turning a pile of numbers into a single picture, or a single figure, you can reason about.

What it means

A data set is just a list of measurements — the heights of everyone in a class, say. Two questions matter most: where is the middle, and how spread out are the values?

For the middle, we use an average. There are three:

  • The mean — add every value and divide by how many there are. It shares the total out evenly.
  • The median — sort the values in order and take the middle one. Half the data sits below it, half above.
  • The mode — the value that appears most often.

For the spread, the simplest measure is the range: the largest value minus the smallest. A small range means the values are bunched together; a large range means they are scattered. A value sitting far away from all the others is an outlier.

A distribution describes the whole shape at once — where values cluster (the centre), how far they reach (the spread), and any outliers. We picture distributions with charts: a dot plot stacks one dot per value; a histogram groups values into bars; a box plot draws the median and the middle half of the data as a box.

A box plot leans on the quartiles: sort the data, then the median splits it in half, and the quartiles split each half again. The lower quartile (Q1) is the value a quarter of the way through, the upper quartile (Q3) three-quarters of the way. The box runs from Q1 to Q3, so it holds the middle half of the data, with a line at the median inside it and “whiskers” reaching out to the smallest and largest values.

A pie chart answers a different question: how a whole splits into parts. The full circle is 360°, standing for the total. Each category takes a slice whose angle is its share of the total. If 10 of 40 children walk to school, walking is 10/40 = 1/4 of the circle, so its slice is 1/4 × 360° = 90°. This is why D6.1 builds on ratio and proportion — a pie slice is a proportion drawn as an angle.

So far each chart shows one quantity. A scatter graph shows two at once: it plots each individual as a dot, with one measurement across (say, height) and another up (say, arm span). This pairing of two variables is called bivariate data. When the dots trend upward together — taller people tending to have longer arm spans — we say the two are correlated. Scatter graphs are where you see whether two things move together, without assuming one causes the other.

Worked examples

Mean, median, mode, range. Test scores: 4, 7, 7, 8, 9.

mean   = (4 + 7 + 7 + 8 + 9) / 5 = 35 / 5 = 7
median = middle of sorted list  = 7
mode   = most frequent value    = 7
range  = 9 − 4                   = 5

A pie chart. 20 pets: 10 dogs, 5 cats, 5 fish.

PetCountFractionAngle
Dogs101/2180°
Cats51/490°
Fish51/490°

The angles add to 360°, filling the circle.

Spot the outlier. Ages 8, 9, 9, 10, 41. The mean is 77/5 = 15.4, but that “average” describes nobody — the 41 is an outlier dragging it up. The median, 9, describes the group far better.

The generative-art connection

A pie chart is generative art with a rule: sweep an arc whose angle is proportional to each count. Feed the same drawing routine different data and you get a different picture every time — the mathematics is the image. In hue-pulse you can watch a point spread into a plane; that same move, turning a number into filled area, is exactly what a pie slice or a histogram bar does.

Distributions are even more visual. In PhET Plinko Probability balls fall through pegs and pile up into a histogram in real time — you watch the centre and spread emerge from randomness rather than being told them. Seeing Theory animates the same idea, letting you drag data points and see the mean and spread shift live. Making these pictures, then changing the numbers behind them, is how the shape of data becomes something you can feel.

Common misconceptions

  • “Average always means the mean.” Median and mode are averages too, and often better. When there is an outlier, the median usually gives the truer middle.
  • “Pie slices are about the count.” A slice shows a proportion of the whole. The same 5 pets are a big slice in a group of 10 and a tiny one in a group of 100.
  • “A bigger range means a bigger average.” Spread and centre are independent. 1, 5, 9 and 4, 5, 6 share the mean 5, yet their ranges (8 and 2) are very different.

What a learner should be able to solve

The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.

  1. For the data set `5, 8, 8, 9, 10, 12`, find the **mean**, the **median**, the **mode** and the **range**.

    Answer

    Mean `= (5+8+8+9+10+12)/6 = 52/6 ≈ 8.67`. Median (even list, two middle values 8 and 9) `= (8+9)/2 = 8.5`. Mode `= 8` (appears twice). Range `= 12 − 5 = 7`.

    Art hook One canvas showing all four at once: a dot plot of the six values, the mode column glowing, a span-bar for the range, and two markers on the axis for mean and median.

  2. `40` people were asked their favourite season: `10` spring, `20` summer, `5` autumn, `5` winter. Work out the pie-chart **angle** for each season and confirm they total `360°`.

    Answer

    Spring `10/40 = 1/4 → 90°`; summer `20/40 = 1/2 → 180°`; autumn `5/40 = 1/8 → 45°`; winter `5/40 = 1/8 → 45°`. Check: `90+180+45+45 = 360°`.

    Art hook Sweep a four-slice pie (90°, 180°, 45°, 45°) with a seasonal colour per wedge; a running tally shows the degrees summing to 360 as each slice is drawn.

  3. A shop records daily sales for a week: `4, 5, 5, 6, 6, 6, 41`. Explain why the **median** describes a typical day better than the **mean**, and give both values.

    Answer

    Mean `= 73/7 ≈ 10.4`; median (middle of sorted list) `= 6`. The `41` is an **outlier** that inflates the mean well above every ordinary day, whereas the median `6` matches a typical day.

    Art hook Number line: six dots bunched near 4-6 and one far-right outlier at 41. Animate the mean marker being dragged toward the outlier while the median marker stays in the cluster.

  4. A **scatter graph** plots 20 pupils with hours of sleep across and reaction-test score up. The dots fall from top-left to bottom-right. Describe the relationship this shows, and explain why it does not prove that more sleep *causes* faster reactions.

    Answer

    As sleep increases, reaction score decreases together in a downward trend — a **negative correlation**. Correlation shows the two move together but does not prove causation; another factor could explain it, so we can only say they are associated, not that one causes the other.

    Art hook Scatter of 20 dots along a falling diagonal with slight noise, plus a faint downward trend line. A toggle flips the trend from rising to falling so pupils compare positive vs negative correlation.

  5. Create a data set of **6 numbers** with a **mode of 7**, a **median of 6** and a **range of 8**. Show that your set meets all three conditions.

    Answer

    One valid set is `2, 4, 5, 7, 7, 10`. Mode `= 7` (the only value that repeats). Median `= (5 + 7)/2 = 6` (mean of the two middle values). Range `= 10 − 2 = 8`. All three conditions hold. (Other correct answers exist — the mode value must appear more often than any other, the two middle numbers must average 6, and largest − smallest must be 8.)

    Art hook Six draggable dots on a number line with live read-outs for mode, median and range; each condition turns green when satisfied, so solving it is a visual puzzle.

Training exercises

Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.

  1. Here is a list of numbers: `3, 5, 5, 5, 7`. Which value appears most often? (That value is called the **mode**.)

    Answer

    The mode is `5` — it appears three times, more than any other value.

    Art hook Draw the list as a dot plot: a horizontal number line 1-10 with one dot stacked above each value. The tallest stack (over 5) is the mode. Colour the tallest column a bright hue and the rest grey so the mode 'lights up'.

  2. Find the **range** of these daily temperatures: `12, 15, 9, 14, 11` (all in °C). Range = largest value − smallest value.

    Answer

    Range `= 15 − 9 = 6`°C. The values stretch across 6 degrees.

    Art hook Place the 5 values as dots on a horizontal number line. Draw a coloured bar spanning from the leftmost dot to the rightmost dot — its length IS the range. Re-run with new data and the bar stretches or shrinks.

  3. Find the **mean** of `6, 8, 10, 12`. (Add them all, then divide by how many there are.)

    Answer

    Sum `= 6 + 8 + 10 + 12 = 36`; there are 4 values, so mean `= 36 / 4 = 9`.

    Art hook Draw 4 towers of dots with heights 6, 8, 10, 12. Then animate the dots 'levelling out' until all four towers are the same height (9) — the mean is the height when everything is shared evenly.

  4. Sort these and find the **median** (the middle value once sorted): `9, 3, 7, 1, 5`.

    Answer

    Sorted: `1, 3, 5, 7, 9`. Five values, so the middle one is the 3rd: median `= 5`.

    Art hook Show the values as dots that slide into sorted order along a number line. Then a marker walks in from both ends, one step at a time, meeting at the middle dot — highlight it as the median.

  5. Find the **median** of an even-sized list: `4, 6, 8, 10`. (With no single middle, take the mean of the two middle values.)

    Answer

    The two middle values are `6` and `8`, so median `= (6 + 8) / 2 = 7`.

    Art hook Four dots on a number line; markers walk in from both ends and stop on the two central dots. Draw a small dot exactly halfway between them to show the median sitting in the gap.

  6. In a survey of `24` children, `6` chose apple as their favourite fruit. What **fraction** of the circle would apple's pie-chart slice be, and what **angle** in degrees?

    Answer

    Fraction `= 6/24 = 1/4`. Angle `= 1/4 × 360° = 90°` — a right-angle slice.

    Art hook Canvas arc: sweep a filled wedge from 0° to 90° on a circle. Let a slider change the count out of 24 and watch the wedge angle update live as `count/24 × 360°`.

  7. A class of `30` split their break like this: `15` played football, `10` read, `5` drew. Work out the pie-chart angle for **each** activity and check they add to `360°`.

    Answer

    Football `15/30 = 1/2 → 180°`; reading `10/30 = 1/3 → 120°`; drawing `5/30 = 1/6 → 60°`. Check: `180 + 120 + 60 = 360°`.

    Art hook Full pie chart: sweep three coloured wedges in turn (180°, 120°, 60°), each starting where the last ended, so the circle fills exactly. Give each slice its own hue from a colour wheel.

  8. **Spot the mistake.** Sam says the median of `2, 8, 3, 9, 5` is `3` because it is the middle number in the list. What did Sam forget, and what is the correct median?

    Answer

    Sam forgot to **sort** first. Sorted: `2, 3, 5, 8, 9`; the true middle is `5`. So the median is `5`, not `3`.

    Art hook Two rows of dots: the unsorted row (wrong middle circled in red) above the sorted row (correct middle circled in green), so the fix is visible at a glance.

  9. A dot plot shows shoe sizes: size 3 has 1 dot, size 4 has 3 dots, size 5 has 4 dots, size 6 has 2 dots. How many children in total, and which size is the **mode**?

    Answer

    Total `= 1 + 3 + 4 + 2 = 10` children. The mode is size `5` (the tallest stack, 4 dots).

    Art hook Rebuild this dot plot on a canvas: a row of columns whose dot-heights are 1, 3, 4, 2. Auto-detect and glow the tallest column as the mode; print the running total as dots are added.

  10. For the ages `7, 8, 8, 9, 28`, find the mean and the median. Which one better describes a typical age, and why?

    Answer

    Mean `= (7+8+8+9+28)/5 = 60/5 = 12`; median (sorted middle) `= 8`. The median (`8`) is better — the `28` is an **outlier** that drags the mean up to an age nobody is near.

    Art hook Number line with the four low dots clustered and one lone dot far right (the outlier). Show the mean marker pulled toward the lone dot while the median marker stays with the cluster — the tug is visible.

  11. A **scatter graph** plots each child as a dot: height across, arm span up. The dots rise from bottom-left to top-right. In your own words, what does this pattern tell you?

    Answer

    The two measurements tend to increase together — taller children tend to have longer arm spans. We say height and arm span are (positively) **correlated**. It does not mean one causes the other.

    Art hook Scatter of ~30 dots seeded along a rising diagonal with a little random scatter. Draw a faint trend line; a slider adds 'noise' so pupils watch the upward pattern blur as correlation weakens.

  12. **Open-ended.** Invent a list of 5 whole numbers whose **mean is 6** and whose **range is 4**. Show it works.

    Answer

    Many answers work, e.g. `4, 5, 6, 7, 8`: mean `= 30/5 = 6`, range `= 8 − 4 = 4`. (Any 5 numbers summing to 30 with max − min = 4 is correct.)

    Art hook An interactive: five draggable dots on a number line. Live-display the mean marker and the range bar; the goal is to nudge the dots until mean = 6 and range = 4 both light green.

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