Statistics, Year 9

A scatter graph

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

为什么Each 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)

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

为什么One 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

提示Say what happens to the price as the years go by.

为什么Moving 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

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

为什么When 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

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

为什么The 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

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

为什么A 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

提示Choose the value that sits fairest between the two ends of the class.

为什么The 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

提示It is how many pieces of data there are altogether.

为什么A 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

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

为什么Grouping 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

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

为什么An 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

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

为什么Bus 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

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

为什么A 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

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

为什么Bags 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

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

为什么A 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

提示Count how much time passes between each neighbouring pair of labels.

为什么The 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

提示Work out how much paper each drawing covers.

为什么The 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

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

为什么Every 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

提示Measure each bar from where the axis begins, not from zero.

为什么The 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

提示Ask how much one square of height is worth.

为什么Without 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.