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Understanding Probability and Confidence Intervals in Statistics

This session delves into the foundational concepts of probability and confidence intervals. It explores essential statistical principles, including the mean, normal distribution, and standard deviation. The session discusses the importance of confidence intervals for measurements, the distinction between population and sample, and the standard error of the mean. Additionally, it covers the philosophical underpinnings of truth in relation to empirical observations and certainty in statistical conclusions. Participants will gain a solid understanding of how to express probability and interpret results confidently.

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Understanding Probability and Confidence Intervals in Statistics

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  1. Session 1Probability to confidence intervals By : Allan Chang

  2. Probability • Mean • The normal distribution • Standard Deviation (of measurements) • Confidence interval of measurements • Population and sample • Standard Error (of the mean) • Confidence interval of the mean • Difference between two means (mean of differences) • Standard Error (of the difference) • Confidence interval of the difference

  3. Probability • Philosophical nature of truth • Aristotle : things can be true or false, but cannot be both • Confucius : everything is in the middle • Science and empiricism • Truth cannot be known, only estimated from observations • Things repeatedly observed are more likely to be true • No theory can be accepted unless supported by repeated observations • Uncertainties in observations • Multiple influences on any observation • Precisions of measurement • Very unlikely to get the same result every time • Probability provides an expression of confidence in the results • Statistics is the set of tools to summarise the observations

  4. How probability is expressed • Convention : a number between 0 and 1. • 0 (0%) = absolutely impossible, 1 (100%) = absolute certainty • Subjective probability :An expression of opinion. • e.g. the probability of my staying awake in this lecture is 10% • Logical probability : mathematical calculations based on assumptions, how most statistical theories are developed • e.g. If the probability of getting heads in tossing a coin is 0.5 • getting 2 heads in 2 tosses is 0.5x0.5=0.25, getting 2 tails = 0.25, and getting 1 head and 1 tail = 0.5 • Getting at least 1 heads in 2 tosses is 0.5 + (0.5x0.5) = 0.75 • Empirical probability : calculated from observations • e.g. if there are 10 women and 20 men in the class, the probability of being male is 20/(10+20) = 0.67 (67%)

  5. Probability • Mean • The normal distribution • Standard Deviation (of measurements) • Confidence interval of measurements • Population and sample • Standard Error (of the mean) • Confidence interval of the mean • Difference between two means (mean of differences) • Standard Error (of the difference) • Confidence interval of the difference

  6. Mean • Havara : a system of insurance amongst Phoenecian traders • Havara -> average -> mean • Mean is the centre of all the measurements

  7. Probability • Mean • The normal distribution • Standard Deviation (of measurements) • Confidence interval of measurements • Population and sample • Standard Error (of the mean) • Confidence interval of the mean • Difference between two means (mean of differences) • Standard Error (of the difference) • Confidence interval of the difference

  8. Gauss De Moivre

  9. Probability • Mean • The normal distribution • Standard Deviation (of measurements) • Confidence interval of measurements • Population and sample • Standard Error (of the mean) • Confidence interval of the mean • Difference between two means (mean of differences) • Standard Error (of the difference) • Confidence interval of the difference

  10. De Moivre Fisher

  11. Fisher p z Z=0; p=0.5 Z=1.29; p=0.1 Z=1.65; p=0.05 Probability p Z=1.96; p=0.025 Z=2.33; p=0.01 Z=2.58; p=0.005 Z=3.09; p=0.001 Standard deviate z

  12. Probability • Mean • The normal distribution • Standard Deviation (of measurements) • Confidence interval of measurements • Population and sample • Standard Error (of the mean) • Confidence interval of the mean • Difference between two means (mean of differences) • Standard Error (of the difference) • Confidence interval of the difference

  13. Confidence interval Half of excluded each side e.g. in 95% CI 5% is excluded 2.5% each side (z=1.96) 95% CI = mean +/- 1.96 SD

  14. Confidence intervalof measurements • The range any single measurement is likely to be within • CImeasurement = mean +/- z SD • 90% CI = mean +/- 1.65 SD • 95% CI = mean +/- 1.96 SD • 99% CI = mean +/- 2.58 SD

  15. Probability • Mean • The normal distribution • Standard Deviation (of measurements) • Confidence interval of measurements • Population and sample • Standard Error (of the mean) • Confidence interval of the mean • Difference between two means (mean of differences) • Standard Error (of the difference) • Confidence interval of the difference

  16. Population • Measure everyone, the results are true of the population • Very expensive and inconvenient to do, in most cases not possible • Sampling • Measure a small but representative portion of the population, and assume that the results are true also of the population as a whole • The results are estimates of the truth, approximations • The degree of approximation (error) is important and needs to be estimated

  17. Probability • Mean • The normal distribution • Standard Deviation (of measurements) • Confidence interval of measurements • Population and sample • Standard Error (of the mean) • Confidence interval of the mean • Difference between two means (mean of differences) • Standard Error (of the difference) • Confidence interval of the difference

  18. Standard Error of the mean • Conceptually, the Standard Deviation of all the mean values, if the samples are taken repeatedly

  19. n mean SD SE 20 -0.14 1.01 0.23 20 0.10 1.30 0.29 20 0.11 1.20 0.27 20 0.44 0.90 0.20 20 -0.13 1.12 0.25 20 0.29 0.90 0.20 20 0.29 1.15 0.26 20 0.36 1.07 0.24 20 0.04 1.41 0.32 20 -0.02 0.90 0.20 mean and SD of the 10 means 10 0.13 0.20 • Standard Error of the mean • SE = SD / sqrt(n) • Random number mean=0 SD=1 • 10 samples of 20 each • SE each sample = 0.2 – 0.32 • SD of the 10 means = 0.2 • SE is an estimate of the SD of the means if repeated sampling of the same size occurs

  20. Standard Error of the mean (SE) • A function of Standard Deviation, and sample size • If the sample size is infinite, or the whole population, then SE = 0, as the mean value has no error • If the sample size =1, then the SE is the same as the Standard Deviation • SEmean = SD / sqrt(n)

  21. SE = SD / SQRT(n) SE as % of SD Sample size

  22. Probability • Mean • The normal distribution • Standard Deviation (of measurements) • Confidence interval of measurements • Population and sample • Standard Error (of the mean) • Confidence interval of the mean • Difference between two means (mean of differences) • Standard Error (of the difference) • Confidence interval of the difference

  23. Confidence interval of the mean • The range within which the true mean value can be expected • CImean = mean +/- z SE • 90% CI = mean +/- 1.65 SE • 95% CI = mean +/- 1.96 SE • 99% CI = mean +/- 2.58 SE • CImean = CImeasurement / SQRT(n)

  24. Probability • Mean • The normal distribution • Standard Deviation (of measurements) • Confidence interval of measurements • Population and sample • Standard Error (of the mean) • Confidence interval of the mean • Difference between two means (mean of differences) • Standard Error (of the difference) • Confidence interval of the difference

  25. Theory of sampling • Difference between the means of samples in two groups can be considered as the mean difference between these two groups in the population • e.g. The difference in the mean weights between 200 men and 200 women represents the mean difference in weight between men and women in the population • The problem is the Standard Error and the confidence interval

  26. Probability • Mean • The normal distribution • Standard Deviation (of measurements) • Confidence interval of measurements • Population and sample • Standard Error (of the mean) • Confidence interval of the mean • Difference between two means (mean of differences) • Standard Error (of the difference) • Confidence interval of the difference

  27. R.A. Fisher devised the Analysis of Variance between groups • Total variations in measurements are partitioned • That attributable to groups difference • That attributable to individual differences (residual, error)

  28. From the Analysis of variance • Individual or random variation is partitioned • What is left are group variance and difference • From this • Further partition to Standard Errors for each group • What is left then is the variation or error of the difference • From this, the Standard Error of the difference is derived • SEdifference is a function of the sample size and SD of the two groups, and the difference in the means between the two groups

  29. Probability • Mean • The normal distribution • Standard Deviation (of measurements) • Confidence interval of measurements • Population and sample • Standard Error (of the mean) • Confidence interval of the mean • Difference between two means (mean of differences) • Standard Error (of the difference) • Confidence interval of the difference

  30. Confidence interval of the difference • The range within which the difference can be expected • CIdifference = mean +/- z SEdifference • 90% CI = difference +/- 1.65 SEdifference • 95% CI = difference +/- 1.96 SEdifference • 99% CI = difference +/- 2.58 SEdifference

  31. Interpretation of CIdifference • CI represent the range within which the “true value” is likely to be • CI that does not traverse the null value means that the difference is not null, or that the difference is statistically significant • CI that are inside of a meaningless difference (tolerance) means that the difference is not of practical consequence, and can be considered to be equivalent

  32. An example of CI • We do two studies • Study 1, the difference in weight between women and men • Study 2, the difference in weight between those named Jim and all those named Bill • We think that a difference of less than 1Kg is probably not big enough to worry about

  33. Tolerance +/- 1Kg men – women n = 20 men – women n = 100 Jim – Bill n = 200 Jim – Bill n = 1000 Difference in weight (Kg)

  34. Terms you need to know • Measurements • Mean, Standard Deviation, confidence interval • Means • Mean, Standard Error of the mean, confidence interval • Difference • Difference between means, mean difference, Standard Error of the difference, confidence interval, significant difference, not significantly different, equivalence

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