Smarties

Tasks
How many smarties do you think are on this poster? How many blue smarties are on this poster?
Can you use the frames provided to estimate the total number of smarties? Which frame do you think is the best choice?
Survey as many people as possible on what they think. What are the range of answers? Using your data, what is your best estimate?
Maths

If we count the number of smarties inside a frame, and then work out how many frames cover the poster, we can estimate the total number of smarties.
Or we can count how many smarties are inside the frame, and how many of them are blue smarties, and estimate the proportion of blue smarties.
It is easier to work out how many frames cover the picture when the frame is square or a triangle.

It is often possible to make a reasonable estimate of a quantity even if your information is incomplete.
The method has been formalised:
1. What will the answer look like, or what will be the units of the answer?
2. What sort of quantities do we need to know about?
3. How do these quantities relate to each other?
4. What figures do we know or can be guessed?
5. What reasonable assumptions can we make?

Using this method, could you estimate how many tennis balls fill a suitcase? Or how about how many packets of popcorn can fill a room?

Another way to estimate the total number is to take a survey of people’s best guesses, and using the average as your estimate. This works because some of these guesses will be too high, and some will be too low, but if we’re lucky the errors will cancel out. This is called Wisdom of the Crowds.
History

This kind of estimation is often called Fermi estimation after the Italian physicist Enrico Fermi. One of his examples was to calculate how many piano tuners are there in Chicago. By using the known population of Chicago, and estimates for the number of households with pianos, Fermi came up with a number that was quite close to the actual number.
A much more serious estimate was to estimate the power of the atom bomb in its first test. He did this by dropping pieces of paper and seeing where they fell as the blast reached his observation point. By combining this with other reasonable assumptions he was able to get an estimate that was within the correct order of magnitude.
In 1906, British statistician Francis Galton attended a county fair. One stall had a contest to guess the weight of an ox and win a prize. 800 people entered the competition, secretly writing their best guess on a piece of paper and submitting their answer. Galton noticed that the average of these guesses was almost exactly correct.
People
Francis Galton 1822-1911
Darwin’s theory of evolution sparked Galton’s interest in measuring variations in human populations, such as height and fingerprints. To do this, Galton invented new methods of collecting and analysing data. He also wrote about the best way to cut a round cake
Enrico Fermi. 1901 – 1954
Fermi was an Italian physicist best known for his work on the first nuclear reactor, in Chicago. He won the Nobel Prize for Physics in 1938. He had emigrated to America to escape the Fascist dictatorship of Mussolini . He made major contributions to the development of quantum theory, nuclear and particle physics and statistical mechanics. He was a member of the Manhattan project that developed the atom bomb, but his work on the nuclear reactor led to the peaceful uses of atomic energy.
Applications

Sometimes we want to count large numbers. Maybe you are a factory making thousands of sweets a day, or want to know how many people in the country wear glasses. Counting large numbers can be difficult, or expensive. So it’s often easier, and cheaper, to take a small sample and then estimate how many things there are in total.
Maths at Home

Try the wisdom of the crowds experiment at home. For example, ask people to guess how many jelly beans are in a jar. The more people you ask the better. What was the largest guess? What was the smallest guess? Was the average close to the correct answer?