The shapes of distributions

The distribution of data says which values are in it and how many times each value appears. In a histogram it shows as a shape: where the bars are tall there are many values, and where they are short, few.

The heights of the 90 pupils of a year group:

• The two sides are almost mirror images of each other round the middle, 170 cm.
• The mean and the median are equal, or almost equal, and the mode is in the middle.

The waiting times of 89 patients at a clinic:

• Most values are small, and a few are very large: the shape stretches to the right in a long tail.
• The large values raise the mean but hardly affect the median: the mean is greater than the median.
• Monthly pay, flat prices and the time spent waiting for a bus are distributed like this too.

The marks of 60 pupils in an easy test:

• Most values are high, and a few are low: the long tail is on the left.
• The small values lower the mean: the mean is less than the median.

• A uniform distribution: every value appears about the same number of times, and the histogram is flat. For example, the results of 600 rolls of a dice.
• A two-peaked distribution: many small values, many large values and few in the middle – two “hills”. Usually these are two different groups measured together, like the marks in a class where half the pupils revised for the test and half did not:

In a two-peaked distribution the mean falls in the middle, where there are few values: it describes neither of the two groups well.

The distribution of the heights, symmetric and bell-shaped, is called a normal distribution. It fits a feature affected by many small causes, each adding a little or taking a little away:

• the height and weight of people of the same age, babies’ birth weights;
• the marks in a large national test;
• the amount a machine fills into a bottle, the life of batteries;
• measuring errors.

Things that are not normally distributed:

• pay, flat prices, waiting times – a right tail;
• the marks in a very easy test – a left tail;
• the results of a dice – a uniform distribution.

The final exam of course A at a university was harder than usual: most marks were low, and few marks were high.
In the final exam of course B the marks were normally distributed.

Which graph fits course A, and which course B? Explain.

In the graph left over – where is the mean compared with the median: greater or less? Explain.

The monthly pay, in shekels, of the seven workers of a small company: 6,000, 6,500, 7,000, 7,000, 7,500, 8,000, 30,000.

Work out the mean pay, to the nearest shekel.

What is the median? Which of the two describes the pay of an ordinary worker in the company better? Explain.

Which of these are probably normally distributed? For the others – what is the shape of the distribution? Explain.

• the weight of apples from the same tree;
• the number of friends teenagers have on a social network;
• the last digit of phone numbers;
• the time it takes to run 2,000 metres, for all the grade 11 pupils in the country.

In a course the marks are normally distributed.
The standard score of the mark 78 is 0.
What is the mean of the marks in the course? And the median? Explain.