1 Introduction

Definition 1.1. Statistics is the science of collecting, organising, analysing and interpreting data in order to draw conclusions and make decisions in the face of uncertainty.

It is convenient to think of the subject in two halves. Descriptive statistics is concerned with summarising and presenting data that has been collected – tables, charts, averages, measures of spread. Inferential statistics goes further, using a sample to draw conclusions about the larger population it came from, and stating how much confidence those conclusions deserve.

The whole of this course follows that division: the first chapter is descriptive, and everything after it is inference.

A statistical investigation ordinarily proceeds through five stages:

1.: Collection – obtaining the data, by survey, experiment or observation.
2.: Presentation – organising it into tables and charts.
3.: Analysis – computing summaries and fitting models.
4.: Interpretation – deciding what the results actually mean.
5.: Decision – acting on them, which is the point of the exercise.

Definition 1.2. Data are the recorded values of some characteristic observed on a set of individuals or items. A single such characteristic is a variable, and each recorded value is an observation.

Data are classified by what can meaningfully be done with the values – whether they can be ordered, and whether the gaps between them mean anything. There are four types:

1.

Nominal Data:
Used for labelling variables without any quantitative value.
Notice that all these labels are mutually exclusive and none of them have numerical significance.

\(\longrightarrow \): What is your nationality?
\(\longrightarrow \): Where do you live?

2.

Ordinal Data
With ordinal data it is the order of the values that matters, but the difference between each one is not really known.

\(\longrightarrow \): How do you feel today?

3.

Interval Data
Numerical data in which we know not only the order but also the exact difference between the values.

4.

Ratio Data
These tell us the order and the exact difference between values, and they have a true zero. There are two types of quantitative data that we are going to study:

(a): Discrete data
(b): Continuous data

In order to develop data analysis skills, it is important to keep the following three points in mind:

1.: For most data sets, there is more than one appropriate and informative way to plot the data.
2.: It is valuable to see the same data plotted in several ways, both for the purpose of learning more about data, and because this will help to gain experience in choosing appropriate method of data display and analysis. Professionals routinely do this in order to explore patterns in their data.
3.: Creative insights are an asset to good graphing and data analysis. If you invent a different kind of graph that successfully makes a point about the data that can be interpreted by someone else, then the graph is useful.

1.1 Discrete Data
1.2 Continuous Data
1.2.1 Choice Of Class Interval
1.2.2 How many classes? The \(2^k\) rule and Sturges’ formula
1.3 Histogram With Unequal Class Intervals
1.3.1 Stem-and-Leaf Plots
1.4 Data Collection
1.5 Measures Of Central Tendencies
1.5.1 Mean
1.5.2 Median and Interquartile Range
1.5.3 Median from a frequency table for discrete data
1.5.4 Median of a Continuous Data
1.5.5 Median for grouped data
1.5.6 Quartiles for grouped data
1.5.7 Mode
1.5.8 Mode for grouped data
1.6 Measure Of Dispersion
1.7 Range
1.8 Standard Deviation
1.8.1 Population and sample: why the divisor changes
1.9 Mean Deviation
1.10 Quartiles
1.10.1 Interquartile Range
1.10.2 Interquartile Range for grouped data
1.11 Percentiles
1.11.1 The five working forms of the variance
1.12 Properties of \(E(X)\)
1.13 Properties of Variance
1.14 Properties of Covariance (Cov\((X,Y)\))
1.15 Moments
1.15.1 Raw moments (about the origin)
1.15.2 Central moments (about the mean)
1.15.3 Converting between them
1.15.4 Making them comparable
1.15.5 Moment Ratios
1.15.6 Skewness
1.16 Practice problems

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