1. Basic Statistics Foundations of Technology Basic Statistics © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology Teacher Resource – Unit 2 Lesson 2

2. The BIG Idea Big Idea: Computers assist in organizing and analyzing data used in the Engineering Design Process. © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

3. Basic Statistics The Mean is the average of a given data set: x = represents the data set ∑ = the sum of a mathematical operation n = the total number of variables in the data set Equation for Mean = ∑x n © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

4. Practice Questions What is the mean for the following data set? 1, 4, 4, 6, 7, 8, 10 Equation for Mean = ∑x n © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

5. Practice Questions What is the mean for the following data set? 1, 4, 4, 6, 7, 8, 12 ∑x = 1 + 4 + 4 + 6 + 7 + 8 + 12 ∑x = 42 n 7 Mean = 6 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

6. Basic Statistics The Median is the middle number in a given ordered data set. Example: 1, 2, 3, 4, 4 If the given data set has an even number of data, the Median is the average of the two center data. Example: (1, 2, 4, 4) Median = (2+4) = 6 = 3 2 2 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

7. Practice Questions What is the median for the following data set? 1, 6, 12, 4, 4, 8, 7 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

8. Practice Questions What is the median for the following data set? 1, 6, 12, 4, 4, 8, 7 Ordered Data Set = 1, 4, 4, 6, 7, 8, 12 Median = 1, 4, 4, 6, 7, 8, 12 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

9. Practice Questions What is the median for the following data set? 1, 6, 12, 4, 4, 7 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

10. Practice Questions What is the median for the following data set? 1, 6, 12, 4, 4, 7 Ordered Data Set = 1, 4, 4, 6, 7, 12 Middle Numbers = 4, 6 = (4+6) = 10 = 5 2 2 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

11. Basic Statistics The Mode is the most frequently occurring number in a given data set. Example: 1, 2, 3, 4, 4 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

12. Practice Questions What is the mode for the following data set? 1, 6, 12, 4, 4, 8, 7 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

13. Practice Questions What is the mode for the following data set? 1, 6, 12, 4, 4, 8, 7 Mode = 4 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

14. Basic Statistics Standard Deviation Standard Deviation (SD) is a UNIT. It is used to measure how "weird" something is. Inches measure distance. Grams measure mass. SD's measure weirdness. Things that equal the norm measure 0 standard deviations. Things above the norm register with positive standard deviations. Things below the norm register with negative standard deviations. © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

15. Basic Statistics Understanding standard deviation: When you construct something you are going to have error. Let's say I have a relatively skilled crew build 100 bridge pilings according to a certain specification. There is going to be some wiggle in the height of the piling. A few mm taller a few mm shorter on each build. Now the mean of these builds should be REALLY REALLY close to what is on the specification. From this set of builds you can determine a standard deviation. © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

16. Basic Statistics Understanding standard deviation: You can now set a tolerance for what a good build is. In some circumstances being within 2 SD of the norm is good enough. If these pilings are holding up a bridge that needs to carry the space shuttle to the launch pad you may want the piling to be to 1/2 of a standard deviation (space shuttles don't deal well with bumps). When a new piling is built you measure it, convert the height to SD and then decide whether it’s in its tolerance. If it is, great! If it isn't you tear down and rebuild the piling. © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology Click on graph to learn more about Standard Deviation.

17. Basic Statistics Calculating Standard Deviation Equation for Standard Deviation = ∑(x i – μ)² √ n - 1 x i = represents the individual data μ = represents the mean of the data set ∑ = the sum of a mathematical operation n = the total number of variables in the data set © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

18. Basic Statistics What is the standard deviation for the following data set? 1, 4, 4, 6, 7, 8, 12 Equation for Standard Deviation = ∑(x i – μ)² √ n - 1 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

19. Practice Questions What is the standard deviation for the following data set? (1, 4, 4, 6, 7, 8, 12) ∑(x i – μ)² √ n - 1 The mean for the data set is 6, therefore μ = 6. ∑(x i – μ)² = ∑(1 – 6)² + (4 – 6)² + (4 – 6)² + (6 – 6)² + (7 – 6)² + (8 – 6)² + (12 – 6)² = ∑(-5)² + (-2)² + (-2)² + (0)² + (1)² + (2)² + (6)² = ∑(25) + (4) + (4) + (0) + (1) + (4) + (36) = 74 ∑(x i – μ)² = 74 = 74 = 12.3 = 3.51 √ n – 1 √ 7 - 1 √ 6 √ © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

20. Basic Statistics The Range is the distribution of the data set or the difference between the largest and smallest values in a data set. Example: 1, 2, 3, 4, 4 Largest Value = 4 and the Smallest Value = 1 Range = (4 – 1) = 3 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

21. Practice Questions What is the range for the following data set? 1, 4, 4, 6, 7, 8, 12 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

22. Practice Questions What is the range for the following data set? 1, 4, 4, 6, 7, 8, 12 Largest Value = 12 and the Smallest Value = 1 Range = (12 – 1) = 11 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

23. Basic Statistics Engineering tolerance is the amount a characteristic can vary without compromising the overall function or design of the product. Tolerances generally apply to the following: Physical dimensions (part and/or fastener) Physical properties (materials, services, systems) Calculated values (temperature, packaging) © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology

24. Basic Statistics Engineering tolerances are expressed like a written language and follow the American National Standards Institute (ANSI) standards. Example: Bilateral Tolerance (1.125 0.025) Example: Unilateral Tolerance (2.575 ) Upper and lower specification limit are derived from the acceptable tolerance. Bilateral and Unilateral are just two examples of how tolerance is expressed using ANSI. © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology +0.005 - 0.005 + –

25. Practice Questions What are the upper and lower specification limit for the examples below? Example: Bilateral Tolerance (1.125 0.025) Example: Unilateral Tolerance (2.575 ) © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology +0.005 - 0.005 + –

26. Practice Questions What are the upper and lower specification limit for the examples below? Example: Bilateral Tolerance (1.125 0.025) Upper Specification Limit = 1.125 + 0.025 = 1.150 Lower Specification Limit = 1.125 – 0.025 = 1.100 The Range should equal the difference between the upper and lower specification limit. Range = 0.050 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology + –

27. Practice Questions What are the upper and lower specification limit for the examples below? Example: Unilateral Tolerance (2.575 ) Upper Specification Limit = 2.575 + 0.005 = 2.580 Lower Specification Limit = 2.575 – 0.005 = 2.570 The Range should equal the difference between the upper and lower specification limit. Range = 0.010 © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology +0.005 - 0.005

28. Resources: Pierce, Rod. (15 Jan 2014). "Standard Deviation and Variance". Math Is Fun. Retrieved 3 Jul 2014 from http://www.mathsisfun.com/data/standard- deviation.html © 2013 International Technology and Engineering Educators Association, STEM  Center for Teaching and Learning™ Foundations of Technology