Statistics

40

提示Undo the sharing: multiply the average by how many values there are.

为什么Mean = total ÷ count, so total = mean × count = 8 × 5 = 40. Working backwards from a mean is a favourite KS3 question, and it is the key to problems such as a sixth number being added and the mean rising to 9.

The range

提示Only the two end values matter to it — everything in between is ignored.

为什么The range describes how far apart the data reaches, so it is a measure of spread rather than an average. One extreme reading can stretch it a long way.

7

提示With an even count the two middle values are averaged, so work backwards from that average.

为什么For six values the median is the mean of the third and fourth, so (x + 11) ÷ 2 = 9, giving x + 11 = 18 and x = 7. That value also keeps the list in order, which is the check worth making at the end.

42

提示The frequency column records how often each outcome came up — combine every entry.

为什么The frequencies add to 8 + 5 + 7 + 4 + 6 + 12 = 42, so the dice was rolled 42 times. Totalling the frequency column tells you the size of the data set, which you need before working out any proportion or estimating a probability from it.

It exaggerates the differences between the bars

提示Only the top slice of each column is drawn, so a small gap fills most of the picture.

为什么Cutting the axis means the heights are no longer proportional to the frequencies, so a rise from 92 to 96 can look like a doubling. The honest fix is to start the axis at zero, or to mark the break in it clearly.

2

提示Cut the value of one whole symbol in the same fraction as the picture is cut.

为什么The key gives the value of a whole symbol, so a part-symbol is worth the matching fraction of it: half of 8 is 4 and a quarter is 2. Read the key before counting anything, or every figure will come out wrong.

How quickly the temperature is changing

提示A steeper climb means the same rise happens in less time.

为什么Gradient on a time graph is a rate: a steep rise means a fast increase, a gentle slope a slow one, and a flat section no change at all. Comparing steepness lets you say when the temperature was changing fastest without doing any calculation.

9

提示Share a full turn out between everyone in the survey.

为什么A full circle is 360°, and dividing that between 40 people gives each one a slice of 9°. Multiplying by a category frequency then gives that slice its angle.

The mode

提示A shoe shop uses it: the size that turns up most often, not a calculated middle.

为什么It is the only average that works for categories such as favourite colour, and it is what a shop needs when deciding which sizes to restock.

It pulls the mean upwards

提示Every reading goes into the total, so an extreme one drags the average with it.

为什么Because all the data is added in, a single extreme reading shifts the answer. The median barely moves, which is why it is preferred for figures such as house prices.

positive correlation

提示As one measurement goes up, the other tends to go up as well.

为什么A line of best fit summarises that trend and can be used to predict, though only within the range of the data collected. Points falling from left to right show the opposite relationship.

The hot weather

提示Ask what else changes outdoors between June and December.

为什么This is the classic example of correlation without causation: temperature drives both figures, and neither one causes the other. Always ask what else could explain a pattern before claiming a cause.