Engineering Probability and Statistics - SE-205 -Chap 6 By S. O. Duffuaa

Lecture Objectives Sample and population Random sample Type of data summarizes Numerical summarizes Diagrams and Tables Graphical summarizes

Numerical Summarizes Sample mean X-bar Population mean µ Sample variance S2 Population variance σ2

Range: Max X – Min X COV = S/(X-bar)

Objective of the Lecture Sample and population Random sample Stem-leaf diagram The frequency distribution Histogram

Population and Sample Population is the totality of observations we are concerned with. Example: All Engineers in the Kingdom, All SE students etc. Sample : Subset of the population 50 Engineers selected at random, 10 SE students selected at random.

Stem-And –Leaf Diagram Each number xi is divided into two parts the stem consisting of one or two leading digits The rest of the digits constitute the leaf. Example if the data is 126 then 12 is stem and 6 is the leaf. What is the stem and leaf for 76

Data Table 1.1 Compressive Strength of 80 Aluminum Lithium Alloy 105 221 183 186 121 181 180 143 97 154 153 174 120 168 167 141 245 228 174 199 181 158 176 110 163 131 154 115 160 208 158 133 207 180 190 193 194 133 156 123 134 178 76 167 184 135 229 146 218 157 101 171 165 172 158 169 199 151 142 163 145 171 148 158 160 175 149 87 160 237 150 135 196 201 200 176 150 170 118 149

Stem-And-Leaf f Stem leaf frequency 7 6 1 8 7 1 9 7 1 10 5 1 2 7 6 1 8 7 1 9 7 1 10 5 1 2 11 5 0 8 3 12 1 0 3 3 13 4 1 3 5 3 5 6 14 2 9 5 8 3 1 6 9 8 15 4 7 1 3 4 0 8 8 6 8 0 8 12 16 3 0 7 3 0 5 0 8 7 9 10 17 8 5 4 4 1 6 2 1 0 6 10 18 0 3 6 1 4 1 0 7 19 9 6 0 9 3 4 6 20 7 1 0 8 4 21 8 1 22 1 8 9 3 23 7 1 24 5 1

Number of Stems Considerations Stem Leaf 6 1 3 4 5 5 6 7 0 1 1 3 5 7 8 8 9 8 1 3 4 4 7 8 8 9 2 3 5

Stem number considerations Stem leaf 6L 1 3 4 6U 5 5 6 7L 0 1 1 3 7U 5 7 8 8 8L 1 3 4 4 8U 7 8 8 9L 2 3 9U 5

Number of Stems Between 20 and 5 Roughly n where n number of data points

Percentiles Pth percentile of the data is a value where at least P% of the data takes on this value or less and at least (1-P)% of the data takes on this value or more. Median is 50th percentile. ( Q2) First quartile Q1 is the 25th percentile. Third quartile Q3 is the 75th percentile.

Percentile Computation : Example Data : 5, 7, 25, 10, 22, 13, 15, 27, 45, 18, 3, 30 Compute 90th percentile. 1. Sort the data from smallest to largest 3, 5, 7, 10, 13, 15, 18, 22, 25, 27, 30, 45 2. Multiply 90/100 x 12 = 10.8 round it to to the next integer which is 11. Therefore the 90th percentile is point # 11 which is 30.

Percentile Computation : Example If the product of the percent with the number of the data came out to be a number. Then the percentile is the average of the data point corresponding to this number and the data point corresponding to the next number. Quartiles computation is similar to the percentiles.

Inter-quartile range Q3 – Q1

Cumulative Relative Frequency   Class Interval (psi) Tally Frequency Relative Frequency Cumulative Relative Frequency 70 ≤ x < 90 || 2 0.0250 90 ≤ x < 110 ||| 3 0.0375 0.0625 110 ≤ x < 130 |||| | 6 0.0750 0.1375 130 ≤ x < 150 |||| |||| |||| 14 0.1750 0.3125 150 ≤ x < 170 |||| |||| |||| |||| || 22 0.2750 0.5875 170 ≤ x <1 90 |||| |||| |||| || 17 0.2125 0.8000 190 ≤ x < 210 |||| |||| 10 0.1250 0.9250 210 ≤ x < 230 |||| 4 0.0500 0.9750 230 ≤ x < 250 1.0000

2 5 20 15 Frequency 10 5 70 90 110 130 150 170 190 210 230 250 Compressive Strength ( psi )

100 150 200 250

Whisker extends to smallest data point within 1 Whisker extends to smallest data point within 1.5 interquartile ranges from first quartile Whisker extends to largest data point within 1.5 interquartile ranges from third quartile Third Quartile First Quartile Second Quartile Outliers Extreme Outliers Outliers 1.5 IQR 1.5 IQR IQR 1.5 IQR 1.5 IQR

100 150 200 250 Strength

120 110 100 Quantity Index 90 80 70 1 2 3 Plant