Chapter 0: Some basic preliminaries

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1 Chapter 0: Some basic preliminaries 0.1 Introduction Unfortunately, I will tend to use some simple common and interchangeable terms that you ll all have heard of but perhaps you don t really know or cannot recall what they are, so its best we try make the basics clear. 0.2 What is a mathematical function? You all know this A function is a device/tool/method/rule that takes an input and transforms it into exactly one output. A function is usually symbolically written as a single output value f(x) where the input value is (x). When the input is left unspecified it is usually called an input variable (x), the output is often called the output variable. A familiar example of a function is f(x) = x 2, given a numeric input x = 2, f(x) = 2 2 = 4 A less familiar example is a function of two numeric inputs f(x, y) = x + y, given two inputs x = 2 and y = 3, f(x, y) = = 5. More generally a function can have many inputs, say n. Then we can define a general sum function of the n inputs as f(x 1, x 2, x 3, x 4,, x n 1, x n ) = x 1 + x 2 + x 3 + x 4, + + x n 1 + x n. Using the sum notation this can be expressed more concisely as f(x 1, x 2, x 3, x 4,, x n 1, x n ) = n i=1 x i. A function does not have to operate on numbers though mostly it s used that way. In fact it can take any input, say a colour and transform it to say an animal, for example: f(red) = dog, or f(blue) = cat, or f(yellow) = mouse, or f(black) = dog In contrast to the sum function observe here we have not specified how the function works, we ve simply stated the input and the output.

2 A function where we specify how the output is produced from the input is called an Algorithm or Program or simply a rule. Note: If we have f(blue) = mouse, we no longer have a function as the same input is not allowed to give two different outputs. A very important function (for this course and many others) is the linear function or straight line relationship. The output y = f(x) is computed from x by the rule y = f(x) = a x + b Here the two numbers a and b are called the slope and intercept parameters respectively these are given fixed quantities. 6 Example graph of the linear function y = f(x) = 2 x Intercept 0 Slope (Rise/Run) x Since there is just 1 input and 1 output we can reverse the relationship and express x as function of y using simple algebra by the rule x = y b a This is sometimes called the inverse relationship. Aside: for the purists among you and being utterly pedantic, the so-called inverse function re-expresses this relationship in terms of the input variable x as follows y = f 1 (x) = x b a

3 0.3 What is an observation or data value and a random variable? You all know this An observation or data value is a record of one event. An observation is usually symbolically written as a quantity or variable (x). A sequence of 1 to say n observations/data values is usually written as x 1, x 2, x 3, x 4,, x n 1, x n When the situation or process or system that generates these observations is unknown (i.e. cannot be specified by a rule) the observations are called random. So, for example the following is a sequence of observations of colours of cloths in a my wife s wardrobe blue, red, black, blue, pink, pink, brown, beige, white, red, blue is random as it changes frequently and cannot be specified by a rule. In contrast the rainbow sequence: red, orange, yellow, green, blue, indigo, violet is definitely not random, it is always the same every time you see a rainbow. Now for the fun bit, you won t really need this but unfortunately I don t trust myself enough to know that I ll never use the term, so for safety sake.. A random variable (rv) is a simply a variable that cannot be specified by a rule; put simply a set of random observations. For convenience we often label the observations as real numbers. So, for the wardrobe example we might re-define the rv as: Observed Colour: blue, red, black, blue, pink, pink, brown, beige, white, red, blue Assigned Number: 1, 2, 3, 1, 4, 4, 5, 6, 7, 2, 1 More often than not, an observation is recorded as a number and then the observed data value itself is then called the random variable. So if you hear me call something a random variable in general I simply mean the variable denoting the observed data value. So, if I measure the circumference of say 10 trees in a forest I might get (in cm s) 120, 150, 130, 40, 70, 190, 120, 120, 80, 70.

4 I call these circumference measurements a rv because there is no rule to tell me beforehand that the first tree will be 120, the second 150 and so on. I only know the measurements after I actually measure the trees. 0.4 What is a population and sample? You all know this. A population is a (possibly infinite) set of all possible (or the universe of) data values. This class is a population of finite size 84, the data values (rv) might be each persons age. A sample is a set of observed data values. A random sample is a sample where the data values are NOT chosen in a particular order or according to a specific rule. Many common samples are NOT random samples. For example, the monthly unemployment data from the Live Register, here the sample itself is a monthly sequence so it cannot be random. But importantly each data value, i.e. the number unemployed in a given month is random. Similarly, the 10 trees considered above are not a random sample, they are a sequential sample of trees as I have given the rule for inclusion in the sample, but the circumferences measurements themselves are a random data values. How do I take a random sample? Simple, label all data values in the population (say 1 to 84 for this class), generate a single random number between 1 and 84 and include the corresponding data value in the sample. Repeat 10 times to get a random sample with 10 data values. How do I generate a random number? a) Use a computer to get (pseudo) random numbers and multiply the result by 84. b) Roll a die 17 times and add up the results to get a random number between 17 and 102 (why not 14 times?). c) Use a clock face with 84 sectors and spin a hand on the face observing which sector it lands in. I devised this approach when I was a student and needed to generate random numbers for an assignment. d) Get a book with 84 pages and open it at some page, record the page number.

5 0.4 What do I mean by the term probability? You all know this it s the so-called frequency definition. The probability of an observed value is the proportion of times that value occurs. So, for the wardrobe colours example we get Observed data value (x) : 1, 2, 3, 4, 5, 6, 7 Frequency (of x): 3, 2, 1, 2, 1, 1, 1 Probability (of x): 3/11, 2/11, 1/11, 2/11, 1/11, 1/11, 1/11 So the probability is the (relative) frequency of a value divided by the total frequency. Aside: when you have a set of observed values these generate a set of probabilities, the observations are said to induce the probabilities. So, if you like, in the frequency definition the probabilities not handed down from God but come about through empirical observation. The full set of probabilities is called a Probability Distribution. The total value of all the probabilities always equals 1. A plot of the frequencies is called Histogram Frequency 4 Histogram of Wardrobe Colours (x)

6 0.5 What is a statistic and the study of statistics? A statistics is a function of the observed data values Or, put in the simple language used above, it is a number output or computed solely from the observed data values. The most commonly used statistic is the AVERAGE. Its formal name is the Sample Mean or mean for short, it is formally written as x called x-bar. It is sum of all the observed data values divided by the count of those values. Mathematically it is written in the same way as the sum function described earlier except for division by n, the number of input observed values, as x = f(x 1, x 2, x 3, x 4,, x n 1, x n ) = (x 1 + x 2 + x 3 + x 4, + + x n 1 + x n )/n. x = n i=1 x i n For the wardrobe example with 11 inputs the mean value is (the function) x (wardrobe) = = = Importantly, the mean value varies randomly as items of different colour are added or removed from the wardrobe. Another well know statistics is the MEDIAN, this simply the middle value of the observed values. Simply sort the observed values by size and select the middle value x med (wardrobe) = middle(1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 7) = 3. Note how the median and mean are close to each other. In this course another statistic that will be very important is the sample standard deviation, it measures the spread or width of the data values and is denoted the small letter s, this statistics computes the square root of the average squared deviation of each data value from the mean. s = (x 1 x ) 2 + (x 2 x ) (x n x ) 2 /n 1. s = (x i x ) 2 n i=1 n 1 Importantly, note the square of the standard deviation, s 2 is called the sample variance.

7 The population (of all data values) also has a population mean, population standard deviation and population variance, these are called population parameters and are denoted by the small Greek letters μ (called mu ) and σ (called sigma ) respectively. Statistics is the study of the variation of a particular type of function (e.g. the mean) of the observed data values as different samples of those observed data values vary randomly. In this course (and most others) the variation properties of the mean function for different samples from the same overall population are emphasised. 0.6 Exercises For the Table below: i x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 x18 x19 x a) Compute, by hand, the mean of the data values for each variables x1, x2, etc to x20; check your calculations with Excel. b) Plot the histogram of the resulting 20 mean values? c) Comment on the variation of the 20 mean values and the shape of the histogram? d) Compute the standard deviation for each of the variables x1, x2, etc to x20; do x1 and x2 by hand and use Excel if you wish for x3 to x Repeat Exercise 1 a) to c) using the median function and contrast with results of Exercise 1.

8 0.6.3 For the following table below of the observed colours of shirts worn by 10 people labelled x1 x10 over 10 separate days. Day (i) x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 1 blue green yellow purple brown green black green green Brown 2 red brown brown blue red white blue brown brown 3 purple pink blue yellow pink blue brown black black 4 pink yellow blue brown blue black green 5 yellow yellow pink brown purple pink pink yellow brown white 6 white red pink purple red blue red 7 yellow yellow blue pink black yellow white green white red 8 pink blue pink green pink red green pink blue red 9 purple red yellow red pink red red yellow brown 10 brown red white blue white yellow yellow black blue a) What is the probability of someone wearing a black or brown shirt? b) Which shirt colour is most preferred (think about this)? Outline Solutions (a) Variable x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Mean Variable x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 Mean Note: the mean is only computed on the NON missing values, so for x1 the mean is x (x 1 ) = = 40 8 = 5. Be careful about assuming that a blank value is zero, always check first with the data collection unit. If we found that the data collection unit should have recorded the blank as zero then the result would be which is quite a bit different to 5. x (x 1 ) = = = 4.

9 Frequency (b) Histogram More Mean (c) The key feature of the histogram is the peak around the middle values at 5 and 6. In general the histogram of a set of mean values tends to peak in the middle around the overall average value. (d) Variable x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 SD Variable x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 SD (a) Variable x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Median Variable x11 x12 x13 x14 x15 x16 x17 x18 x19 x20 Median x (x 1 ) = middle(1.1.2,4,4,6,8,14) = 4. Note: Once again the median is only computed on the NON missing values, if we had assumed the blanks were zero, we would incorrectly get x (x 1 ) = middle(0,0,1.1,2,4,4,6,8,14) = 3. The middle of the middle two values of 2 and 4. Don t assume blanks are zero, always check with the data provider.

10 Frequency (b) 10 Histogram More Medians (c) Once again we see the histogram peaks around the middle value 4.75, so aside from the slightly different location (bias) of the central peak the results are similar to Exercise (The Excel Histogram option in the Data Analysis Toolpack Add-in was used to compute the histogram of the medians accordingly, you may get slightly different results by hand) (a) I m recoding the colours by their intensity (darkest to lightest) to numbers, accordingly we get Black Brown Purple Blue Red Pink Green Yellow White Aside: This is a simple recoding to a number scale facilitates computing the average shade of garment in the wardrobe. Interestingly, engineers might recode the colours to phase angles of θ = 2nπ, n = 0,, 8, labelling the colours and then using a complex number scale (e.g. e iθ = cos θ + i sinθ) in order to compute the average shade of garment in the wardrobe. In general this coding is lightly to reveal more tone among the colours. 9 The probability of a black or brown shirt is the number of occurrence of black or brown divided by the total number of occurrences, or the relative frequency, simply count these colours in the columns in the table of colours and compute the total Colour type x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Total Black or Brown All colours

11 Frequency So the probability of a black or brown shirt is 18/90 = 0.2; this person has a preference for brighter colours, or maybe the sample was taken in the summer when brighter colours are more prevalent. (b) The shirt colour most preferred is the one with the highest relative frequency or probability, this is called the MODE of the distribution. Histogram Colour The histogram of colours shows that colours 2, 4, 5, 6 and 8 are all equally the most probable. Interpreting the colour that is most preferred needs some thought as sometimes the average or median (middle) value is often taken as the preferred value. For distributions that show a central tendency like the histogram of a set of mean values, the mean, median and the mode are close or identical (e.g. Normal Distribution, next chapter).