Some Material on the Statistics Curriculum

Transcription

1 Some Material on the Curriculum A/Prof Ken Russell School of Mathematics & Applied University of Wollongong

2 involves planning the collection of data, collecting those data, describing, analysing and interpreting those data, and then reporting on your conclusions. Most data are collected in the form of numbers, but not all numbers are created equally. Many errors arise from applying incorrect techniques to your data.

3 We usually recognise four types of numbers: nominal or categorical (e.g., labels ) How did you come here this morning? 1: drove, 2: public transport, 3: cycled, 4: walked, 5: other. ordinal (e.g., rankings) Julia Gillard will be a great PM. 1. Strongly Disagree, 2. Mildly Disagree, 3. Neither Disagree or Agree, 4. Mildly Agree, 5. Strongly Agree. interval (e.g., measurements of temperature in C); ratio (e.g., most physical measurements).

4 If a and b are two numbers that are nominal, then a < b, a b and a/b make no sense ordinal, then a < b makes sense, but a b and a/b make no sense interval, then a < b and a b make sense, but a/b makes no sense ratio, then a < b, a b and a/b all make sense.

5 Exercise Classify the following measurements as nominal, ordinal, interval or ratio: Number of mistakes made by a pupil in a test Grade given for that test: A = 1, B = 2, C = 3, F = 4. Sex of the person sitting on your right: 0 = F, 1 = M, 2 = N/A Proportion of this talk that has been interesting so far: , , Proportion of people in this room who are over 40 years of age.

6 Should not generally add or multiply nominal or ordinal data and expect the answer to make sense. Should not generally multiply interval data and expect the answer to make sense. This does not stop people doing so. You need to exercise caution when seeing the results of such operations.

7 Summarizing data Tables and bar graphs are suitable for summarizing nominal and ordinal data. Display either the numbers of observations (frequencies) of each score, or the proportions ( relative frequencies ). Relative frequencies normally required for comparison purposes.

8 Example Travel to the Maths Teachers Day (frequency) Drove Public Transport Cycled Walked Other Total No good for comparisons across years!

9 Example, continued Travel to the Maths Teachers Day (rel. frequency) Drove Public Transport Cycled Walked Other

10 We can also show the relative frequencies by a bar graph Rel. frequency Drove Public Method of Transport

11 Discrete vs Continuous Interval or ratio data are classified as either discrete or continuous. Discrete data take separate ( discrete ) values on the number line (usually integers); e.g., the number of Labor MPs who would have voted for Kevin Rudd last week. Continuous data take values anywhere in one or more intervals; e.g., the time we wait from now until there is a new PM.

12 The (relative) frequency of discrete data can still be plotted on a conventional bar graph, especially if there is a constant difference between possible consecutive values. However, a line graph is better. Suppose that I have 25 children in the class, and I get each of them to toss a coin twice and count the number of heads. I may get the results Number of Heads Frequency Rel. freq Total

13 Line graph You could draw the following line graph: This is perfectly adequate.

14 Intervals of unequal width Bar graphs can also be drawn for continuous data if the possible values are divided into intervals of equal width. However, if the intervals are of unequal width, the image displayed by a bar graph can be very misleading. The eye is distracted by the area of a bar, not the height, so we can t use relative frequency on the vertical axis.

15 Example You have collected data on a variable, X, that can take values between 0 and 2. You have the following results: Interval Rel. freq 0 x < x < x < x A naive user would put relative frequency on the vertical axis.

16 Not good! This is not satisfactory.

17 Histograms Instead, we let the vertical axis show density, and we select the height so that it is the area ( = base height) of the bar that shows the relative frequency. Such a graph is called a histogram.

18 Exercise You have collected data on a variable, X, that can take values between 0 and 2. You have the following results: Interval Rel. freq 0 x < x < x < x If you want the last bar to be 17 mm high, how high should the other three bars be?

19 Working it out Interval Rel. freq 0 x < x < x < x We know that area equals relative frequency, and that 17 (mm) 1 = So the vertical scale is: 1 mm = 0.01.

20 Interval Rel. freq 0 x < x < x < x First interval: Area = h 0.2 = 0.32 h = 0.32/0.2 = 1.60 = 160 mm. Similarly, height of second bar is mm, and height of third bar is 46 mm.

21 The correct histogram

22 Random variables A random variable is just a variable (i.e, its value can vary) where we cannot predict with certainty what value we will take when we do something. It is often introduced simply to save us having to write something over and over. For example, say let X be the number of Heads from two tosses of a coin, and then you can just say X instead of the number of Heads from two tosses of a coin.

23 It is convenient to write the values of X as numbers, but these values can be nominal, ordinal, interval or ratio data. Different types of data require different techniques. I will just talk about interval and ratio data.

24 Samples and populations Statisticians distinguish between samples and populations. A sample is just a limited number of observations on the value of X. For example, we saw the results of 25 exercises in tossing a coin twice and counting the number of Heads. This was a sample of 25 observations. A population is the theoretical set of values you would get if you could toss the coin twice, and then do it over and over. As the number of pairs of tosses heads towards, the relative frequency settles down to a probability.

25 Discrete random variables For discrete random variables, we can still draw a line graph, where the height of each line represents the probability (rel. frequency as n ) of obtaining that value of X. A display of the possible values of a discrete random variable, and the probability of obtaining each one, is called a probability function, whether or not it is actually written as a function.

26 Example Let X be the number of Heads from two tosses of a fair coin. Each of the following displays the probability function of X : and Pr(X = x) = ( 2 x x Pr(X = x) 1/4 1/2 1/4 ) ( 1 2 ) x ( ) 1 2 x = 2 ( ) 2 x ( ) 1, x = 0, 1, 2. 4

27 Continuous Random Variables As we increase the number of observations made on a continuous random variable (e.g., the time between serious accidents on Picton Road), we can make the bars in a histogram thinner and thinner - until we get a smooth curve. This is called the probability density function (c.f. probability function for a discrete random variable). It is still the case that areas under the curve represent probabilities.

28 4 3 prob. density function x As the number of observations increases, the histogram tends to the p.d.f.

29 x Calculating probabilities by integration If we have an integrable formula for the p.d.f, e.g. f (x) = { 3 8 (2 x)2, 0 < x < 2, 0, elsewhere, we can calculate probabilities by integration. For example, Pr(0.5 < X < 1.5) = (2 x)2 dx. prob. density function

30 Calculating probabilities from tables If we can t integrate the probability density function, we have to use tables obtained by numerical integration to calculate probabilities. The Normal distribution (the bell-shaped curve) is the standard example of this.

31 Statistical computing The Syllabi all say: Use of technology The Shape of the Australian Curriculum Mathematics states that available technology should be used for teaching and learning situations. Technology can include computer algebra systems, graphing packages, financial and statistical packages and dynamic geometry. These can be implemented through either a computer or calculator.

32 The R package One statistical resource is the package R, available freely from the website (or just type CRAN into Google). Definitely not for everyone, but nerds and others comfortable with computing will find it great. Lots of online help available, and a plethora of books.