Lecture 3 Chapter 2. Studying Normal Populations

 It is often the case that collected data have a distribution with the characteristic shape of the Normal distribution.  Let’s have a look at an example…

Example – Female Haematocrit  Haematocrit measures the percentage of blood volume occupied by packed red blood cells.  Measurements taken from 126 female medical students are as follows…

Female haematocrit measurements  Let’s look at the shape of the distribution of this data using a histogram…

Example – Female Haematocrit

 These data show the characteristic shape of the Normal Distribution.  It is characterised by the symmetrical “bell shape”, which corresponds to values near the mean being more common, while values further away “tail off” in terms of their frequencies.  A perfect Normal distribution curve looks like….

 In order to understand what it really means for data to be Normally distributed, we first need to consider the idea of probability…

Probability  Probability is used to measure the likelihood of an event occurring.  Definition Suppose we were to repeat a particular experiment over and over again. Then the probability of a particular outcome A is defined as the proportion of the total number of repeats in which A would actually occur, if we were to keep on repeating the experiment. We denote this probability by Pr(A).

Probability Examples 1.Rolling a fair die We roll a standard six-sided die. Let event A be that the die lands with three spots face up. Then the probability of the event A is: Pr(A) = 1/6 ≈ because in the long run, the proportion of times that A happens will be 1/6. Note that in this experiment there are six equally likely outcomes, all with probability 1/6.

Probability Examples 2.Tossing a fair coin You toss a fair coin once. Let event A be that the coin lands heads up. Then the probability of the event A is: Pr(A) = ½ = 0.5 because in the long run, the proportion of times that A happens will be 1/2. This time there are two possible outcomes with equal probability. Note that the Probability scale runs between 0 and 1 inclusive. The higher the number, the more likely the event.

Probability Examples 3.Buying a ticket for the UK National Lotto You buy a single ticket for one draw of the UK National Lotto. The event A is that your six numbers exactly match the six main numbers drawn from 1, …, 49, so that you win a share of the jackpot. Then the probability of the event A is: Pr(A) = 1 / 13,983,816 ≈ Pr(A) = 1 / 13,983,816 ≈ because there are 13,983,816 equally likely outcomes for the six main numbers.

 Probability measurements only really make sense for discrete outcomes, i.e. when we can make a list of all the possible outcomes.  When the measurements are on a continuous scale, such as the haematocrit measures, then there are infinitely many possible outcomes, and it is not possible to list them.  The distribution of haematocrit outcomes has roughly the Normal distribution shape: