Normal Distribution and The Empirical Rule Introduction to Summary Statistics Introduction to Engineering Design Unit 3 – Measurement and Statistics Normal Distribution and The Empirical Rule Get out your notes. Open Note Unit 3 Test on Friday.

Learning Objectives Be able to recognize a normal distribution curve Be able to use the Empirical Rule to predict measurements that are normally distributed. Apply the Empirical Rule to predicting the quality of measured blocks.

How can you determine if it is a Normal Distribution? Review How can you determine if it is a Normal Distribution? Bell Shaped Curve Peaks at the mean Curve decreases on both sides away from the mean Curve is symmetric. Looks the same on both side.

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   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.   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   Data Elements

Example  

Example Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Example   [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.]

Review What is the Empirical Rule? Given a mean value of 10 with a standard deviation of 2: What range of values would 68% of the data fall in? What range of values would 95% of the data fall in? What range of values would 99.7% of the data fall in? Can you apply the Empirical rule to all sets of data?

Tasks Friday’s Test is Open Notes (Your notebooks), not internet/cell phone/Smith’s Website Complete Worksheet 3.3 (the cube measuring worksheet) When finished you can check past PowerPoints and add information to your notes to be used on Friday’s exam. Test Topics US/SI Measurements Unit Conversions Using Dial Calipers Statistics: Mean, Median, Mode, Standard Deviation, Histogram Using Excel to calculate statistics Normal Curves The Empirical Rule Practice Test