2. Medical Statistics (full English class) Ji-Qian Fang School of Public Health Sun Yat-Sen University

3. 9.2 Normal Distribution and t Distribution 9.2 Normal Distribution and t Distribution

6. 1. Normal distribution 3.8 4 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 5.8 RBC(10 /L) 0 5 10 15 20 25 30 Frequency

7. 2.The Property of Normal Distribution a. Symmetric -3 -2 -1 0 1 2 3

8. b. Location of the peak – center, mean, median

9. c. Shape of the curve -- a bell

10. d. The area under the curve

11. The probability density function of a normal distribution: Denoted with If Z=(X-μ)/σ Z follows standard normal distribution

12. 3. Reference range (1) What is reference range? Example: WBC 4-10 (10 9 /L) Lead in urine <0.08mg/L a. Medical measurement X (such as physiological, Biochemical, … measurements) b. “Normal persons” c. The interval which covers the values of most normal persons.

13. (2) How to work out a reference range? a.Well define “normal person” b.Needs grouping or not? (male-female? Age groups?…) c.Determine sample size d.Random sampling e.Measurement (instrument, method, quality control…) f.Two sides? One side? g.“Most person”? (99%? 95%? 90%?…) h.Statistical method?

14. Statistical method? If the frequency distribution is close to a normal distribution: Two side (1- α) range One side (1- α) range Example 9-1 Height data of 130 girls (14 years old ) 143.084±1.96×6.58=130.18~155.98(cm) or

15. If the frequency distribution is skew: By percentiles: Two side (1- 0.05) range One side (1- 0.05) range Example The reference range of latent time P 95 =16+4/6(108×95%-99)=18.4(days)

16. 9.3 Statistical inference Estimation of population parameter Hypothesis testing

17. 1.The sampling error and standard error of mean Sampling error: Sample → sample mean (different from population mean) Different samples → Different sample means (different from each other )

18. (1) Sampling error is related to the variation of the population Example: The sample means of systolic blood pressure For adult population (age 18~90) -- vary substantially For young population (age 18~25) -- not vary too much

19. Sampling error is also related to sample size If sample size = population size there is no sampling error! If sample size = 1 Sampling error ~ variation of population!

21. Sampling from N(4.6602, 0.5746 2 )

22. 123456789 (a) 1234578 n=5 (b) 123456789 n=10 (c) 123456789 n=20 (d) 123456789 n=30 (e) Sampling from

23. (2) The distribution of sample mean If the variable ~ a normal distribution sample means ~ a normal distribution If the distribution of variable -- skew, For small sample distribution of sample mean – skew For large sample sample mean close to a normal distribution

24. (3) Standard error Standard deviation of the population Standard deviation of the sample mean or Standard error of sample mean or Standard error In any case: Standard error of sample mean = or

25. 2. Student ’ s t distribution For a normal distribution follows a standard normal distribution ---N(0,1). When is unknown, follows a t distribution.

26. υ=∞(standard normal distribution) υ=5 υ=1 Table 9-9 (p.289) is the Table for t distribution

27. 3. Confidence Interval of Population Mean Point estimation of population mean -- sample mean Interval estimation of population mean -- (1-α) confidence interval Confidence level: 1-α, such as 95% or 99%

28. Whenever we get a mean and standard deviation from a sample, put them into then is called (1-α) confidence interval of population mean The two extreme values are called confidence limits.

29. Example Systolic blood pressures of 20 health males were measured. 95% confidence interval of the population mean? 95% confidence interval:

30. What does “confidence” mean?