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STAT 1201
Aug 28, 2023
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Statistics (STAT1201) WEEK 1 Error variability Penguins come in different shapes and sizes and so we expect to get different body mass measurements. This is just the natural variability of the quantity we are measuring. The measurement process may be prone to error, giving measurement variability . Wrangling a penguin to measure its body mass is not trivial - here the researchers used spring scales and a weigh bag. Even if they weigh a penguin twice they may well come up with different answers. If different people make the measurements then this problem is compounded. However, measurements can be made more accurate in a systematic way, such as by having a clear protocol for how mass is to be measured, or improving equipment. There is usually no way to distinguish between measurement variability and natural variability in the data. We cannot tell whether the first two Adélie penguins, with body masses of 4675 g and 4050 g, are different because their body masses are really different or were in fact of the same but there were measurement errors. We collectively call this variability the error variability since it gets in the way of making inferences from our data. The presence of error variability makes it necessary to replicate our experiments. Taking a single penguin and measuring their body mass tells us very little about body masses in general. Having 10 observations not only gives us information about the typical body masses, it gives us information about the nature and magnitude of the variability present in them. Group variability In this example we also find differences in our observations because the two groups , Adélie and Gentoo, do tend to have different body masses. This is the variability we would like to understand and make inferences about. Note that we cannot make any conclusion like "Gentoo penguins have greater body masses than Adélie penguins" because this statement is not universally true. There is an Adélie penguin who has a body mass that is greater than that of two of the Gentoo penguins. Instead we will talk about means, so we might claim more correctly that "the mean body mass of Gentoo penguins is greater than the mean body mass of Adélie penguins". For this data, the mean for Gentoo is 4935 g while for Adélie it is 3895 g, a 1040 g difference.
Sampling variability This difference of 1040 g is a straightforward calculation and we wouldn't expect there to be any variability in the result. But there is! This is because if we took another 5 Adélie and 5 Gentoo and carried out the measurements again then we would most likely get different mean body masses. The difference would most likely not be 1040 g again. The mean we calculate depends on the sample . This is a very important point. What we would like to do is to use the means we calculate in our study to say something about penguins in general, such as " Gentoo penguins tend to have greater body masses than Adélie penguins". But if we did the study again then we might get different data which say something else! How can we ever make conclusions? Fortunately, we are able to quantify this sampling variability , particularly if the study has been properly designed. A focus of this course will be on understanding and characterising sampling variability in a range of contexts. As a result of this, statistics can be viewed as a communication skill . If a researcher wants to communicate her findings to someone then she has to use the language of statistics in order to incorporate sampling variability. Most research articles in the biological sciences, particularly in medical and other human-related settings, are full of statistical statements and conclusions. A variable is a characteristic that we can record about the subjects or objects in a study. These can be measurements we make, like a forearm length or blood pressure, or can be attributes, like sex or age. Quantitative variables represent measurements, such as the height of a person or the temperature of an environment. Quantitative variables are quite often continuous , taking any value over some range. Continuous variables capture the idea that measurements can always be made more precisely. Discrete variables have only a small number of possibilities, such as a count of some outcomes or an age measured in whole years. Note that systolic blood pressure is shown in the above image as a whole number, suggesting it might be discrete, but it is really a continuous quantity.
Categorical variables represent groups of objects with a particular characteristic. For example, recording the sex of subjects is essentially the same as making a group of males and a group of females. Variables like sex are called nominal because they are arbitrary categories with no order between them. Ordinal variables are those whose categories do have an order. A common example of this is in recording the age group someone falls into. We can put these groups in order because we can put ages in order. In most of this course we will not make much of the distinction between nominal and ordinal variables. The randomisation test in the video is one example of statistical hypothesis testing . In the caffeine analysis, the first explanation for the observed difference in pulse rates between the groups is referred to as the null hypothesis of the test. The null hypothesis will usually be a statement of "no effect". For example, if we were trying to show that a new drug helped a medical condition then our null hypothesis would be that it had no benefit. Note that this sense of "hypothesis" is quite different to a scientific hypothesis. Here, our null hypothesis is that the mean increase in pulse rate is the same for caffeinated and caffeine-free cola. The null hypothesis is usually denoted when discussing the theory of hypothesis testing but you will rarely find this notation appearing in scientific papers that use hypothesis tests. In fact it is rare for authors to specify the null hypothesis at all, though it is usually easy to infer what it was, based on the statement of results.
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