School Level Links > Sample vs Population Statistics - An Exercise > Sample Vs Population Statistics - An Exercise

Sample vs Population Statistics - An Exercise

This exercise developped by Deryn Griffiths. Comments and enquiries may be directed to her at D.Grifftihs@bom.gov.au.

This exercise is provided with the expectation that teachers would modify it to suit their own needs. Choose a day 1-2 months away that you would like to forecast the weather for. Determine what observations from near your school can be used. In this exercise, it is assumed that Dunedoo Central School has a sports day in September (and that it is now sometime in the first half of the year).

Make a rainfall prediction for your school sports day (or similar). If the sports day is in the next week or so, weather forecasts may be able to help. Beyond that, the best information will be the climate, that is looking at what usually happens for your area at that time of year.

Daily rainfall from the last year is easily available from the Bureau of Meteorology website and the exercise is based on these. If you have longer rainfall records (your own, or obtained from the Bureau) you could use these as part of an extension exercise (see below).

The Probability of getting 1mm of rain on a day in September is related to the Population Statistic which refers to a figure based on the entire (past and future) record (whether recorded or not). A Sample Statistic works out the chance last September (or over several Septembers) and uses that to estimate the Probability or Population Statistic, the chance of getting 1mm of rain on a day in next September.

To get data from last September, go to the Bureau of Meteorology, the Climate Services section, then the link to Recent months' observations (past 13 months). Within that section look for Dunedoo. If your town is not there, look for somewhere nearby. For September 2007, Dunedoo recorded exactly 0mm of rain. The Sample statistic suggests that the chance of getting 1mm of rain on a day in September is 0%. The Probability is certainly not 100%, but it may be greater than 0%. (Archived copy of Dunedoo 2007 record)

Investigations can be done to determine the true chance of getting 1mm of rain on a September day, or at least, find a range within which it is likely to lie.

Investigation 1.

Assume the true chance of getting 1mm of rain on a September day is 100p% (so p is the probability, a number lying between 0 and 1).

In this case, the chance of receiving no rain on all 30 days in September is (1-p)³⁰ (assumptions discussed in Note 1 below). Let us solve (1-p₁)³⁰ = 0.05. We find p₁≈ 0.1. That is, if the chance of getting 1mm of rain on a September day is greater than 10%, a completely dry month would happen less than 5% of the time. The Sample Statistic has given us an estimate of 0% chance, and we think it is likely that the true value is less than 10%. Can we do better than this? Is it true that the chance is less than 10%?

How would I have found values with 95% confidence in another situation? (See Note 2 below)

Investigation 2.

Take a larger sample. This can be done by taking Septembers over more years, looking at more than one observation site or by looking at the other months of the year. Of these, taking more than one observation site is the least satisfactory, as we know that if it was an unusally dry September, it will probably have been dry at nearby places (that is, the extra information is not independent, so is not adding as much extra information as we would like). Taking Septembers over more years would be the best option. However, what we have at hand is the data from August and October last year, so let's start by looking at that.

In August 2007 Dunedoo recorded 4 days with ≥1mm of rain. That is about a 13% chance of getting rain on any one day in August. In October 2007, Dunedoo also recorded recorded 4 days with ≥1mm of rain. Unless we can think of a good reason for the chance of rain to jump around from August to September to October, we can make a new estimate by averaging the chance of rain from those three months. That is, 8 out of 92 days with ≥1mm of rain, or about 9%. Our new estimate of ≥1mm of rain on a particular day in September is 9%, but the true value may be larger or smaller than that.

Investigation 3.

Use more Septembers. The Bureau of Meteorology has done this for us. From the Climate Services page, use the link to Climate Data Online (data, statistics & maps) and look for Dunedoo Postoffice. You can use a table of stations or the map to help you find Dunedoo. Look at the table, and the row labelled "Mean number of days of rain ≥1mm". For Dunedoo you will find "5.3". So, an estimate of 100 x 5.3 / 30, or 18% chance of rain on any particular day in September. Look along the row. How many years were used in putting together this figure? Does the number of years increase your confidence in the figure being correct? It can be very misleading to look at just one year! Use the "Plot" link just beyond the list of years to plot the monthly data. Does the variation from month-to-month give an indication of the confidence of the value of 18%? In this case, the variation within the July to December period may not be meaningful. The small variation in the July to December period indicates that the true chance of at least 1mm on a September day is very likely 17, 18 or 19%. This is based on 95 years of data, and comparing months either side.

Review.
We have found that the chance of rain in September is much greater than that indicated by the 2007 September data. If we accept that the true chance of getting 1mm of rain on a September day is about 18%, then the chance of getting a completely dry month is (1- 0.18 )³⁰ = 0.0026, or only 0.26% chance. That is, the 2007 September was truely an unusual month, the sort that would usually only occur about once out of every 400 months, or once every 33 years (if all months had a similar chance of rain to September).

Note 1:

I have assumed that the chance of rain on various days is independent. This is not true, but is a reasonable approximation for this exercise.

Note 2:

If I had selected October as my month, I would have (in Option 1) found a Sample Statistic of 13%. That is, rain fell on 4 days of 30 so we estimate the likelihood of rain to be 4/31 = 0.13 but we can’t be sure it’s exactly 0.13. We can use the table below to say that we are 95% confident the "real" chance (the Population Statistic) is between 5% and 29%. The mathematics is complicated and the easiest option is to use a look-up table. For example, the modified Jeffreys table in Brown, Cai and DasGupta an extract of which is produced here.

For a sample of 30 days, an observation of an event on x of those days gives a 95% confidence interval for the probability of the event as shown.

x=number of events out of 30 possibilities	0	1	2	3	4	5	6	7	8
lower likely p	0	0	0.01	0.03	0.05	0.07	0.09	0.11	0.14
upper likely p	0.12	0.15	0.20	0.24	0.29	0.33	0.37	0.40	0.44

x=number of events out of 30 possibilities	9	10	11	12	13	14	15
lower likely p	0.16	0.19	0.21	0.24	0.27	0.30	0.33
upper likely p	0.48	0.51	0.55	0.58	0.61	0.64	0.67