1. Freshman Engineering Clinic II
Statistics II
Freshman Engineering Clinic II

2. 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

3. 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

4. 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 mg/dl porphyrin in their bloodstream?
Z1 = ( )/12 = 2.46; Probability from Z-table =
Z2 = ( )/12 = ; Probability from Z-table =
Final answer = =

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

6. 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

7. 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)

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

9. 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

10. 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”

11. 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!

12. 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.

13. 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.

14. 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

15. Example 1: 95% CI

16. 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

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

18. 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 + (8-5.2)2 + (10-5.2)2] /(6-1)
= 11.8 NTU2
Excel: Variance – VAR

19. Standard Deviation, s Square-root of variance
If phenomena follows Normal Distribution (bell curve), 95% of population lies within 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

20. Standard Deviation

21. 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

22. Standard Error