Freshman Engineering Clinic II Statistics II Freshman Engineering Clinic II

Course Reminders & Deadlines Pathfinder Before exercises (on Intellectual Property) due by 10:30 am Wed. March 1st 3D Game Lab 2nd deadline of 700 XP midnight Fri. March 10th Heart Lung Project Re-write of Literature Review due by Mon. February 27th

Review of Last Class – Key Concepts Statistics I Area under the bell curve always equals 1 More similar your data the larger the peak of the bell curve Z-statistic is used to determine probabilities with normally distributed populations 𝑍= 𝑋−𝜇 𝜎

Review of Last Class – Example Problem Porphyrin is a pigment in blood protoplasm and other body fluids that is significant in body energy and storage. In healthy Alaskan brown bears, the amount of porphyrin in the bloodstream (in mg/dl) has approximate normal distribution with a mean of 38 mg/dl and a standard deviation of 12. What proportion of these bears have between 27.5 and 67.5 mg/dl porphyrin in their bloodstream? Z1 = (67.5-38)/12 = 2.46; Probability from Z-table = 0.9931 Z2 = (27.5-38)/12 = -0.875; Probability from Z-table = 0.1908 Final answer = 0.9931-0.1908 = 0.8023

Class Overview Mean, Median, Variance, Standard Deviation, Standard Error 95% Confidence Interval (C.I.) Error Bars Comparing Means of Two Data Sets

Basic Stats Review (NTU) 1 3 6 8 10 Using the data to the left, calculate the following: (NTU) 1 3 6 8 10 Mean Median Variance Standard Deviation Standard Error

95% Confidence Interval (C.I.) for Mean A 95% C.I. is expected to contain the population mean 95 % of the time (from 100 samples, 95 will contain population mean if expressed as ) t95%,n-1 is a statistic for 95% C.I. from sample of size n In EXCEL: t95%,n-1 = TINV(0.05,n-1) Where 0.05 = (100-95)/100 & n = sample size For a sample size of 6, t95%,5 = TINV(0.05, 5) = 2.57 If n ≥ 30, then t95%,n-1 ≈ 1.96 (Normal Distribution)

Mean Confidence Interval (NTU) 1 3 6 8 10 2.57*1.4 NTU = 5.2 NTU 3.6 NTU Note: 95% confidence intervals is typically larger than +/- standard error interval

Determine the 95% C.I. for each filter Filter Example Determine the 95% C.I. for each filter Turbidity Data Test 1 Test 2 Test 3 NTU Filter 1 2.1 2.2 Filter 2 3.2 4.4 5 Filter 3 4.3 4.2 4.5

Error Bars Show data variability on plot of mean values Types of error bars include: Max/min, ± Standard Deviation, ± Standard Error, ± 95% C.I. “Significant Difference”

Using Error Bars to compare data Standard Deviation Demonstrates data variability Standard Error If bars overlap, any difference in means is not statistically significant If bars do not overlap, indicates nothing! 95% Confidence Interval If bars overlap, indicates nothing! If bars do not overlap, difference is statistically significant We’ll use 95 % CI by default Any time you have 3 or more data points, determine mean, standard deviation, standard error, and t95%,n-1, then plot mean with error bars showing the 95% confidence interval But if you want to conclude samples are the same or different, you have to use the right error bar!

Standard Error Bars No overlap: cannot be sure that the difference is statistically significant. Overlap: can be sure that the difference is not statistically significant.

Confidence Interval Error Bars No overlap: Can be sure that the difference is statistically significant. Overlap: Can not be sure that the difference is not statistically significant.

Adding Error Bars to an Excel Graph Create Graph Column, scatter,… Select Data Series In Layout Tab-Analysis Group, select Error Bars Select More Error Bar Options Select Custom and Specify Values and select cells containing the values

Example 1: 95% CI

Key Takeaways: How to calculate confidence intervals How to read the t-test chart How to calculate variance How to calculate standard error Difference between error bars with standard error and confidence interval What confidence interval means

Review: Measures of Central Tendency Mean = = = (1 + 3 + 3 + 6 + 8 + 10) / 6 = 5.2 NTU (NTU) 1 3 6 8 10 Median = m = Middle number Rank - 1 2 3 4 5 6 Number - 1 3 3 6 8 10 For even number of sample points, average middle two = (3+6) NTU/2 = 4.5 NTU Excel: Mean – AVERAGE; Median - MEDIAN

Variability Variance, s2 Example (cont.) sum of the square of the deviation about the mean divided by degrees of freedom Example (cont.) s2 = [(1-5.2)2 + (3-5.2)2 + (3-5.2)2 + 6-5.2)2 + (8-5.2)2 + (10-5.2)2] /(6-1) = 11.8 NTU2 Excel: Variance – VAR

Standard Deviation, s Square-root of variance If phenomena follows Normal Distribution (bell curve), 95% of population lies within 1.96 standard deviations of the mean Error bar is s above & below mean -1.96 1.96 95% Excel: standard deviation – STDEV Standard Deviations from Mean

Standard Deviation

Standard Error of Mean Also called St-Err For sample of size n taken from population with standard deviation estimated as s As n ↑, sxbar estimate↓, i.e., estimate of population mean improves Error bar is St-Err above & below mean

Standard Error