Introductory Statistics: High Throughput Data Dispersion Analysis

1. Introductory Statistics for Laboratorians dealing with High Throughput Data sets Centers for Disease Control

2. Problem 7: Dispersion • Prepare 2 line graphs, one for males and one for females using the data presented below. • Put both line graphs on the same axes.

3. Problem 7: Dispersion

4. Problem 7: Dispersion

5. Problem 7: Dispersion

6. Problem 7: Dispersion • How can we quantify the difference between the men and the women in this problem. • Compute the mean (average) for the men. • Compute the mean (average) for the women.

7. Problem 7: Dispersion • What are the highest and lowest scores for the men? • What are the highest and lowest scores for the women? • Count the number of scores from lowest to highest. This number is called the Range of the scores. • In this case the Range doesn’t help us describe the difference between the males and the females. We need better measures of dispersion.

8. Problem 8: Dispersion • For the following data: • What is the highest and lowest score? • What is the Range? (count the number of scores from the lowest to the highest.) • What is the Mean (average)? • How far is each person from the Mean? (Fill in the column. Always subtract the mean from the score. )

9. Problem 8: DispersionData Table

10. Problem 8: Dispersion • Compute the “Sum of Squared Deviations from the Mean” (SS) for this data set (or sample or whatever you call it). • Compute the variance of the sample. • Compute the standard deviation of the sample.

11. Dispersion Definitions • The range is the number of scores from the smallest to the largest. • Deviation Score = Score – Mean • Always subtract the mean from the score • Always preserve the sign (positive or negative) • The total of the deviation scores is always zero • Sum Squares = Total of the squared deviation scores. (SS) • Variance = SS/N • Standard Deviation = square root of variance

12. Standard Deviation • Surely there is an easier way to measure dispersion than using all this squaring and square rooting. • Turns out, the standard deviation is the exact point on a normal curve where the second derivative is zero. • If you were skiing down the slope, it would get steeper and steeper then it would start to flatten out. That point is the standard deviation. • That’s why it is the preferred measure of dispersion.

13. Standard Deviation

14. Problem 9 • Given the following collection of scores: 2, 3, 5, 6, 6, 8 • Calculate the range of the scores • Calculate the sum of squares • Calculate the variance • Calculate the standard deviation

15. Problem 9Data Table

16. Normal distributions Normal—or Gaussian—distributions are a family of symmetrical, bell- shaped density curves defined by a mean m (mu) and a standard deviation s (sigma): N (m, s). x x e = 2.71828… The base of the natural logarithm π = pi = 3.14159…

17. A family of density curves Here the means are the same (m = 15) while the standard deviations are different (s = 2, 4, and 6). Here the means are different (m = 10, 15, and 20) while the standard deviations are the same (s = 3).

18. All Normal curves N (m, s) share the same properties • About 68% of all observations are within 1 standard deviation (s) of the mean (m). • About 95% of all observations are within 2 s of the mean m. • Almost all (99.7%) observations are within 3 s of the mean. Inflection point mean µ = 64.5 standard deviation s = 2.5 N(µ, s) = N(64.5, 2.5) Reminder: µ (mu) is the mean of the idealized curve, while is the mean of a sample. σ (sigma) is the standard deviation of the idealized curve, while s is the s.d. of a sample.

19. Definitions: Statistical Symbols • In an actual sample • Scores are represented by • Mean = • Deviation Score • Standard Deviation = s • Variance = s2 • In a theoretical distribution (density curve) • Mean = μ • Standard Deviation = σ • Variance = σ2