Centre, spread and chance

The centre and the spread

What was a typical day? Find the three averages. Then look at how spread out the days are.

Which average?

The data has no extreme values: the days run from $27$ to $35$ degrees, with no day far away. So the mean and the median are close, and either is fine. If one day had been $50\ ^\circ\text{C}$, the median with the IQR would be the better pair.

Chance

We cannot count equally likely outcomes for the weather. We use the relative frequency from the month, as in Course 5.1.

It rained on $10$ of the $30$ days, so the relative frequency of a rainy day is $\dfrac{10}{30} \approx 0.33$. In the next $30$ days, we expect about $10$ rainy days. It is an estimate, from one month only.

You should find that a lower line counts more days as hot, so the estimate of the chance depends on how you define "hot". A report must say which definition it used.

Be careful with the chance of two days

If the chance of a dry day is $\dfrac{2}{3}$, are two dry days in a row $\dfrac{2}{3} \times \dfrac{2}{3} = \dfrac{4}{9} \approx 0.44$? That rule is for events that do not affect each other. Weather is not like that: a rainy day is often followed by another rainy day. In this month, $16$ of the $29$ pairs of days in a row were both dry, which is $0.55$, more than $0.44$. The days are linked.

Practise with new numbers

Common mistakes

• Giving one average and no spread. "The mean is 30.7 °C" hides that days ran from 27 °C to 35 °C.
• Treating one month's share as exact. It is an estimate from 30 days.
• Multiplying the chances of linked days. Weather comes in spells.
• Forgetting the definition. Say what counts as a hot day.

Remember these

• The centre: mean, median and mode. A set of numbers can have two modes.
• The spread: range, IQR and standard deviation.
• The relative frequency estimates a chance, and it also predicts how many to expect.
• Days of weather are linked, so be careful when you multiply their chances.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. It rained on $6$ of $20$ days. What is the relative frequency of rain? How many rainy days do you expect in $50$ days?

2. Why can the median be a better centre than the mean for data with one extreme value?

3. A list of temperatures has the modes 30 °C and 31 °C. What does that mean?

4. Why is the chance of two dry days in a row in real weather often more than $\left(\dfrac{2}{3}\right)^2$?