2. Statistical analysis

3. Why?? (besides making your life difficult …)  Scientists must collect data AND analyze it  Does your data support your hypothesis? Is it valid?  Statistics helps us find relationships between sets of data.  You are the scientist now, you must be comfortable with analysis of your data

4. Let’s look at two sets of data  Sample 1  -10, 0, 10, 20, 30  Sample 2  8, 9, 10, 11, 12 What can you tell me about this data???

5. Mean: the “average” of the data or the central tendency Sample 1 -10, 0, 10, 20, 30  -10 + 0 + 10 + 20 + 30 5 Sample 2 8, 9, 10, 11, 12  8 + 9 + 10 + 11 + 12 5 Mean = 10 Is this analysis complete??? NO!

6. Range: how far is the spread? Largest # - smallest # Sample 1 -10, 0, 10, 20, 30  30 – (-10) Sample 2 8, 9, 10, 11, 12  12 - 8 Range = 40 Range = 4 Does this data help? Yes, Sample 1 is more dispersed Obvious? Perhaps, but now shown mathematically

7. Something more … standard deviation  SD is a measure to show how individual data points are dispersed around the mean

8. Assuming normal data distribution (bell curve)  68% of all collected values lie within +/- 1 SD  95% of all collected values lie within +/- 2 SD  So what???

9. Standard deviation  A small SD indicates the data values are clustered around the mean  May also indicate few exteme data points  A large SD indicates the data values are spread out  May also indicate extreme data points  Outliers??

10. Standard deviation

11. Let’s practice …

13. Let’s compare …  Sample 1  SD = 15.8  Sample 2  SD = 1.58 How can I use this in my lab?

14. Error bars  Error bars represent the variability of your data  STANDARD DEVIATION  range  measurement uncertainties

15. Error bars  On a bar graph, the bar represents the mean of your data and the error bars represent +/- 1 sd mean sd

16. Error bars  On a line graph, the point represents the mean of your data and the error bars represent +/- 1 sd mean sd

17. t-test  t-test determines statistical significance between 2 sample means  Is the difference significant?  Is the difference due to your variable?? Or is it random chance??  How valid is your data?  t-test determines the probability that difference is due to random chance  A p value (probability) of 0.05 (5%) shows a 5% chance of randomness, but a 95% chance of confidence … Key word!!!!! You want 95% or higher! your difference IS DUE TO YOUR VARIABLE

18. t-test  For tests, you do NOT need to calculate t- values, but you must be able to read a t- chart!!  For internal assessments, you may use calculators or excel to calculate t-values

19. This is the range you are hoping for The difference between your samples has a HIGH probability of being due to your variable (and not chance) Need to be able to calculate degrees of freedom

20. Calculating degrees of freedom  df = (n 1 + n 2 ) - 2 Size of sample 1 Size of sample 2 # of samples

21. Calculating degrees of freedom  df = (n 1 + n 2 ) – 2  Population 1  -10, 0, 10, 20, 30  n 1 = 5  Population 2  8, 9, 10, 11, 12  n 2 = 5  df = (5 + 5) -2  df = 8

22. Using the t-table  If df = 8 and t = 3.5, is this a significant difference?  Less than 1% probability difference in data is due to chance  Therefore, greater than 99% probability difference in data is due to our variable

23. Other options, less commonly used in our class  Median  The middle #, when arranged in numeric order  Sample 1  -10, 0, 10, 20, 30  Median = 10  Sample 2  8, 9, 10, 11, 12  Median = 10  Mode  The # that occurs most often  Sample 1  -10, 0, 10, 20, 30  No mode  Sample 2  8, 9, 10, 11, 12  No mode

24. Some practice: looking at plant height Height in sun (cm)Height in shade (cm) 124131 12060 153131 98160 124212 141117 156131 12895 139145 117118  Calculate the mean for both samples  Sun = 130 cm  Shade = 130 cm

25. Some practice: looking at plant height Height in sun (cm)Height in shade (cm) 124131 12060 153131 98160 124212 141117 156131 12895 139145 117118  Calculate the range for both samples  Sun = 58 cm  Shade = 152 cm

26. Some practice: looking at plant height Height in sun (cm)Height in shade (cm) 124131 12060 153131 98160 124212 141117 156131 12895 139145 117118  Calculate the median for both samples  Sun = 126 cm  Shade = 131 cm If even # of samples, find the average of the two middle numbers

27. Some practice: looking at plant height Height in sun (cm)Height in shade (cm) 124131 12060 153131 98160 124212 141117 156131 12895 139145 117118  Calculate the mode for both samples  Sun = 124 cm  Shade = 131 cm

28. Some practice: looking at plant height Height in sun (cm)Height in shade (cm) 124131 12060 153131 98160 124212 141117 156131 12895 139145 117118  Calculate the sd for both samples  Sun = 17.56 cm  Shade = 39.85 cm

29. Some practice: looking at plant height Height in sun (cm)Height in shade (cm) 124131 12060 153131 98160 124212 141117 156131 12895 139145 117118  Sun: sd = 17.56 cm  Low sd indicates even (close) distribution of data points  More valid  Shade: sd = 39.85 cm  High sd indicates wide spread of data points  MAY indicate a problem with your experimental design

30. Some practice: looking at plant height Height in sun (cm)Height in shade (cm) 124131 12060 153131 98160 124212 141117 156131 12895 139145 117118 If t = 1.5, is this a significant difference? No

31. Be careful: correlation vs. cause  Observations (and carefully chosen data) may imply a CORRELATION, but does NOT necessarily demonstrate a cause  The average global temperature has increased over the past 100 years.  The number of pirates in the world has decreased over the past 100 years.  Therefore, decreased number of pirates causes increased global temperatures

32. Be careful: correlation vs. cause no no !

33. Be careful: correlation vs. cause  To discern a CAUSE, a valid EXPERIMENT must be done  Other scientists must also be able to repeat your experiment

34. Last word …  Remember, it is ALWAYS better to PROVE your experiment failed to support your hypothesis, than to lie about it being a success!!!

35. Any questions?