Quality Improvement – Besterfield Summary 4 FUNDAMENTALS OF STATISTICS OVERVIEW What Statistics Means There are two commonly understood definitions of the term statistics: 1.a compilation of quantitative data about any topic or subject, particularly when the information is methodically obtained and compiled. Blood pressure readings, football game statistics, job data, and accident statistics are a few instances of this meaning. 2.the branch of science that deals with quantitative data collection, tabulation, analysis, interpretation, and presentation. It should be emphasized that while the second definition also has to do with data collecting, it is more expansive than the first. The divisions of gathering, tabulating, analyzing, interpreting, and presenting the quantitative data are all part of the second and larger definition of statistics as they relate to quality. Every division depends on the precision and thoroughness of the one before it. A technician assessing a plastic part's tensile strength or a market researcher ascertaining customer color preferences can both gather data. It may be tallied with a computer or with basic paper and pencil methods. A quick visual inspection or in-depth computations may be part of an analysis. The final results are presented and evaluated to help with excellent decision-making. Statistics are divided into two stages: 1.Descriptive statistics, often known as deductive statistics, aim to characterize and examine a subject or group. 2.Inductive statistics aim to draw significant conclusions about a much greater quantity of data (population) from a small sample of data. Probability is a common phrase employed since these findings or inferences cannot be expressed with perfect certainty. The statistical foundations covered in this chapter are necessary to comprehend the quality control techniques that follow. grasp quality and many other disciplines, for that matter, requires a grasp of statistics. Gathering the Information Direct observation is one method of gathering data; indirect methods include asking written or spoken inquiries. Public opinion pollsters and researchers that work in market research frequently employ the latter method. Direct observation is used to gather data for quality purposes, which are then categorized as either variables or attributes. Measurable attributes of a product, such weight measured in grams, are known as variables. Conversely, attributes are those quality characteristics—like a "go/no go gage"—that are categorized as either meeting or failing to meet criteria. A variable is considered continuous if it may be subdivided to any extent. One example of a continuous variable is the weight of a gray iron casting, which, depending on the precision of the measurement device, can be measured as 11 kg, 11.33 kg, or 11.3398 kg (25 lb). Continuous data includes measurements like feet (meters), gallons (liters), and pounds per square inch (pascals). Discrete variables are those that show gaps in their data. A travel trailer may have any whole number of nonconforming rivets, for example, 0, 3, 5, 10, 96. ....; yet, a certain trailer cannot include, say, 4.65 non-conforming rivets. Generally speaking, discrete data cannot be counted, while continuous data can. Verbal or nonnumerical data may occasionally be more convenient when treated as variables. For instance, a piece of furniture's surface finish might be rated as nice, mediocre, or terrible. The numbers 1, 2, or 3 can be used in place of the bad, average, or good categorization, respectively. Similarly, schools designate the numerical values 4, 3, 2, 1, and 0 for the letter grades A, B, C, D, and F, respectively, and utilize those discrete numerical values as discrete variables for communication. putative objectives. There are several qualities that need to be expressed as attributes, even if many of them are expressed as variables. Qualities that are assessed through eye inspection are often categorized as attributes. The words on this page are spelled correctly or wrongly, the switch is on or off, the electric motor's wire is hooked to the terminal or it is not, and the response is accurate or incorrect. The preceding statement provides examples of both conformity and nonconformance to a certain standard. At times, it is preferable to classify variables as characteristics. It is common for factory workers to want to know if the product they are making meets the requirements. For instance, knowing that the weight of a packet of sugar is within the allowed ranges may be more significant than the weight's exact values. As a result, the weight of the sugar packet is used to collect data, which is then reported as either meeting or exceeding criteria. The quantity of figures that are collected depends on how the data is going to be used. For instance, when gathering data It is appropriate to record 995.6 hours for the life of light bulbs; however, documenting 995.632 hours is superfluous and excessively precise. In a similar vein, data would be gathered to the nearest 0.001 mm and rounded to the nearest 0.01 mm if a keyway specification had lower and upper bounds of 9.52 mm (0.375 in. and 9.58 mm, respectively). The more figures a measuring device has to the right of the decimal point, generally speaking, the more advanced it is. Because of issues with accuracy and precision, measuring devices might not provide an accurate reading. Because the average of these repeated measurements is near the genuine value, which is located in the middle, Figure 1(a) displays an accurate sequence of measurements. The repeated measurements in the series in Figure 1(b) are accurate (quite close together), but they are not very near to the actual value. The sequence of repeated measurements, which are exact and accurate, are densely clustered around the correct value in Figure 1(c). A team trying to address an issue without a suitable measuring method often fails. Therefore, the first thing to do is make that variety better. There are specific norms that must be adhered to when rounding data. In rounding the integers 0.9530, 0.9531, 0.9532, 0.9533, and 0.9534 to the nearest thousandth, the result is 0.953, since all the values are closer to 0.953 than they are to 0.954. Additionally, the result of rounding the values 0.9535, 0.9536, 0.9537, 0.9538, and 0.9539 is 0.954 as every one of the numbers is closer to 0.954 than to 0.953. Stated otherwise, the value is rounded up if the final digit is 5 or higher.2 A rounded number, according to this rounding rule, is an estimate of the actual amount. Consequently, the rounded number 6.23, which is represented as 6.23, is between 6.225 and 6.235. 6.235 6 … 6.23 6.225 The accuracy, or the difference between 6.225 and 6.235, is 0.010. Greatest potential error (g.p.e.) is a related phrase that is equal to half the precision, or 0.010 x 2 = 0.005. Significant figures hold great significance while working with numerical data. A number's important figures are its digits minus any leading zeros that are required to get the decimal point. 3.69, for instance, contains three significant digits; 36.900, five; 2700, four; 22.0365, six; and 0.00270, three. Leading zeros are not considered important, whereas trailing zeros are. Working with whole numbers becomes challenging due to this rule since the number 300 might have one, two, or three significant figures. Scientific notation can be used to get around this problem. As a result, 3.0 × 102 has two significant figures, 3.00 × 102 has three significant figures, and 3 × 102 has one. Leading zero numbers can be expressed as 2.70 × 10−3 for 0.00270. Counting numbers can contain an infinite number of significant digits. For example, the counting number 65 can have two different writing systems: 65 and 65.000. ......Observing these guidelines will prevent inconsistent responses from qualified staff; yet, discretion may occasionally be needed. Significant figure and rounding decisions can be assessed at the conclusion of a series of calculations. Regardless, the ultimate response can only be as precise as the received information. For instance, the final response must be to two decimal places (x.xx) if the input data is to two decimal places. An explanation of the data The volume of data that has been gathered is enormous in the fields of business, industry, healthcare, education, and government. One item, like a major organization's daily billing mistake rate, might reflect so much data that it can be more confusing than useful. Take a look at the information in Table 1. It is obvious that these data, in their current format, are hard to utilize and inadequate for characterizing the data's properties. To demonstrate what value or values the data tend to cluster around and how the data are distributed or spread out, some method of summarizing the data is required. There are two methods for achieving this data summarization: graphical and analytical. A plot or image representing a frequency distribution, which summarizes how the data points (observations) occur within each subdivision of observed values or groupings of observed values, is the graphical approach. Analytical methods calculate a measure of dispersion and a measure of central tendency in order to summarize data. Both the graphical and analytical methods are applied sometimes. The sections of this chapter that follow will provide descriptions of these strategies. TELEPHONICAL DISTRIBUTION Ungrouped Information Grouped data show how the observed values are grouped together, whereas ungrouped data are a list of the observed values. The data may be continuous, as in the next section, or discrete, as in this one. Unorganized data are essentially worthless, hence some sort of data processing technique is required. The idea will be shown using Table 1. It would be difficult for an analyst examining the data as presented in this table to understand the significance of the information. To gain a more comprehensive understanding, one might calculate the frequency of each value, as indicated in Table 2. Creating an array, or a collection of unprocessed numerical data arranged in either ascending or descending order of magnitude, is the initial stage. In Table 2, the first column displays an array arranged in ascending order from 0 to 5. The next action is to tabulate each value's frequency by inserting a tally mark in the relevant row and beneath the tabulation column. The first set of numbers is 0, 1, 1, 2, etc. ....tally marks of Table 1 and keep doing so until all the data have been tabulated. The frequency, or the number of tallies, is indicated numerically in the final column of Table 2. Table 2 analysis demonstrates that the data distribution may be seen. The table that results when the "Tabulation" column is removed is categorized as a frequency distribution as it displays the frequency of values in each category through data layout. The frequency distribution is a fundamental statistical concept and a helpful tool for data visualization. Understanding that a collection of numbers has a distribution is fundamental to resolving quality-related issues. Frequency distributions come in a variety of forms, and the kind of distribution might reveal the strategy used to solve the problem. When better clarity in visual representation is required, frequency distributions are given graphically. There are several methods for displaying the frequency distribution. A collection of rectangles that show the frequency in each category make up a histogram. The observed values' frequencies are graphically shown. A histogram of the information in Table 2 is shown in Figure 2(a). In theory, a vertical line rather than a rectangle would have been appropriate because this is a discrete variable (see Figure 5). But the rectangle is a widely utilized shape. The relative frequency distribution is a different kind of visual depiction. In this context, relative refers to the percentage or part of the whole. To calculate relative frequency, divide the frequency for each data value (number nonconforming in this example) by the total, which is the sum of the frequencies. The third column of Table 3 displays the results of these computations. In Figure 2(b), a graphic depiction is displayed. One benefit of relative frequency is that it may be referred to. For instance, 0.29 is the percentage of 15 nonconforming units. Instead of using fractions for the vertical scale, some practitioners prefer to utilize percents. The frequency of each data value is added to the total of the frequencies for the preceding data values to determine the cumulative frequency. The cumulative frequency for 0 nonconforming units is 15, as indicated in Table 3's fourth column; for 1 nonconforming unit, it is 15 + 20 = 35; for 2 nonconforming units, it is 35 + 8; and so on. The number of data points that are equal to or less than a data value is known as cumulative frequency. This number, for instance, for two or less nonconforming units is 43. In Figure 2(c), a graphic depiction is displayed. By dividing the cumulative frequency for each data item by the total, one may get the relative cumulative frequency. Table 3's fifth column displays the results of these computations, and Figure 2(d) provides a graphical depiction of the results. According to the graph, 0.83, or 83%, of billing mistakes had two or less nonconforming components. There was just one discrete variable with six values in the example before it. This example covers the frequency distribution idea well enough for a basic introduction, but it doesn't give a comprehensive understanding of the topic. Grouping is necessary since most data are continuous rather than discrete. Combined Data There are often more categories when creating a frequency distribution for aggregated data, which makes the process more difficult. The idea is demonstrated via a continuous variable example issue. 1.Compile information and create a tally sheet. Table 4 displays the weight data that was gathered for 110 steel shafts. As seen in Table 5, the first step is to total the values. The weights are coded from 2.500 kg in order to increase efficiency. This is a data-simplifying strategy. As a result, 2.531 kg (2.500 + 0.031) is the equivalent of a weight of 31. Although Table 5's analysis provides the analyst with more information than Table 4's data did, the overall picture is still a little hazy. There are 45 categories in this issue, which is too many. To limit the number of categories, organize them into cells.3 A cell is a collection of observed values along the horizontal axis of the histogram that falls within a set of boundaries. Although the distribution is shown more simply when the data is grouped by cells, some information is lost. When there are many cells, the distribution's real picture is obscured since some of the cells have none or very few items. Alternatively, a skewed actual representation of the distribution results from an excessive concentration of objects in a small number of cells when the number of cells is minimal. A frequency distribution's cell or group count is mostly determined by the analyst's assessment. The best number of cells may need to be determined by trial and error because this decision is depending on the quantity of observations. Generally speaking, there should be five to twenty cells. Generally speaking, you should utilize 5 to 9 cells when there are less than 100 observations, 8 to 17 cells when there are between 100 and 500 observations, and 15 to 20 cells when there are more than 500 observations. The rules have overlapping numbers of cells to allow for flexibility. It should be noted that these recommendations are flexible and can be changed as needed to provide a frequency distribution that is acceptable. 2.Establish the range. According to the formula, it is the difference between the greatest and lowest observed values. 3.Calculate the spacing between cells. As seen in Figure 3, the cell interval is the separation between neighboring cell midpoints. To ensure that the midpoint values are to the same number of decimal places as the data values, it is advised to choose an odd interval, such as 0.001, 0.07, 0.5, or 3. Using Sturgis' rule is the easiest method. A other method makes use of trial and error. The spacing between cells The formula, h = R>i, relates the number of cells (h) to (i). Since both h and i are unknown, the interval that will satisfy the requirements is found via trial and error. 4.Identify the midpoints of each cell. To include the lowest data value in its cell, the lowest cell mid-point has to be found. Choosing the lowest data point is the easiest method. This formula cannot be used to round the answer. A midpoint value of 2.533 may be utilized for the first cell as there are five data values in each cell due to the interval's 0.005 value. In the first cell, where the data values are 2.531, 2.532, 2.533, 2.534, and 2.535, this value will have the lowest data value (2.531). Choosing a midpoint involves making a subjective decision; in this instance, a midway of 2.533 was chosen, resulting in 9 cells. Even while it wouldn't have been wrong, choosing a different midpoint would have resulted in 10 cells in the frequency distribution. Different frequency distributions can be produced by choosing different midpoint values; in this case, five are conceivable. The cell interval is added to the preceding midpoint to determine the midpoints for the remaining 8 cells: 0.005 + 2.533 = 2.538, 0.005 + 2.538 = 2.543, 2.543 + 0.005 is equivalent to 2.548. ...., and 2.573 = 2.568 + 0.005. These In Table 6, midpoints are displayed. If a cell has a significant number of observations and a relatively small variation in borders, the midpoint value is the most representative value within the cell. The number of observations above and below a cell's midpoint will often be equal, even in the absence of this requirement. Furthermore, even if there is an imbalance in one direction between the number of observations above and below a cell's midpoint, it will most likely be balanced out by an imbalance in the other direction of another cell. The precision of the midpoint values ought to match that of the first observations. 5.Establish the borders of the cells. Cell borders, often known as the upper and lower limits, are the extreme or limit values of a cell. That specific cell contains all observations that lie between the upper and lower limits. Set boundaries to ensure that the location of an observation is unambiguous. As a result, the boundary values are more accurate than the observed values by one significant figure or additional decimal point. There will be an equal amount of data values on either side of the midpoint due to the odd interval. There will be two values on each side for the first cell, which has a midpoint of 2.533 and an interval of 0.005. Consequently, the numbers 2.531, 2.532, 2.533, 2.534, and 2.535 will be present in that cell. The genuine borders are stretched almost halfway to the following number in order to prevent any gaps, yielding values of 2.5305 and 2.5355. The following number line serves as an example of this idea: The borders should be kept at the same number of decimal places as the data, according to some researchers. As long as the cell interval is odd and it is acknowledged that the genuine borders are extended halfway to the next number, there should be no problems with this technique. This book and Excel both follow this procedure. Table 6 displays it. As a result, 2.531 is the first cell's bottom limit. The borders for the remaining cells are generated by adding the cell interval one after the other until the boundaries for one cell are formed. Consequently, 2.531 + 0.005 = 2.536 and 2.536 + 0.005 = 2.541 are the lower bounds. ...., + 0.005 * 2.566 = 2.571. The upper bounds are acquired. similarly, and are displayed in the top column of Table 6. 6.following the cell frequency. Each cell's number count is posted to Table 6's frequency column. Table 5 examination reveals that there are one 2.531, two 2.532, one 2.533, two 2.534, and zero 2.535 in the lowest cell. Consequently, the lowest cell has a total of six values, and the cell with a 2.533 midpoint has a frequency of six. The quantities are calculated similarly for the other cells. Table 6 displays the final frequency distribution. Compared to disorganized data or a tally sheet, this frequency distribution provides a clearer understanding of the core value and how the data are distributed around that value. In Figure 4, the histogram is displayed. With one exception, the information on the creation of the relative frequency, cumulative frequency, and relative cumulative frequency histograms for grouped data is identical to that for ungrouped data. The number indicated on the abscissa is the genuine upper cell boundary for the two cumulative frequency histograms. The reader is left to construct these histograms for the example problem as an exercise. Different Kinds of Graphs for Frequency Distribution With the information from Table 1, the bar graph may also depict frequency distributions, as seen in Figure 5(a). The bar graph is rarely often utilized, although being technically accurate for discrete data, as was previously mentioned. Another graphical method for illustrating frequency distributions is the polygon, often known as the frequency polygon, which is demonstrated in Figure 5(b) using data from Table 6. To create it, place a dot at the height specified for each frequency above the midway of each cell. To encompass the figure, the arc is stretched at both ends. The most often used graphical representation is the histogram, which is thought to offer a better image than the polygon as it displays the area in each cell. A cumulative frequency, often known as an ogive, is a graph that shows the frequency of all values that are smaller than the top cell border of a certain cell. A cumulative frequency distribution curve for the data in Table 6 is displayed in Figure 5(c). On the graph, the cumulative value for every cell is shown and connected by a straight line. Except for the first cell, which also contains the genuine lower boundary, all of the true upper and lower cell boundaries are identified on the abscissa. Features of Graphs with Frequency Distributions Smooth curves are used in Figure 6's graphs in place of the rectangular forms connected to the histogram. A sample frequency distribution is shown by a histogram, whereas a population frequency distribution is represented by a smooth curve. This chapter's later portion covers the distinction between a population and a sample. There are several features that make frequency distribution curves recognizable. One feature of the distribution pertains to the data's symmetry or lack thereof. On either side of the center value, are the data dispersed evenly, or Which way is the data skewed—left or right? The quantity of peaks or modes in the data is another characteristic. One mode, two modes (bimodal), or several modes are possible. One last feature relates to how "peaked" the data is. Leptokurtic refers to a curve that is extremely peaked, whereas platykurtic refers to a curve that is flatter. Without additional study, frequency distributions can reveal enough details about a quality control issue to serve as a foundation for decision-making. As shown in Figure 7, distributions can also be compared with respect to location, dispersion, and form.