Topic 1: Introduction to Statistics 1.1 Introduction, examples and definitions of terms 1.1.1 Introduction We begin the module with some basic data analysis. Since Statistics involves the collection, organization, presentation, analysis and interpretation of numerical data to make meaningful conclusions, we must first know how to understand, display and summarize large amounts of quantitative information, before undertaking a more sophisticated analysis. Statistical analysis of quantitative data is important throughout the pure and social sciences. For example, during this module we will consider examples from fields like: Biology, Medicine, Agriculture, Economics, Business, Engineering and Meteorology amongst others. Statistics can be grouped into two main categories: Descriptive and Inferential statistics. Descriptive statistics involves the study of the population in totality in order to reveal the main characteristics. Whereas inferential involve drawing inferences about the population based on the findings of a study of a section of the population. 1.1.2 Examples Survival of cancer patients: A cancer patient wants to know the probability that he will survive for at least 5 years. By collecting data on survival rates of people in a similar situation, it is possible to obtain an empirical estimate of survival rates. We cannot know whether or not the patient will survive, or even know exactly what the probability of survival is. However, we can estimate the proportion of patients who survive from data. Car maintenance: When buying a certain type of new car, it would be useful to know how much it is going to cost to run over the first three years from new. Of course, we cannot predict exactly what this will be — it will vary from car to car. However, collecting data from people who bought similar cars will give some idea of the distribution of costs across the population of car buyers, which in turn will provide information about the likely cost of running the car. 1.1.3 Definitions of terms used in statistics The quantities measured in a study are called random variables, and a particular outcome is called an observation. Several observations are collectively known as data. The collection of all possible outcomes is called the population. In practice, we cannot usually observe the whole population. Instead we observe a sub-set of the population, known as a sample. In order to ensure that the sample we take is representative of the whole population, we usually take a random sample in which all members of the population are equally likely to be selected for inclusion in the sample. For example, if we are interested in conducting a survey of the amount of physical exercise undertaken by the general public, surveyin people entering and leaving a gymnasium would provide a biased sample of the population, and the results obtained would not generalize to the population at large. Variables are either qualitative or quantitative. Qualitative variables have non-numeric out-comes, with no natural ordering. For example, gender, disease status, and type of car are all qualitative variables. Quantitative variables have numeric outcomes. For example, survival time, height, age, number of children, and number of faults are all quantitative variables. Quantitative variables can be discrete or continuous. Discrete random variables have outcomes which can take only a countable number of possible values. These possible values are usually taken to be integers, but don’t have to be. For example, number of children and number of faults are discrete random variables which take only integer values, but your score in a quiz where “half” marks are awarded is a discrete quantitative random variable which can take on non-integer values. Continuous random variables can take any value over some continuous scale. For example, survival time and height are continuous random variables. Often, continuous random variables are rounded to the nearest integer, but they are still considered to be continuous variables if there is an underlying continuous scale. Age is a good example of this. Measurement refers to assigning numerical values to observations in a systematic manner. Measurement can be done in four levels/ scales namely; Ordinal. Nominal, Interval and Ratio scale. 1.1.4 Need for statistics The demanding need of statistics use in the current world is surpassing. Statistics is applied or used in all walks of life. The following is a summary of the need for statistics. i). Planning purposes ii). Evaluation purposes iii). Prediction or forecasting purposes iv). Justification or assertion of a hypothesis 1.1.5 Work to do {Assignment 1} 1. Find out what each level/ scale of measurement entails? 8mks 2. Giving at least three examples in each case outline the differences between a). Discrete and Continuous variables 6mks b).Ordinal and Nominal variables 6mks 2.0 Data presentation 2.1 Introduction A set of data on its own is very hard to interpret. There is lots of information contained in the data, but it is hard to see. We need ways of understanding important features of the data, and to summarize it in meaningful ways. The use of tables, graphs and summary statistics for understanding data is an important first step in the undertaking of any statistical analysis. For example, it is useful for understanding the main features of the data, for detecting outliers, and data which has been recorded incorrectly. Outliers are extreme observations which do not appear to be consistent with the rest of the data. The presence of outliers can seriously distort some of the more formal statistical techniques to be examined in the second semester, and so preliminary detection and correction or accommodation of such observations is crucial, before further analysis takes place. 2.2 Frequency tables It is important to investigate the shape of the distribution of a random variable. This is most easily examined using frequency tables and diagrams. A frequency table shows a tally of the number of data observations in different categories. For qualitative and discrete quantitative data, we often use all of the observed values as our categories. However, if there are a large number of different observations, consecutive observations may be grouped together to form combined categories. Example For Example, 1 (germinating seeds), we can construct the following frequency table. No.germinating 85 3 Frequency 86 1 87 5 88 2 89 3 1 90 6 91 11 92 4 93 4 94 1 Since we only have 10 categories, there is no need to amalgamate them. For continuous data, the choice of categories is more arbitrary. We usually use 8 to 12 nonoverlapping consecutive intervals of equal width. Fewer than this may be better for small sample sizes, and more for very large samples. The intervals must cover the entire observed range of values. Example For Example, 2 (survival times), we have the following table. Range 0 — 39 40 — 79 80 — 119 120 — 159 160 — 199 200 — 240 Frequency 11 4 1 1 2 1 n = 20 N.B. You should define the intervals to the same accuracy of the data. Thus, if the data is defined to the nearest integer, the intervals should be (as above). Alternatively, if the data is defined to one decimal place, so should the intervals. Also note that here the underlying data is continuous. Consequently, if the data has been rounded to the nearest integer, then the intervals are actually 0 39:5, 39:5 79:5, etc. It is important to include the sample size with the table. 2.3 Histograms Once the frequency table has been constructed, pictorial representation can be considered. For most continuous data sets, the best diagram to use is a histogram. In this the classification intervals are represented to scale on the abscissa (x-axis) of a graph and rectangles are drawn on this base with their areas proportional to the frequencies. Hence the ordinate (y-axis) is frequency per unit class interval (or more commonly, relative frequency — see below). Note that the heights of the rectangles will be proportional to the frequencies if and only if class intervals of equal width are used. 2 Example The histogram for Example 2 is as follows. 6 4 0 2 Frequency 8 10 Raw frequency histogram (n=20) 0 40 80 120 160 200 240 Times Note that here we have labelled the y-axis with the raw frequencies. This only makes sense when all of the intervals are the same width. Otherwise, we should label using relative frequencies, as follows. 3 008 Frequency Relative 0.012 Relative frequency histogram (n=20) 0 40 80 120 160 200 240 Times The y-axis values are chosen so that the area of each rectangle is the proportion of observations falling in that bin. Consider the first bin (0–39). The proportion of observations falling into this bin is 11/20 (from the frequency table). The area of our rectangle should, therefore, be 11/20. Since the rectangle has a base of 40, the height of the rectangle must be 11=(20 40) = 0:014. In general therefore, we calculate the bin height as follows: 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 (𝑓) 𝑏𝑖𝑛𝑤𝑖𝑑𝑡ℎ Height = This method can be used when the interval widths are not the same, as shown below. 4 0.005015 0 . 0 0. Relative Frequency Relative frequency histogram (n=20) 0 20 50 80 160 240 Times Note that when the y-axis is labelled with relative frequencies, the area under the histogram is always one. Bin widths should be chosen so that you get a good idea of the distribution of the data, without being swamped by random variation. Example Consider the leaf area data from Example 3. There follows some histograms of the data based on different bin widths. Which provides the best overview of the distribution of the data? 5 0 20 40 Frequency 60 80 100 120 140 Histogram of Leaves 0 20 40 60 80 100 120 140 Leaves 0 20 Frequency 40 60 80 Histogram of Leaves 0 20 40 60 80 Leaves 6 100 120 140 0 5 Frequency 10 15 20 25 Histogram of Leaves 20 40 60 80 100 120 100 120 Leaves 0 2 Frequency 4 6 Histogram of Leaves 20 40 60 80 Leaves 7