Statistics

40

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

The whyMean = 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

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

The whyThe 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

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

The whyFor 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

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

The whyThe 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

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

The whyCutting 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

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

The whyThe 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

HintA steeper climb means the same rise happens in less time.

The whyGradient 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

HintShare a full turn out between everyone in the survey.

The whyA 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

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

The whyIt 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

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

The whyBecause 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

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

The whyA 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

HintAsk what else changes outdoors between June and December.

The whyThis 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.