Statistics, Year 9

A scatter graph

HintEach person or item becomes one dot, and you look at the cloud of dots.

The whyEach point carries two measurements of the same person or thing, such as height and mass. The points are not joined up: it is the overall drift of the cloud that tells the story.

(150, 45)

HintThe horizontal axis is read first, exactly as with any coordinates.

The whyOne pupil gives one point. Go across to 150 on the height axis, then up to 45 on the mass axis. Plotting each pair this way turns a table of paired data into a picture.

Older cars tend to be worth less

HintSay what happens to the price as the years go by.

The whyMoving right means more years; the points getting lower means less money. "Tend to" matters, because the points make a loose band and not a perfect line: a few cars will not fit the pattern.

There is no relationship between them

HintDescribe what the cloud of points does, and do not invent a pattern it does not show.

The whyWhen the points show no drift either way, knowing one measurement tells you nothing about the other. Saying so plainly is the correct conclusion, not a failed one.

320 cm; 16 cm

HintLet every seedling stand at the middle of its class, add up all twenty heights, then share the total out.

The whyThe midpoints are 5, 15 and 25. Each class contributes midpoint × frequency: 5 × 4, 15 × 10 and 25 × 6, which are 20, 150 and 150. Their sum is the estimated total, and dividing by the 20 seedlings gives the estimated mean.

The exact values inside each class are unknown

HintThink about what was thrown away when the raw numbers were sorted into groups.

The whyA table that says "10 seedlings between 10 cm and 20 cm" has lost the ten actual heights. The method pretends they all sit at the midpoint, which is a fair guess but not the truth.

45

HintChoose the value that sits fairest between the two ends of the class.

The whyThe midpoint is halfway between the class boundaries: (40 + 50) ÷ 2. Some real values will be above it and some below, so across the whole class the errors tend to balance out.

frequency

HintIt is how many pieces of data there are altogether.

The whyA mean shares a total equally among every piece of data. A table with 4 classes and 50 people describes 50 values, so the total is divided by 50.

The modal class

HintWith grouped numbers you cannot name the single most common value, only the busiest interval.

The whyGrouping hides the individual values, so the mode itself cannot be found. The modal class is the best that can be said: the interval that holds more of the data than any other.

A measure of spread, such as the range

HintOne number says what is typical; you also need one that says how much the values vary.

The whyAn average tells you which set is higher overall; the spread tells you which is more varied. A full comparison makes one statement about each, written in the context of the question.

Bus A — its range is much smaller

HintBeing quick on a typical day is not the same as being dependable.

The whyBus B is usually a little quicker, but its times vary by a quarter of an hour, so a bad day could be very late. Bus A barely varies. The spread, not the average, answers a question about reliability.

On average, sprinter A is faster

HintDecide whether a bigger number is good or bad in this sport.

The whyA comparison should say what the numbers mean, not just which is bigger. For race times a lower figure means a quicker run, so the lower median belongs to the faster sprinter, by half a second.

No — the gap is tiny beside the spread

HintSet the 0.1 g between the two averages against how much individual bags vary.

The whyBags from either machine differ from one another by up to 6 g, sixty times the gap between the means. A difference that small could vanish or reverse with the next batch, so no real difference has been shown.

consistent

HintBunched closely together, so one result is much like the next.

The whyA small range means the values are close to one another. In context that reads as reliable, steady or predictable, depending on what was measured.

Equal gaps stand for unequal numbers of years

HintCount how much time passes between each neighbouring pair of labels.

The whyThe last step covers ten years but is drawn the same width as the five-year steps, so any change in that decade looks twice as steep as it really was. On an honest scale, equal distances mean equal amounts.

It has four times the area, not twice

HintWork out how much paper each drawing covers.

The whyThe eye judges a picture by its area. Doubling both the width and the height multiplies the area by 2 × 2, so a rise of two times is made to look like a rise of four times.

It shows only a chosen part of the data

HintAsk what the chart would look like if it started five years earlier.

The whyEvery point plotted may be accurate and the picture can still deceive, because the start date was picked to hide the fall. The company benefits: a reader sees growth where the fuller record shows decline.

3

HintMeasure each bar from where the axis begins, not from zero.

The whyThe bars are drawn 96 − 95 = 1 unit and 98 − 95 = 3 units tall. So a value only about 2% larger is drawn as a bar three times the height. Redrawn from zero, the two bars would look almost level.

scale

HintAsk how much one square of height is worth.

The whyWithout numbers a gap between two bars could stand for 1 or for 1,000. Adverts sometimes leave them off so that the shape makes the impression and nobody can test it.