UNIT 1: ANALYTICAL METHODS FOR ENGINEERS

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1 UNIT 1: ANALYTICAL METHODS FOR ENGINEERS Unit code: A/601/1401 QCF Level: 4 Credit value: 15 OUTCOME 4 - STATISTICS AND PROBABILITY TUTORIAL 1 STATISTICAL DATA 1 Tabular and graphical form: data collection methods; histograms; bar charts; line diagrams; cumulative frequency diagrams; scatter plots Central tendency and dispersion: the concept of central tendency and variance measurement; mean; median; mode; standard deviation; variance and interquartile range; application to engineering production Regression, linear correlation: determine linear correlation coefficients and regression lines and apply linear regression and product moment correlation to a variety of engineering situations Probability: interpretation of probability; probabilistic models; empirical variability; events and sets; mutually exclusive events; independent events; conditional probability; sample space and probability; addition law; product law; Bayes theorem Probability distributions: discrete and continuous distributions, introduction to the binomial, Poisson and normal distributions; use of the Normal distribution to estimate confidence intervals and use of these confidence intervals to estimate the reliability and quality of appropriate engineering components and systems This section of the syllabus should have been covered at national level so the same material is repeated here and should be skipped if you are already familiar with the work. On completion of this tutorial you should be able to do the following. Explain the use of raw data. Present data as frequency polygons, bar graphs and histograms. Explain and find the mean and median values. Explain and plot ogives. Explain and find quartiles. D.J.Dunn 1

2 1. INTRODUCTION Statistics are used to help us analyse and understand the performance and trends in various areas of work. These might be financial trends, things to do with the population or things to do with manufacturing. 2. SAMPLE VALUES The samples are taken from a POPULATION and the samples form a SET within that population. For example a set of 100 people might be taken from a population of The samples must be truly random with no factors to bias the results to one extreme or the other. If the set is large enough and truly random then we might expect any information we get to apply equally well to the larger population. The data collected will vary about a mean or average value. The data is a variable such as the height of people, the values of resistors coming off a production line or the diameter of a machined component coming off a machine tool. We find out how many of these things f fall within defined bands (also called bins in modern statistics) and these are simply ranges of values denoted x. We need to discuss what x means. If you were throwing a dice over and over again you would get a score of exactly 1, 2, 3, 4, 5 or 6. Hence x can only be these exact numbers and if you throw the dice repeatedly you can measure how many times a particular number comes up. This kind of data is called a DISCRETE data set because the result can only be a whole number and this is easier to deal with. In the case of variables such as height, weight, size and so on, the values of x is a CONTINUOUS VARIABLE and will be a decimal number and depends on the accuracy of measurement. In the extreme we would get one sample for each exact decimal value and this would be no use. We could do one of two things. We could round off the values so that all those close together will be counted as the same, or we could count the number of samples f that fall within a specified range with x being the value at the middle of the range. Let s use an example to get started. 3. RAW DATA Consider a set of statistics compiled for the height of children of age 10. First we would compile a table of heights. This would be the raw data. Note that the larger the sample we take, the more meaningful the results will be. By measuring the heights to an accuracy of 0.01 m we may get children with the same height but this is unlikely with a small set. This is the table of raw data for 10 year old children Sample Height (m) Sample Height (m) BANDS If we used more accurate the measurements it is unlikely we would find two children exactly the same height but the more we round off the values, the more likely it becomes. When handling large numbers of samples, we need to simplify the table by creating sensible bands within which the measurements fall. The number of children within each band is the frequency. Next we would have to go through the laborious task of counting how many there are in each band. If we found a child with a height exactly on the edge of the band edge, we might decide to allocate a half to each band on either side. D.J.Dunn 2

3 This results in a frequency values that are not whole numbers. The result is a FREQUENCY DISTRIBUTION TABLE. Height Mid Point Freq PLOTS If we plot frequency vertically against height horizontally, we get a frequency distribution graph and this can be drawn in different ways. Notice that the points are drawn for the middle of the band. HISTOGRAMS are another way of plotting data. The histogram is made of vertical columns and the area of the columns represents the frequency. The vertical height is the frequency density. The horizontal divisions are commonly known as BINS. In our example the bins are all equal and have a width of 0.1 m. The height of each column is hence the frequency divided by the bin so the first column is = 20, the next is = 50 and so on. You will find many examples that plot the frequency vertically and not the frequency density. If you do this it is a bar chart and not a histogram. 6. MEAN This is one of the more common statistics you will see and it's easy to compute. All you have to do is add up all the values in a set of data and then divide that sum by the number of values in the dataset. For our example, let the height be represented by the variable x and the frequency be f. Sample number Height (m) Sample number Height (m) Total D.J.Dunn 3

4 The mean value is denoted x and x = 24.34/17 = m We can do this a bit more simply using the grouped frequency distribution table but this is only an approximation. X (mid pt) f total f x The mean value is x = 24.25/17 = m. This is not quite as accurate as the previous answer because the values have been taken at the mid point of the band. 7. MEDIAN While the mean or average is a useful statistic it is also useful to know the median value. This is easy to determine because the median is literally the value in the middle. If the data is evenly spread about the mean value then the median is going to be similar to it. If the data is lop sided then the median and mean will be quite different and this tells us useful things. For example if the mean is much larger than the median we have a lot of tall children. If the median is larger than the mean then we have a lot of short ones. In order to find it, you just line up the values in your set of data from largest to smallest. The one in the dead-centre is your median. Our table would look like this. Sample number Height (m) Sample number Height (m) Total The mid point in the table is point number 9 with so the median value is 1.44 m. If we had an even number of samples, say 18, then there would be two values in the middle and we should average the two to get the median. To do it mathematically a method that works for any number of samples n is (n+1)/2 and this gives the position of the median. In our example we have 17 results. The median is at the (17+1)/2 = 9 th result. This works for an odd or even number of results. We can also use the grouped frequency distribution table when dealing with a lot of data but there are problems with this kind of data when the data is continuous and not discrete (meaning only whole numbers). X (mid pt) f total f x Point number 9 is between 1.35 and 1.45 and we can t say it is one or the other. We can show this with a graph called an OGIVE. D.J.Dunn 4

5 8. OGIVE This is another way of plotting the data with the cumulative frequency plotted vertical. For a small number of samples we can plot the raw table against sample number. In effect this is using bands of 0.01 m and the sample number becomes the accumulative frequency. For large amounts of data we need the grouped frequency table and we add a new row containing the cumulative frequency as shown. x Total f = n f x cum. f If the graph is to have any meaning, we must plot the upper edge of the band or bin so that the point indicates how many are less than that value. The first point must be at zero. The red plot is the grouped frequency and the blue is the ungrouped. The median at point 9 is almost the same. The ninth sample divides the ogive into two vertically and the projection down to the x axis gives the median value as Examining the graph we see that the banded data plot does not correlate perfectly with the un-banded plot that gives us the correct median. D.J.Dunn 5

6 9. QUARTILES and PERCENTILES If we divide the vertical scale into 4 equal parts (each 4.5 in this case) and project the lines down to the x axis we have 4 divisions on the x axis. Q1 is the first or lower Quartile and represents the median value (height) of the first quarter. In this case it is around Q2 is the median for the whole range. Q3 is the median for the last quarter or Upper Quartile and is around The inter-quartile range is the difference between Q1 and Q2 ( = 0.18) These tell us something about how the samples are spread around the median but a better method of doing this is to use the standard deviation. If we repeated the process by dividing the vertical axis into 100 parts, then we can look up how many children have a height less than a given percentage. These are called the PERCENTILES. The median is the 50 th percentile. The first quartile is the 25 th percentile. D.J.Dunn 6

7 WORKED EXAMPLE No.1 A company manufactures steel bars of nominal diameter 20 mm and cuts them into equal lengths. The diameter of each length is measured at the middle for the purpose of quality control. The results for 20 bars are given below. Produce a frequency distribution table using bands of 0.1 mm. Calculate the mean of the samples. Draw a histogram. Plot the Ogive. Determine the median, the upper and lower quartiles and the semi- inter-quartile. Use the tables and plots to show your solutions Sample diameter Sample diameter SOLUTION Total = Total samples n = 20 mean = 398.3/20= Rank the data in order smallest to largest. Sample diameter Sample diameter n = 20 Mean = 398.3/20= The Median is at (n+1)/2 = (20+1)/2 = 10.5 so it is between 10 and 11. Average of these is 19.9 so the median is 19.9 FREQUENCY DISTRIBUTION TABLE Bin width is 0.1 d Totals Mid f f d cum.f f Den D.J.Dunn 7

8 median = Upper quartile = Lower quartile = Inter-quartile range = = 0.22 Semi-inter-quartile = 0.22/2 = 0.11 D.J.Dunn 8

9 SELF ASSESSMENT EXERCISE No.1 The diameters of a number of components are measured to the nearest 0.05 mm. These are grouped into bins of 0.1 mm as shown. Diameter mm Number Draw the histogram and the Ogive and deduce the mean, the median, the upper and lower quartile and the semi-interquartile range. Answers Mean = mm Median is 10 upper quartile Q3 = lower quartile Q1= 9.91 semi-interquartile range = = 0.18 D.J.Dunn 9