I have six books on statistics in my personal library. All of them are bigger than Bulmer's book, but none have been read as many times, and none are as tattered, marked up, and cross-referenced. Simply put, Bulmer's book is the most useful and complete book on basic statistics that I have. It's a nice package in a reasonably sized book with all the most important stuff for dealing with basic statistical problems that many engineers are likely to encounter in a day's work.
Chapter 1 is a short blurb on the concept of probability. This is very useful because it places the rest of the text on a very specific and concise footing. Essentially there are two concepts of probability. One is the relative frequency with which an event occurs in the long run. An example of this is the tossing of a coin many times and counting the number of times it comes up heads. The author describes this as statistical probability.
The second concept of probability is what the author calls inductive probability. Inductive probability is "the degree of belief which it is reasonable to place on a proposition on given evidence." The essential difference between the two concepts of probability is that statistical probability is an empirical concept, while "inductive probability is a logical concept." Bulmer closes chapter 1 by saying, "It has been reluctantly concluded by most statisticians that inductive probability cannot in general be measured and, therefore, cannot be ............" Read chapter 1 to find some interesting arguments in support of this proposition - a proposition that may be surprising to some people. As a result (and as the book's title suggests) Bulmer keeps his book almost exclusively in the domain of statistical probability.
Chapter 2 introduces two simple law of probability. The first relates to the addition of probabilities of mutually exclusive events. The second relates to the multiplication of probabilities. Simple in concept, Bulmer illustrates these two laws by several examples including tables of measurements made on real experiments, and some from Mendel's laws of heredity.
Chapter 3 is pivotal. It develops the mathematical expressions for random variables and probability distributions. Chapter 3 is relatively short, but lays the groundwork for chapter 4, which describes the properties of distributions. Chapter 4 has many useful equations, including those for the mean, variance, measures of dispersion, moments, etc.
Chapter 5 introduces the notion of expected values for both discrete and continuous variables. These are determined not only for single distributions, but also for distributions that are combined in algebraic ways through multiplication, addition, division, etc., which also leads (naturally enough) to the moment-generating function.
Chapter 6 highlights some important distributions (the Binomial, Poisson, and Exponential) and discusses their statistical properties (mean, variance, skewness, and kurtosis). Bulmer adds additional insight into these distributions by describing how they arise in real-world situations. [As a note here, this chapter is useful and interesting, but it could easily be many pages longer. For example, when I was investigating polarization mode dispersion in optical fibers I wanted to know the statistical properties of the Maxwellian distribution. Bulmer did not have it - and I eventually found what I was looking for in "The Handbook of Mathematical Functions," by Milton Abramowitz and Irene A. Stegun.]
The normal distribution is not covered in chapter 6. Instead, as the granddaddy of all statistical distributions, it gets its own chapter - chapter 7. Here Bulmer derives the moments, variance, and a couple of proofs relating to the Gaussian or normal distribution. He also has a nice discussion on the central-limit theorem - which explains why the normal distribution is found in so many places. Chapter 8 continues the theme of distribution functions by considering the Chi-squared, t, and F functions.
Chapter 9 leaves the subject of distributions and describes tests of significance. This is an extremely important chapter for anyone involved in experimental science where the uncertainty of experimental results must be understood and reported. Chapter 10 deals with a related subject - namely statistical inference. In these two chapters Bulmer develops the tools and techniques needed to properly interpret, understand, and report on statistical data - including non-statistical data with statistical noise.
The book ends with a discussion on regression and correlation. Again, this is a very useful chapter with equations for the slope and intercept for linear regression, as well as variance for the slope and intercept. Bulmer includes the derivations for these equations, making it easy and straightforward to extend the analysis to provide regression and correlation for any polynomial fit. This chapter - along with those on tests of significance and statistical inference - will probably be the most useful to students in the sciences.
The book ends with several tables. Many of these tables were generated before the age of calculators, so they may be less needed today than in days gone by. Still, you don't always have a calculator handy. The tables include the probability and density functions for the standard normal distribution, the cumulative probability function for the normal distribution, percentage points of the t distribution, Chi-squared distribution, and the five-percent and one-percent points of the F distribution.
The book has an adequate index (though I'd like it to be longer) and each chapter has problems - with answers in the back. This makes the book ideal for individual study, and the problems often provide greater insight by helping the student extend ideas found in the book.
Overall, this is one of the most used books in my library. And for the price, it's an absolute steal. If you've been wanting a short, concise, yet relatively complete book on statistics - and one that is well-written and easy to follow, yet mathematically involved - but still practical, I highly suggest Bulmer's book.