Understanding Standard Deviation Copyright © 2010 Raytheon Company. All rights reserved. Customer Success Is Our Mission is a registered trademark of Raytheon Company.

Page 2 Introduction to Variation There is variation in EVERYTHING! – If you order 25 pepperoni pizzas, does each slice have the same amount of pepperoni, cheese and sauce? – If you try on 10 pairs of jeans, the same brand, the same style, do they all fit the same? – When students take a physics exam, do they all get the same grade? Variation in the pepperoni count isn’t a big deal, but what happens if there is a lot of variation in – Potency of medicine? – Airplanes performance during landings? – Concrete quality in bridges and buildings?

Page 3 Introduction to Variation In order to understand a process or product’s performance, you have to understand the variation in the process or products Once you understand the variation, then you can focus on reducing it, which can save time, materials, and / or money

Page 4 Reminder: Mean One way to understand a data set is by examining the mean The mean is often called the average of the data set. Think about your grade point average in school. Some grades are higher, some are lower, and the mean is somewhere in between, based on the following calculation: Mean = X 1 + X 2 + …. X N Where N = the # of grades contributing to the average N So, if you had class grades of: 78, 95, 100, 67 (what?!), 82, 90, and 89, the mean = =

Page 5 Reminder: Median Another way to understand a data set is by examining the median The median is the central value of the data set. So half of the data is greater than the median and half of the data is less than the median value. So, if you had class grades of: 78, 95, 100, 67, 82, 90, and 89, the median would be: 89 67, 78, 82, 89, 90, 95, 100

Page 6 We need more than the Mean and Median! While the mean and median give useful insight into the data, they do not give us the complete picture. Two sets of data can have the same mean and median, but be dramatically different. If the two histograms above represented the time it took using two different routes to get to school in the morning, and you absolutely had to be on time for the free breakfast pizza (yes, pizza), which process would you prefer to use and why? The means and medians are the same Frequency Minutes to get to school, option A Minutes to get to school, option B

Page 7 Standard Deviation In order to understand the distribution or spread of the data around the mean, the Standard Deviation is calculated. Data Set A has a smaller Standard Deviation than Data Set B, as it is grouped tighter around the mean of the data set Product Quality, option A Product Quality, option B

Page 8 Standard Deviation Standard Deviation is thought of as the average distance of data points from the mean, and is calculated as follows: Standard Deviation = Where N = the number of elements in the data set, and X = the mean of the data set i = 1 ( x i – x ) 2 N 1 N

Page 9 Standard Deviation Standard Deviation for our set of grades is: Standard Deviation = = = 10.3 i = 1 ( x i – x ) 2 N 1 N ((67 – 85.9) 2 + (78 – 85.9) 2 + (82 – 85.9) 2 + (89 – 85.9) 2 + (90 – 85.9) 2 + (95 – 85.9) 2 + (100 – 85.9) 2 ) 7

Page 10 Standard Deviation If you have an understanding teacher, who lets you retake the test where you got a 67, and your new grade ends up being a 79, how does that impact the Standard Deviation? New Data Set: 78, 95, 100, 79, 82, 90, and 89 New Mean: 87.6 New Standard Deviation: 7.9 So, the Standard Deviation decreased from 10.3 to 7.9. A smaller Standard Deviation means tighter distribution (less variation) in the data.