Certified Quality Engineer Programme (CQE)Module 6 Quantitative Methods Part 1ByAssociate Professor Dr Shari M. YusofFaculty of Mechanical EngineeringUniversiti Teknologi Malaysia, Skudai, Johor
Basic Concepts Of ProbabilityProbability is a measure that describes the chance that an event will occur.Dimensionless number ranges from zero to one - with 0 meaning an impossible event and 1 refer to event that is certain to occur.Probability of 0.5 means the event is just as likely to occur as not.
Basic Concepts Of StatisticsThe word statistics has two generally accepted meaning:A collection of quantitative data pertaining to any subject or group, especially when the data are systematically gathered and collated.The science that deals with the collection, tabulation, analysis, interpretation and presentation of quantitative data.
Basic Concepts Of StatisticsThe use of statistics in quality engineering deals with the second meaning and involves CollectingTabulatingAnalyzingInterpretingPresenting data
Collecting And Summarizing DataDescriptive Statisticsto describe and analyze a subject or groupanalytical techniques summarize data by computinga measure of central tendencya measure of the dispersion.
Measure of Central TendencyA measure of central tendency of a distribution is a number that describes the central position of the data or how the data tend to build up in the center.Three measures commonly used : 1) average2) median3) mode
AverageIt is the sum of the all the observations divided by the number of observations3 different techniques available for calculating the average 1) ungrouped data2) grouped data3) weighted average
AverageUngrouped data.This method is used when the data are unorganized.The average is represented by the symbol x, which is read as x bar and is given by the formula;x = xi / n = (x1 + x2 +.+xn)/nwherex = averagen = observed values x1, x2,...,xn = observed value identified by the subscripts 1,2,..n or general subscript i = symbol meaning sum of
ExampleA food inspector examined a random sample of 7 cans of tuna to determine the percent of foreign impurities. The following data were recorded :1.8, 2.1, 1.7, 1.6, 0.9, 2.7 and 1.8Compute the sample mean.x = xi / n = (1.8+2.1+1.7+1.6+0.9+2.7+1.8)/7 = 1.8% impurities
ExerciseIn studying the drying time of a new acrylic paint, the data in hours, were coded by subtracting 5.0 from the observation. Find the sample mean and sample standard deviation (s) for the drying times of 10 panels of wood using the paint if the coded measurements are :1.4 , 0.8, 2.4, 0.5, 1.3, 2.8, 3.6, 3.2, 2.0, 1.9
Grouped data.When data have been grouped into frequency distribution, the following technique is applicable. Formula for the average of grouped data x = (fiXi)/n = (fiX1 + f2X2 + +fhXh) / (f1 + f2++fh )where n = sum of the frequencyfi = frequency in a cell or frequency of an observed valuexi = cell midpoint or an observed valueh = no. of cells or no. of observed values
Example Frequency Distribution for Weights of 50 components
Class IntervalWeight (g)Class BoundaryClass mid-point (xi)No of pieces (fi)fixifixi27 96.5 9.58210 129.5 12.511813 1512.5 15.5141416 1815.5 18.5171919- 2118.5 21.5207Totals ()50
When a number of averages combined with different frequencies, a weighted average can be computedThe formula for the weighted average is given by :xw = wixi wi wherexw = weighted averagewi = weight of the i th average
Example weighted averageOn a trip a family bought 21.3 litres of gasoline at 1.21 per litre, 18.7 litres at 1.29 cents per litre, and 23.5 litres at 1.25. Find the mean price per litre.
MedianMedian is the middle value for a set of data arranged in an increasing or decreasing orderCase 1 - when the number of data in the series is odd middle valueCase 2 - when the number of data is even - median is the average of the two middle numbersExample (case 1) 5 test results 82, 93, 86, 92, 79What is the median? Arrange data. Answer = 86 Example (Case 2) The nicotine contents for a random sample of 6 cigarettes of a certain brand are found to be 2.3, 2.7, 2.5, 2.9, 3.1 and 1.9If we arrange in increasing order of magnitude , we get = 1.9 2.3 2.5 2.7 2.9 3.1 , and the median is the mean of 2.5 and 2.7.Therefore, x = (2.5+2.7)/2 = 2.6 milligrams
MedianGrouped Data When data grouped into frequency distribution, the median is obtained by finding the cell that has the middle number and then interpolate within the cell.Formula for computing median :Md = Lm + I
whereMd = medianLm = lower boundary of the cell with the mediann = total no. of observationsfm = frequency of median cellcfm = cumulative frequency of all cells below LmI = cell interval
ModeMode of a set of numbers (data) is the value that occurs with the highest frequencyPossible for mode to be nonexistent in a series of numbers or to have more than one value.A series of numbers is referred to as unimodal if it has one mode, bimodal if it has two modes and multimodal if there are more than two modes.Data grouped into frequency distribution, the midpoint of the cell with the highest frequency is the mode, since this point represents the highest point (highest frequency) of the histogram
Measure of DispersionMeasures of dispersion describe how the data are spread out from the average or scattered on each side of the central value.Common measuresRange (simplest)Standard deviationVariance
RangeRange of a series of numbers is the difference between the largest and smallest values or observations.R = Xh - Xl where R = rangeXh = highest observation in a seriesXl = lowest observation in a seriesExample The temperature for a process recorded 40.2 , 38.7, 42.5, 39.6, 40.9. What is the value of range?
Standard DeviationStandard deviation - numerical value in the units of the observed values that measures the spreading (variation) of the data.Large standard deviation - greater variability of the data than smaller standard deviation, given by the formula:s = (xi x)2 / (n-1)where s = sample standard deviationxi = observed value ith x = averagen = number of data (observed values)It is reference value that measures the dispersion in the data
ExerciseA car manufacturer tested a random sample of 10 steel-belted tyres of a certain brand and recorded the following tread wear: 48000, 53000, 45000, 61000, 59000, 56000, 63000, 49000, 53000 and 54000 kilometers. Find the standard deviation of this set of data.
Collecting AndSummarizing DataConsider the data below which represents the lives of 40 similar car batteries recorded to nearest tenth of a year. What can you learn from these numbers?
Frequency distributionGroup large number of data into different classes (groups) and determining the number of observations that fall into each groupDecide no of classes too few lose information, too many also no meaningUsually choose 5 20 classesLet us choose 7 classes class width must be enough to put in all the dataApproximate width find range divide by no of classes = (4.7-1.6)/7 = 0.443 should have same no of significant places as data, therefore choose the value 0.5
Frequency distributionDecide where to start bottom interval start at 1.5 and lower boundary is 1.45. Then add width 1.45 +0.5 = 1.95 continue for the othersMidpoint is (1.5+1.9)/2 = 1.7 Count the no of observations and record in the tableTotal the frequency to check all data has been counted
Class intervalClass boundariesClass midpointFrequency1.5 1.91.45 1.951.722.0 2.41.95 2.452.212.5 2.92.45 2.952.743.0 3.42.95 3.453.2153.5 3.93.45 3.953.7104.0 4.43.95 4.45 4.254.5 4.94.45 4.954.73
Class intervalClass boundariesClass midpointFrequencyClass boundariesFrequency
1.5 1.91.45 1.951.721.45 1.952
2.0 2.41.95 2.452.211.95 2.451
2.5 2.92.45 2.952.742.45 2.954
3.0 3.42.95 3.453.2152.95 3.4515
3.5 3.93.45 3.953.7103.45 3.9510
4.0 4.43.95 4.454.253.95 4.455
4.5 4.94.45 4.954.734.45 4.953
General steps for Constructing FDDecide number of class intervals (groups) requiredDetermine the rangeDivide the range by no. of classes to estimate approximate width of intervalList lower class limit of bottom interval and lower class boundary. Add lower class width to lower class boundary to get upper class boundaryList all the class limits and class boundaries by adding class width to the limits and boundaries of previous intervalDetermine the class marks (midpoint) by averaging the class limits or class boundariesTally the frequencies for each classSum the frequency column and check against total no. of observations
PROBABILITY DISTRIBUTIONDiscrete DistributionSpecific values such as the integers 0, 1, 2, 3 are used.Typical discrete probability distributions are, binomial and Poisson.
Binomial Probability DistributionApplicable to discrete probability problems that have an infinite number of items or that have a steady stream of items coming from a work center.The binomial is applie