Introduction to Summary Statistics Introduction to Engineering Design Unit 3 – Measurement and Statistics The Empirical Rule (Normal Distributions) Introduction to Engineering Design © 2012 Project Lead The Way, Inc.

Introduction to Summary Statistics Empirical Rule Applies to normal distributions Almost all data will fall within three standard deviations of the mean A normal distribution has a bell shaped probability density function and is informally known as a bell curve. This image shows a graph of examples of the density functions for several normal distributions. To determine whether a data is distributed normally, take a look at the histogram or dot plot of the data. [3 clicks]

Introduction to Summary Statistics Empirical Rule If the data are normally distributed: 68% of the observations fall within 1 standard deviation of the mean. 95% of the observations fall within 2 standard deviations of the mean. 99.7% of the observations fall within 3 standard deviations of the mean. Many quantities tend to follow a normal distribution – heights of people, test scores, errors in measurement, etc. Given normally distributed data, 68% of the data values should fall within 1 standard deviation of the mean, 95% should fall within 2 standard deviations of the mean and 99.7 % should fall within 3 standard deviations of the mean. This is referred to as the Empirical Rule. Of course, with small samples/populations, these percentages may not hold exactly true because the number of values will not allow divisions to this precision.

Empirical Rule Example Introduction to Summary Statistics Empirical Rule Example Data from a sample of a larger population Mean = x = 0.08 Standard Deviation = s = 1.77 (sample) Let’s assume that this data was gathered from a sample taken from a larger population. Assume that we are interested in finding statistics for the larger population. The sample data values are fairly evenly distributed about the mean. Approximately half of the values that are not mean values are less than the mean and approximately half are greater than the mean. And, the frequency of occurrence decreases as the value of the data point moves farther away from the mean. The data appears to form a bell shaped curve. [click] This data set looks to be normally distributed. The mean is 0.08. The sample standard deviation formula is used to estimate the standard deviation of the larger population and is found to be 1.77.

Introduction to Summary Statistics Normal Distribution 0.08 + 1.77 = 1.88 0.08 + - 1.77 = -1.69 68 % s -1.77 s +1.77 Since the data appears to be normally distributed, we can estimate that approximately 68% of the population data will fall within one standard deviation of the mean. [4 clicks] That is, about two thirds of the data will be between 1.69 and 1.88. x 0.08 Data Elements

Introduction to Summary Statistics Normal Distribution 0.08 + -3.54 = - 3.46 0.08 + 3.54 = 3.62 95 % And, again, because the data is assumed to be normally distributed, we can estimate that 95% of the population data will fall within 2 standard deviations of the mean. That is, approximately 95% of the data will fall between 3.46 and 3.62 2s - 3.54 2s + 3.54 x 0.08 Data Elements

Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Your Turn Revisit the data you collected during the Fling Machine Instant Challenge. Assume that you repeated launch cotton balls with your device. Using the mean and sample standard deviation of your data: Predict the range of travel distances within which 68% of cotton balls would fall Predict the range of travel distances within which 95% of cotton balls would fall Allow students to apply the Empirical Rule to the data they collected for their Fling Machine. Example calculations are presented on the following slides.

Example Assume that a statistical analysis resulted in the following: Mean = x = 2.35 ft. Sample standard deviation = s = 0.76 ft Predict the range of travel distances within which 68% of cotton balls would fall x ± s : 2.35 - 0.76 = 1.59 ft 2.35 + 0.76 = 3.26 ft Prediction: Approximately 68% of the launches will result in a travel distance between 1.59 ft and 3.26 ft.

Example Assume that a statistical analysis resulted in the following: Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Example Assume that a statistical analysis resulted in the following: Mean = x = 2.35 ft. Sample standard deviation = s = 0.76 ft Predict the range of travel distances within which 95% of cotton balls would fall x ± 2s : 2.35 – 2(0.76) = 0.83 ft 2.35 + 2(0.76) = 3.87ft Prediction: Approximately 95% of the launches will result in a travel distance between 0.83 ft and 3.86 ft. [Students should calculate the travel distance ranges and compare their results to those of others on their Instant Challenge team.] [Ask students how the range they predicted in the Conclusion questions of the Instant Challenge compare to the calculations they made here.]