Data Analysis, Presentation, and Statistics Fr Clinic I

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)

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!

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

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

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

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.

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

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

Mean and Median Turbidity of Treated Water (NTU) 1 3 6 8 10 Mean = Sum of values divided by number of samples = (1+3+3+6+8+10)/6 = 5.2 NTU 1 3 6 8 10 Median = The 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 Excel: Mean – AVERAGE; Median - MEDIAN

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

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

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

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

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

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

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

Example 2

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.

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