1. Statistics: Data Presentation & Analysis Fr Clinic I

2. Overview Tables & Graphs Populations & Samples Mean, Median, & Variance Error Bars – Standard Deviation, Standard Error & 95% Confidence Interval (CI) Comparing Means of Two Populations Linear Regression (LR)

3. Warning Statistics is a huge field, I’ve simplified considerably here. For example: – Mean, Median, and Standard Deviation There are alternative formulas – 95% Confidence Interval There are other ways to calculate CIs (e.g., z statistic instead of t; difference between two means, rather than single mean…) – Error Bars Don’t go beyond the interpretations I give here! – Comparing Means of Two Data Sets We just cover the t test for two means when the variances are unknown but equal, there are other tests – Linear Regression We only look at simple LR and only calculate the intercept, slope and R 2. There is much more to LR!

4. Tables Table 1: Average Turbidity and Color of Water Treated by Portable Water Filters Consistent Format, Title, Units, Big Fonts Differentiate Headings, Number Columns

5. Figures 11 Figure 1: Turbidity of Pond Water, Treated and Untreated 20 10 7 5 1 11 Consistent Format, Title, Units Good Axis Titles, Big Fonts

6. Populations and Samples Population – All possible outcomes of experiment or observation US population Particular type of steel beam Sample – Finite number of outcomes measured or observations made 1000 US citizens 5 beams Use samples to estimate population properties – Mean, Variance E.g., Height of 1000 US citizens used to estimate mean of US population

7. Central Tendency Mean and Median Mean = xbar = Sum of values divided by sample size = (1+3+3+6+8+10)/6 = 5.2 NTU 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)/2 = 4.5 1 3 6 8 10 Excel: Mean – AVERAGE; Median - MEDIAN

8. Variability Variance, s 2 – sum of the square of the deviation about the mean divided by degrees of freedom – s 2 = n (x i – xbar) 2 /(n-1) – Where x i = a data point and n = number of data points Example (cont.) – s 2 = [(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 NTU 2 Excel: Variance – VAR

9. Error Bars Show data variability on plot of mean values Types of error bars include: Max/min, ± Standard Deviation, ± Standard Error, ± 95% CI

10. 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 -1.961.96 95% Standard Deviations from Mean Excel: standard deviation – STDEV

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

12. 95% Confidence Interval (CI) for Mean A 95% Confidence Interval is expected to contain the population mean 95 % of the time (i.e., of 95%-CIs from 100 samples, 95 will contain pop mean) t 95%,n-1 is a statistic for 95% CI from sample of size n – t 95%,n-1 = TINV(0.05,n-1) – If n  30, t 95%,n-1 ≈ 1.96 (Normal Distribution) Error bar is above & below mean

13. Using Error Bars to compare data Standard Deviation – Demonstrates data variability, but no comparison possible 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 in this class – Any time you have 3 or more data points, determine mean, standard deviation, standard error, and t 95%,n-1, then plot mean with error bars showing the 95% confidence interval

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. What can we do? Lift weight multiple times using different solar panel combinations (or hyrdoturbines, or gear boxes) and plot mean and 95 % Confidence interval error bars. – If error bars overlap between to different test conditions, indicates nothing! – If error bars do not overlap, difference is statistically significant

17. T Test A more sophisticated way to compare means Use t test to determine if means of two populations are different E.g., lift times with different solar panel combinations or turbines or…

18. Comparing Two Data Sets using the t test Example - You lift weight with two panels in series and two in parallel. – Series: Mean = 2 min, s = 0.5 min, n = 20 – Parallel: Mean = 3 min, s = 0.6 min, n = 20 You ask the question - Do the different panel combinations result in different lift times? – Different in a statistically significant way

19. Are the Lift Times Different? Use TTEST (Excel) Fractional probability of being wrong if you claim the two populations are different – We’ll say they are significantly different if probability is ≤ 0.05

20. Marbles

21. Linear Regression Fit the best straight line to a data set Right-click on data point and select “trendline”. Select options to show equation and R 2.

22. R 2 - Coefficient of multiple Determination R 2 = n (ŷ i - ybar) 2 / n (y i - ybar) 2 – ŷ i = Predicted y values, from regression equation – y i = Observed y values – Ybar = mean of y R 2 = fraction of variance explained by regression – R 2 = 1 if data lies along a straight line