Statistics and probability, Year 10: quartiles, correlation and independent events

The interquartile range

HintIts name says it measures the stretch between two markers that cut the data into quarters.

The whyIt is the spread of the middle half of the data, so it is not affected by a few extreme values.

7

HintFind the middle value, then the middle of each half on either side of it.

The whyThe median is the 4th value, 8. The lower quartile is the middle of 3, 5, 7, which is 5, and the upper quartile is the middle of 10, 12, 15, which is 12. Interquartile range = 12 − 5 = 7.

20 and 60

HintThe quartiles sit one quarter and three quarters of the way through the data.

The whyLower quartile: 1/4 of 80 = 20. Upper quartile: 3/4 of 80 = 60. Read across from each to the curve and down to the data axis; subtracting the two readings gives the interquartile range.

It ignores the extreme mark

HintThink about which part of the ordered list each measure looks at.

The whyThe range uses only the highest and lowest values, so a single outlier changes it greatly. The interquartile range covers the middle 50% of the data, between the quartiles, and is not affected by one unusual value.

Class A's marks are more consistent

HintA small spread means the middle half of the results sit close together.

The whyThe interquartile range measures spread, not how high the marks are. A smaller value means the middle half of the marks are closer together. To compare how well the classes did, you would compare their medians.

About as many above it as below it

HintThe line should run through the middle of the cloud of points, following its direction.

The whyA line of best fit is a single straight line that follows the trend, passing as close as possible to the points with a roughly even balance on each side. It is only drawn when the points show a correlation.

Interpolation

HintThe prefix is the same as in "interior" and "internal".

The whyInterpolated estimates are usually reliable, because the line there is supported by real data on both sides.

extrapolation

HintThe prefix is the same as in "exterior" and "external".

The whyExtrapolated estimates are risky: there is no data out there to show that the trend continues in the same way.

The trend may not continue beyond the data

HintAsk what really happens to a person's height once they are grown up.

The whyThe line describes only ages 5 to 15. Extending it assumes people keep growing at the same rate for ever, which gives an impossible height; in fact growth stops in the late teens.

50

HintSix hours is exactly halfway between the two revision times you know.

The whyBetween the two points the score rises by 40 over 8 hours, which is 5 marks per hour. From 2 hours to 6 hours is 4 more hours: 30 + 4 × 5 = 50. Six hours is inside the range of the data, so this is interpolation.

causation

HintThe missing word names one thing bringing about another.

The whyA scatter graph can show that two quantities are linked, but not why. A third factor, the reverse direction or plain coincidence can all produce a correlation.

Age

HintThink about what separates the youngest classes from the oldest ones.

The whyOlder children have bigger feet and have also had more years of practice at reading. Shoe size and reading score rise together because both depend on age, not because feet help reading.

Coincidence (chance)

HintAsk whether there is any sensible way the two could be connected at all.

The whyWith enough pairs of quantities, some will show a pattern purely by accident, especially in a small group. When no reasonable link or shared cause can be found, that is the most likely explanation.

Happier people may choose to exercise more

HintSwap which of the two things comes first.

The whyThe data cannot show which comes first. It may be that feeling well makes people more active, or that both directions play a part.

It treats a correlation as proof of a cause

HintThe data show two things going together; the headline claims rather more.

The whyThe figures show only that breakfast and good grades tend to go together. Something else, such as a settled home routine, could lie behind both, so the data alone cannot show that breakfast produces the grades.

Independent events

HintThe everyday meaning of the word is "not relying on anyone else".

The whyTwo tosses of a coin are an example: the first result has no effect on the second. If one event does change the chance of the other, the events are dependent.

Still 1/6 — each roll is independent of the ones before

HintAsk what the dice "knows" about its earlier throws.

The whyThe dice has no memory: nothing about it changes between rolls, so the chance of each face stays at 1/6. Long runs without a six are unusual, but they do not make the next roll any different.

independent

HintAsk whether the first pick changes what is in the bag for the second.

The whyPutting the counter back restores the bag, so the probability of red is 3/5 on both picks whatever happened first. Without replacement the bag changes and the picks become dependent.

1/2

HintRecount the bag, both the reds and the total, after the first counter has gone.

The whyAfter one red is removed the bag holds 2 red and 2 blue, 4 counters in all. P(red) = 2/4 = 1/2. The first pick changed the chance for the second, so the two events are dependent.

That they are independent

HintThink about whether one of them being late could make it more likely for the other.

The whyMultiplying two probabilities gives the chance of both only when neither event affects the other. If Asha and Ben catch the same bus, a late bus makes both late together and the multiplication no longer holds.