Quality Improvement – Besterfield Summary 5 FUNDAMENTALS OF STATISTICS 2 Examination of Histograms A histogram's analysis can reveal details about a specific quality issue, the population frequency distribution's shape, and specifications. A histogram representing the proportion of wash concentration in a steel tube cleaning process before painting is displayed in Figure 8. The crosshatched rectangle indicates the optimal concentration range, which is 1.45% to 1.74%. Concentrations below 1.45% yield subpar quality; concentrations over 1.75%, although yielding superior quality, are expensive and hence lower output. It is evident from simple statistics that corrective actions are required to get the distribution's spread closer to the desired 1.6% value. Additional information for solving problems is produced by enhancing the graph with specifications. The graph will display the level of capability as the distribution's spread is a reliable indicator of the process's capability. Last Words The Pareto diagram is another kind of distribution that resembles a histogram. A Pareto analysis is a highly useful method for pinpointing the location of significant quality issues. There are two key distinctions between a frequency distribution and a Pareto graph. Instead of using data values, the abscissa is represented by categories, which are arranged descendingly from the highest frequency to the lowest, as opposed to numerically. A frequency distribution's inability to display the sequence in which the data were produced is one of its drawbacks. Stated otherwise, all of the original data may be on one side, and the subsequent data on the other. The interpretation of the frequency distribution will change in this scenario. An analysis might benefit from the use of a time series plot, often known as a run chart, which displays the data's production sequence. An study of the data source or sources is an extra tool. CHESTRAL TENDENCY MEASUREMENTS In many cases, a frequency distribution is enough for quality issues. However, a graphical method is either unsuitable or requires the additional information offered by analytical techniques for a wide range of situations. One benefit of using analytical methods to describe a set of data is that they take up less space than graphs. They also offer the benefit of making it possible to compare different data sets. They also permit further computations and deductions. Measures of central tendency and measures of dispersion are the two main analytical techniques used to characterize a set of data. Measures of central tendency are covered in this part; the latter is discussed in the next section. A numerical number that characterizes the central position of the data or how the data tend to accumulate in the center is called a measure of central tendency of a distribution. The average, median, and mode are the three metrics that are most frequently used. On average The total number of observations divided by the total number of observations yields the average. It is the most often used central tendency metric. The average may be determined using three alternative methods: (1) ungrouped data, (3) Weighted average, (2) aggregated data. 1.Ungrouped data. When the data are disorganized, this method is applied. The formula yields the average, which is denoted by the notation X and reads as "X bar." Problem Example No. 1 Five coils are tested for resistance, and the results are recorded in ohms (V) as follows: X1 = 3.35, X2 = 3.37, X3 = 3.28, X4 = 3.34, and X5 = 3.30. Once the inputs are entered, the majority of electronic hand calculators can compute the average automatically. 2.grouped information. The next method can be used once the data have been sorted into a frequency distribution. The steel shaft problem (Table 6) illustrates that the method works when the grouping is based on cells that have multiple observed values per cell. It may also be used in situations where every observed value, Xi, has a unique frequency, fi, as the billing-error problem (Table 1) illustrates. Here, h denotes the total number of observed values. Put another way, Xi is the cell midpoint and fi is the number of observations in a cell that has been created from the frequency distribution. Xi is the observed value and fi is the number of times that observed value occurs in the data, if the frequency distribution has been sorted by individual observed values. This procedure is applicable to both continuous and discrete variables. The middle of each cell serves as the cell's representative value. The products are combined, divided by the total number of observations, and the midpoint is multiplied by its cell frequency. The first three columns in the following example problem represent a typical frequency distribution. The fourth column, "fiXi," is the result of multiplying the midpoint (second column) and frequency (third column) by each other. There may be a little variation between an average computed using this method and one computed using the ungrouped methodology. The unequal distribution of observations inside each cell is the reason for this discrepancy. In practical use, the disparity will not be significant enough to compromise the problem's correctness. 3.Average weighted. A weighted average is generated by combining multiple averages with various frequencies. Center The value that divides a set of ordered observations so that the number of items above and below it is equal is known as the median, and it is another way to quantify central tendency. 1. Ungrouped methodology. When calculating the median of an ungrouped data series, there are two conceivable scenarios: an odd series number and an even series number. The midway of the numbers is known as the median when the number in the series is odd. Consequently, the median of the ordered collection of integers 3, 4, 5, 6, 8, 8, and 10 is 6, and The median of the collection of numbers 22, 24, 24, 24, and 30 is 24. The median is the average of the two middle numbers in a series if the number is even. As a result, the median of the ordered set of integers 3, 4, 5, 6, 8, and 8 is equal to the product of 5 and 6, or (5 + 6)/2 = 5.5. Because (24 + 24)/2 = 24, the average of the two middle numbers is still calculated if both middle numbers are the identical, as in the ordered set of numbers 22, 24, 24, 30, and 30. Before calculating the median, the reader is advised to confirm that the values are in the correct order. 1.group approach. Finding the cell with the middle number and then interpolating inside it yields the median when data are grouped into a frequency distribution. Table 7 data will be utilized to demonstrate how the formula is applied. The halfway point (320 / 2 = 160) is achieved in the cell with a midpoint value of 37.0 and a lower limit of 35.6 by counting up from the lowest cell (midpoint 25.0). The frequency of the median cell is 58, the cell spacing is 3, and the cumulative frequency (cfm) is 154. The interpolated amount is removed from the upper limit and the cumulative frequency is counted up to the cell upper limit if the counting starts at the top of the distribution. It is more typical, nevertheless, to begin counting at the bottom of the distribution. There isn't much value for the median of clustered data. Moderate The value that appears the most frequently in a group of integers is its modal (Mo). In a set of numbers, the mode could have more than one value or not exist at all. The numbers 3, 3, 4, 5, 5, 5, and 7 have a mode of 5, the numbers 22, 23, 25, 30, 32, and 36 have no mode, and the numbers 105, 105, 105, 107, 108, 109, 109, 110, and so on do not have a mode. The two modes of 112 are 105 and 109. If a set of numbers has one mode, it is called unimodal; if it has two modes, it is called bimodal; and if it has more than two modes, it is called multimodal. The midpoint of the cell with the highest frequency, which indicates the highest point (largest frequency) of the histogram, is the mode when data are grouped into a frequency distribution. By using an approach akin to that of the median interpolation, a more accurate estimate of the mode may be obtained. More precision than the cell midpoint is not necessary, though, as the mode is mostly used as an inspection tool for figuring out the central ten- dency. Connection Between the Central Tendency Measures Figure 9's smooth polygons display the differences between the three central tendency measurements. The values for the average, median, and mode are the same in a symmetric distribution; in a skewed distribution, the values differ. The most often used way to measure central tendency is the average. It is employed in situations when the distribution is symmetrical or not noticeably skewed to the right or left, where extra statistics, such control charts, measures of dispersion, and so on, need to be calculated using the average, and when inductive statistics require a steady value. In cases when the distribution is favorably (to the right) or negatively (to the left) skewed, the median becomes a useful indicator of the central tendency. When a distribution's precise midpoint is required, it is employed. Extreme values in a distribution will negatively impact the average while having no effect on the median. For this reason, the median and average of a set of numbers like 12, 13, 14, 15, 16, and so on are identical and equal to 14. If the initial value is altered to 2, the average drops to 12 but the median stays at 14. A median-based control chart is an easy-to-use and effective tool for assessing quality. When one needs a fast and approximative estimate of the central tendency, one uses the mode. As a result, it is simple to determine a histogram's mode by visual inspection. The most usual value of a distribution, such as the modal age of a certain group, is also described by the mode. The geometric mean, harmonic mean, and quadratic mean are further metrics for central tendency. These measurements aren't applied with precision. DISPERSION MEASURES Overview Methods for characterizing the central tendency of data were covered in the section before this one. The measures of dispersion, which characterize how the data are dispersed or spread out on each side of the center value, make up a second statistical tool. To characterize a set of data, metrics of central tendency and dispersion are required. As an example, the average weekly pay for the workers at a factory's plating and assembly departments is $625.36. On the other hand, the plating department's high and low points are $630.72 and $619.43, respectively, while the assembly department's high and low points are $680.79 and $573.54, respectively. Compared to the plating department, the assembly department's statistics are dispersed or more widely distributed from the average. This section discusses variance, standard deviation, and range as measures of dispersion. In quality, other metrics like mean deviation and quartile deviation are not employed. Range The difference between the greatest and lowest values or observations is the range of a set of numbers. It is provided, symbolically, by the formula The reference value used to measure the dispersion in the data is the standard deviation. The easiest way to think of it is as an index that the formula defines. Because the distribution is more tightly packed around the center value, the quality increases with a smaller standard deviation. Furthermore, the standard deviation aids in population definition. Only when the process is steady can this approach be used. The optimal techniques include s = s/c4 or s = R/d2.6. Connection Between the Dispersion Measures One of the main control charts makes use of the range, which is a widely popular way to quantify dispersion. The range's main benefits are its simplicity and ability to reveal the whole distribution of the data. Additionally, it is useful when there is little or insufficient data to support the computation of a more accurate measure of dispersion. A measure of central tendency has no bearing on the range. Because it is easier for exceptionally high or low measurements to occur, the accuracy of the range falls as the number of observations grows. It is recommended that a maximum of 10 observations be made when using the range. THE IDEA OF A SAMPLE AND A POPULATION It is now desirable to investigate the notions of a population and a sample. A tiny percentage, or sample, is chosen to represent all of the steel shafts in order to create a frequency distribution of the weights of steel shafts. In a similar vein, just a small percentage of all passenger automobiles were represented in the data gathered about car speeds. The population is the whole set of measurements; in the cases that came before, the populations were all of the passenger cars and steel shafts. Statistics refers to the computation of averages, standard deviations, and other measurements from samples. The estimated statistics will differ from their real population values, or parameters, by an amount due to sample composition fluctuations. Parameters are defined as the best estimate of these values that is available at a given moment or as established reference (standard) values. A day's worth of steel shaft manufacture is one example of a limited amount of things that the people may own. It might be virtually endless, like the number of rivets used in a year's worth of jet airplane production. Depending on the specific circumstances, there are several ways to describe the population. Therefore, the population of an hour's production, a week's production, 5000 pieces, and so forth might all be included in a product research. A sample is chosen since it is rarely feasible to measure the entire population. Sampling is essential in situations when measuring the entire population may be impracticable, observing all the data would be prohibitively expensive, the product may be destroyed during the required inspection, or testing the entire population would be too risky, like with a new medical medicine. In actuality, sampling may provide more accurate results than analyzing the full population. It has been demonstrated that when the percentage of nonconforming is very low, 100% manual inspection is less reliable than sampling. This is most likely because inspectors prejudge each thing they check as being because they are bored or tired. The amount of variation with respect to the mean decreases with decreasing value. Normative Curvature Synopsis Despite the fact that there are as many distinct populations as there are situations, they may all be broadly categorized into a few categories. The Gaussian distribution, sometimes known as the normal curve, is one popular form of population. The distribution with the mean, median, and mode all having the same value is called a symmetrical, unimodal, bell-shaped distribution, or normal curve. A frequency histogram is the starting point for creating a population curve or distribution. An increasing sample size in a histogram results in a decreasing cell interval. The histogram will resemble a smooth polygon or a curve that represents the population when the sample size is really big and the cell spacing is extremely tiny. Figure 13 displays a curve of the normal population of 1000 observations of an electrical device's resistance in ohms with a population mean (m) of 90 Æ and a population standard deviation (s) of 2 Æ. One standard deviation, s, equals the space between dotted lines. The frequency distribution of the normal curves is followed by a large portion of variance seen in nature and industry. As a result, there will be a normal curve for the changes in elephant weight, antelope speed, and human height. It is also reasonable to anticipate that industrial variables, such the weight of gray iron castings, the lifespan of 60-watt light bulbs, and the size of a steel piston ring, will follow a normal curve. A small percentage of people should be very tall, and a small percentage should be extraordinarily low, when it comes to human heights, with the bulk of people's heights grouping around the average figure. Many strategies are based on the normal curve because it provides an excellent representation of the variances that occur in the majority of quality criteria in industry. Connection to the Standard Deviation and Mean There is a clear correlation between the mean, standard deviation, and normal curve, as demonstrated by the standardized normal curve formula. Three normal curves with varying mean values are depicted in Figure 15; the position is the only variable that varies. Three normal curves with distinct standard deviations but the same mean are displayed in Figure 16. The graphic demonstrates the idea that a curve with a greater standard deviation (indicating data that are broadly spread) would be flatter than one with a lower standard deviation (indicating data that are narrowly dispersed). There is no curve and all values are equal to the mean when the standard deviation is zero. The population mean and population standard deviation provide a complete definition of the normal distribution. Furthermore, Figures 15 and 16 demonstrate the independence of these two characteristics. Stated differently, alterations made to one do not impact the other. Figure 17 illustrates the link between the area under the normal curve and the standard deviation. According to the figure, 68.26% of the things fall within the range of m + 1 s and m - 1, 95.46% of the items fall within the range of m + 2 and m - 2 s, and 99.73% of the items fall within the range of m + 3 s and m - 3 s in a normal distribution. Between the boundaries +q and -q, all of the items are included 100% of the time. Regardless of how the normal curve is shaped, these percentages remain accurate. Control charts are based on the fact that 99.73% of the items are included between t3s.