Analyzing Data - Data Characteristics

DATA CHARACTERISTICS • Mean or average or the arithmetic average. It is an indication of the expected behavior of a data set. • median is a value such that half the data values lie above and half lie below. • mode is the value that occurs with the greatest frequency within a data set. • min and the max simply indicate the extremities of the data set. • variance provides an indication of the degree of spread in the data. The greater the variance, the greater the spread. • standard deviation also provides a measure of spread in the data set. It is simply the square root of the variance. Analysis ToolPak • Excel allows you to generate a summary of all applicable descriptive statistics for a given data set. • This feature requires the Descriptive Statistics feature found in the Analysis ToolPak. • Figures 6.1-6.4 Example 6.1 Analyzing a Data Set • A car manufacturer wishes to determine how accurately the cylinders are being machined in several engine blocks. The design specifications call for a cylinder diameter of 3.500 inches, with a tolerance of ±0.005 inch. • To determine the accuracy of the cylinders, several engine blocks were taken from the assembly line during manufacture and one cylinder was measured in each block. For consistency, the measurement was always perpendicular to the axis of the engine block (i.e., perpendicular to the straight line connecting the centers of the cylinders). • Analyze the data by placing them in an Excel worksheet and then calculating the mean, median, mode, min, max, and standard deviation. Also, generate a table showing all summary statistics for the given data. The data obtained for 20 engine blocks, selected at random, are shown in the following table. Note that four of the individual values exceed the allowable tolerance of ±0.005 inch (three above, one below). summary statistics • the standard error is a measure of the amount of error in a predicted y-value corresponding to a given x-value, based upon a curve fit of the data • The skewness is a measure of the asymmetry of the data with respect to the mean, and • the kurtosis is a measure of “flatness” or “pointedness” of the data distribution. HISTOGRAMS • to plot the data in a manner that illustrates how the values are distributed within their range. • This type of plot is called a histogram, or a relative frequency plot. • From the relative frequency plot, we can obtain a plot of the cumulative distribution, which allows us to estimate the likelihood that a data value associated with an item drawn at random is less than or greater than some specified value. Histogram Fundamentals • first subdivide the range of the data into a series of adjacent, equally spaced intervals. • The first interval must begin at or below the smallest data value (the min), and the last interval must extend to or beyond the largest data value (the max). • Each interval will have some lower bound, xi, and an upper bound, xi+1, where xi+1 = xi + Δx and Δx represents the fixed interval width. • These intervals are sometimes referred to as class intervals. • determine how many data values fall within each interval. The relative frequencies are then obtained as • where fi represents the relative frequency for the ith interval, ni represents the number of data values in the ith interval, and n represents the total number of data values. Note: if k is the number of intervals, then • The histogram is usually expressed as a bar graph showing the values of the interval counts (ni) or the relative frequencies ( fi). Example 6.2 Constructing a Histogram • A car manufacturer wishes to determine how accurately the cylinders are being machined in several engine blocks. The design specifications call for a cylinder diameter of 3.500 inches, with a tolerance of ±0.005 inch. • Choose 10 equally spaced intervals ranging from x = 3.490 to x = 3.510 inches. If a data value falls on an interval boundary, assign the data value to the lower interval. The data obtained for 20 engine blocks, selected at random, are shown in the following table. Note that four of the individual values exceed the allowable tolerance of ±0.005 inch (three above, one below). The first interval will extend from 3.490 to 3.492 inches. The second interval will extend from 3.492 to 3.494 inches, though it will actually include only those values that exceed 3.492 inches because a value of exactly 3.492 inches will be assigned to the first interval, and so on. For simplicity, however, we will write the interval bounds to four significant figures. Histogram Generation in Excel using the Histogram feature included in the Data Analysis Tools (Step 1) • Enter the basic data and the interval bounds into the worksheet. (The intervals are called bins in Excel.) • Typically, the data are entered in a single column (or a single row), and the interval bounds are entered in another column (or another row). • Only the right interval bounds are entered. • Hence, the last interval (rightmost interval) will be open-ended; that is, it will contain all data values that exceed the largest bound. Example 6.3 Generating a Histogram in Excel select Data Analysis/Histogram from the Analyze group within the Ribbon’s Data tab. This results in the Histogram dialog box. Note that the intervals in column I are labeled Bin, and the interval values in column J are labeled Frequency. The values tabulated in the Frequency column are integer sums, not the relative frequencies A histogram is customarily plotted as a bar graph (i.e., an Excel Column Chart). To create a bar graph automatically, simply select Chart Output at the bottom of the Histogram dialog box Representing the histogram as a bar graph • differs from a traditional histogram in two ways. First, the numerical labels shown beneath every interval (the bin values) actually correspond to the interval right boundaries (which are shown as tick marks) rather than the intervals themselves • The other difference concerns the manner in which the vertical bars are drawn within the bar graph. Because the vertical bars represent contiguous intervals, they should be drawn adjacent to each other, without any intervening space. e intervals themselves. Right-click on any of the vertical bars and choose Format Data Series from the drop-down menu. Legend removed, edited labels, title, border around vertical bars As a rule, a choice of 10 to 15 intervals works well with many data sets. Fewer intervals might be better, however, if the number of data values (n) is relatively small. If n is large, the square root of n often provides a useful starting point. CUMULATIVE DISTRIBUTIONS • to determine the likelihood that a particular value drawn at random is less than or greater than some specified value. • For example, we might wish to determine the likelihood that a cylinder diameter within a randomly selected engine block is greater than some specified value—say, 3.500 inches. • first construct a histogram and determine the relative frequency associated with each of the intervals • then determine the following cumulative values: • Note that the last term, Fk, will always equal 1 Example 6.4 Constructing a Cumulative Distribution Example 6.5 Generating a Cumulative Distribution in Excel • go to the Ribbon’s Data tab, and select Data Analysis/Histogram from the Analyze group. • When the Histogram dialog box appears, we select both the Cumulative Percentage and Chart Output features at the bottom of the dialog box Reading • Example 6.5 Generating a Cumulative Distribution in Excel • Example 6.6 Plotting the Cumulative Distribution • Example 6.7 Drawing Inferences from the Cumulative Distribution