6-1 Numerical Summaries Definition: Sample Mean 6-1 Numerical Summaries Example 6-1 6-1 Numerical Summaries Fulcrum The sample mean as a balance point for a system of weights. 6-1 Numerical Summaries Population Mean For a finite population with N measurements, the mean is The sample mean is a reasonable estimate of the population mean. 6-1 Numerical Summaries Definition: Sample Variance Why (n-1)? (1) xi tend to be closer to the sample mean than the true mean . To compensate for that and avoid obtaining a variance that's often smaller than the true variance 2 we use (n-1). (2) The number of degrees of freedom of the sum is (n-1). 6-1 Numerical Summaries Example 6-2 The table below displays the quantities needed for calculating the sample variance and sample standard deviation for the pull-off force data. 6-1 Numerical Summaries Example 6-2 Computation of s2 6-1 Numerical Summaries Population Variance When the population is finite and consists of N values, we may define the population variance as The sample variance is a reasonable estimate of the population variance. Definition: Sample Range 6-2 Stem-and-Leaf Diagrams A stem-and-leaf diagram is a good way to obtain an informative visual of a data set x1, x2, ..., xn, where each number xi consists of at least two digits. To construct a stem-and-leaf diagram, use the following steps: 6-2 Stem-and-Leaf Diagrams Data of compressive strengths in pounds per square inch of 80 specimens of a new aluminum-lithium alloy under going evaluation As a possible material for aircraft structural elements. 6-2 Stem-and-Leaf Diagrams Stem-and-leaf diagram for the compressive strength data in Table 6-2. Units: Stem: Tens and hundreds (psi) Leaf: Ones (psi) 6-2 Stem-and-Leaf Diagrams The last column is the frequency count of the number of leaves associated with each stem. Inspection of this display reveals that most of the compressive strength lie between 110 and 200 psi and that a central value is somewhere between 150 and 160. The stem-and-leaf diagram enables us to determine quickly some features of the data that were not immediately obvious in the original display in the table. 6-2 Stem-and-Leaf Diagrams Example 6-5: Too few stems, moderate number of stems, or too many? It does not provide much information 6-2 Stem-and-Leaf Diagrams Example 6-5: Too few stems, moderate number of stems, or too many? L: 0,1,2,3, & 4 U: 5,6,7,8, & 9 It provides more information 6-2 Stem-and-Leaf Diagrams Example 6-5: Too few stems, moderate number of stems, or too many? z (0,1), t (2,3), f(4,5), s(6,7), e (8,9) It does not tell much about the shape of the data. 6-2 Stem-and-Leaf Diagrams Ordered Stem-andleaf diagram produced by Minitab software. Median: measure the central tendency 40th and 41st values (160+163)/2 =161.5 Sample Mode: most frequent occurring Sample =158 6-2 Stem-and-Leaf Diagrams Data Features The median is a measure of central tendency that divides the data into two equal parts, half below the median and half above. If the number of observations is even, the median is halfway between the two central values. In previous slide, the 40th and 41st values of strength as 160 and 163, so the median is (160 + 163)/2 = 161.5. If the number of observations is odd, the median is the central value. 6-2 Stem-and-Leaf Diagrams Data Features The range is a measure of variability that can be easily computed from the ordered stem-and-leaf display. It is the maximum minus the minimum measurement. From previous slide, the range is 245 - 76 = 169. 6-2 Stem-and-Leaf Diagrams Data Features When an ordered set of data is divided into four equal parts, the division points are called quartiles. The first or lower quartile, q1 , is a value that has approximately onefourth (25%) of the observations below it and approximately 75% of the observations above. The second quartile, q2, has approximately one-half (50%) of the observations below its value. The second quartile is exactly equal to the median. The third or upper quartile, q3, has approximately three-fourths (75%) of the observations below its value. As in the case of the median, the quartiles may not be unique. 6-2 Stem-and-Leaf Diagrams Data Features The compressive strength data in Figure 6-6 contains n = 80 observations. Minitab software calculates the first and third quartiles as the (n + 1)/4 and 3(n + 1)/4 ordered observations and interpolates as needed. For example, (80 + 1)/4 = 20.25 and 3(80 + 1)/4 = 60.75. Therefore, Minitab interpolates between the 20th (143) and 21st (145) ordered observation to obtain q1 = 143.50 and between the 60th and 61st observation to obtain q3 =181.00. 6-2 Stem-and-Leaf Diagrams Data Features The interquartile range (IQR) is the difference between the upper and lower quartiles (q3,q1), and it is sometimes used as a measure of variability. IQR = q3-q1 In general, the 100kth percentile is a data value such that approximately 100k% of the observations are at or below this value and approximately 100(1 - k)% of them are above it. k takes any value in the range 0.0-1.0 6-3 Frequency Distributions and Histograms A frequency distribution is a more compact summary of data than a stem-and-leaf diagram. To construct a frequency distribution, we must divide the range of the data into intervals, which are usually called class intervals, cells, or bins. number of bins is usually between 5 and 20. the number of bins increases as the number of observations increases. Rule of thumb: number of bins = (n)0.5. 6-3 Frequency Distributions and Histograms Histogram of compressive strength for 80 aluminum-lithium alloy specimens. It looks like a bell shape!! 6-3 Frequency Distributions and Histograms A histogram of the compressive strength data from Minitab with 17 bins. It is recommended to use the histograms if the number of observations is more than 75. 6-3 Frequency Distributions and Histograms A histogram of the compressive strength data from Minitab with nine bins. 6-3 Frequency Distributions and Histograms A cumulative distribution plot of the compressive strength data from Minitab. 6-3 Frequency Distributions and Histograms ~ : Median x x : Mean Negative or left skew Symmetric Positive or right skew mode >median>mean mode =median=mean mode