1. Data Analysis, Presentation, and Statistics
Fr Clinic I

2. Overview Tables and Graphs Populations and Samples
Mean, Median, and Standard Deviation
Standard Error & 95% Confidence Interval (CI)
Error Bars
Comparing Means of Two Data Sets
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
Standard Error and the 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!
Linear Regression
We only look at simple LR and only calculate the intercept, slope and R2. There is much more to LR!

4. Should I Use a Table or Graph?
Tables
Presenting large amount of different data
Comparing multiple characteristics
Graphs
Visual presentation quickly gives information
Compare one or two characteristics
Showing trends

5. Tables 4 5 12 Consistent Format, Title, Units, Big Fonts
Table 1: Average Turbidity and Color of Water Treated by Portable Water Filters
Consistent Format, Title, Units, Big Fonts
Differentiate Headings, Number Columns

6. Figures Consistent Format, Title, Units Good Axis Titles, Big Fonts 2010 7 5 1 11 11
Figure 1: Turbidity of Pond Water, Treated and Untreated

7. Graphing Suggestions
1, 2, 5 rule –
Set gradations so smallest division of the axis is a positive integer power of 10 times 1, 2, or 5.
Huh?
Set your scale up so that the smallest division is an integer increment.

8. Graphing Suggestions Labels Points, lines, curves
All axes should be labeled
Include units on the label
Points, lines, curves
Play around with options
Color can be your friend
Color can be your enemy

14. Populations and Samples
All of the possible outcomes of experiment or observation
US population
Particular type of steel beam
Sample
A finite number of outcomes measured or observations made
1000 US citizens
5 beams
We use samples to estimate population properties
Mean, Variability (e.g. standard deviation), Distribution
Height of 1000 US citizens used to estimate mean of US population

15. Mean and Median Turbidity of Treated Water (NTU) 1 3 6 8 10
Mean = Sum of values divided by number of samples
= ( )/6
= 5.2 NTU 1 3 6 8 10
Median = The middle number
Rank
Number
For even number of sample points, average middle two
= (3+6)/2 = 4.5
Excel: Mean – AVERAGE; Median - MEDIAN

16. Variance Measure of variability
sum of the square of the deviation about the mean divided by degrees of freedom
n = number of data points
Excel: variance – VAR

17. Standard Deviation, s Square-root of the variance
For phenomena following a Normal Distribution (bell curve), 95% of population values lie within 1.96 standard deviations of the mean
Area under curve is probability of getting value within specified range
-1.96
1.96
95%
Excel: standard deviation – STDEV
Standard Deviations from Mean

18. Standard Error of Mean Standard deviation of mean Of sample of size n
taken from population with standard deviation s
Estimate of mean depends on sample selected
As n , variance of mean estimate goes down, i.e., estimate of population mean improves
As n , mean estimate distribution approaches normal, regardless of population distribution

19. 95% Confidence Interval (CI) for Mean
Interval within which we are 95 % confident the true mean lies
t95%,n-1 is t-statistic for 95% CI if sample size = n
If n  30, let t95%,n-1 = 1.96 (Normal Distribution)
Otherwise, use Excel formula: TINV(0.05,n-1)
n = number of data points

20. Error Bars Show data variability on plot of mean values
Types of error bars include:
± Standard Deviation, ± Standard Error, ± 95% CI
Maximum and minimum value

21. 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

22. Example 1
Create Bar Chart of Name vs Mean. Right click on data. Select “Format Data Series”.

23. Example 2

24. Linear Regression Fit the best straight line to a data set
Right-click on data point and use “trendline” option. Use “options” tab to get equation and R2.

25. R2 - Coefficient of multiple Determination
ŷi = Predicted y values, from regression equation
yi = Observed y values
R2 = fraction of variance explained by regression (variance = standard deviation squared)
= 1 if data lies along a straight line