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Biostatistics-Measures-Distributions-and-Inference.pptx

biostatistics

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Biostatistics: Measures, Distributions & Inference A comprehensive guide to the statistical tools used in health sciences — from measures of central tendency and dispersion to hypothesis testing, correlation, and regression. Understanding these principles is essential for every medical and dental professional who consumes or conducts research. HEALTH SCIENCES STATISTICS EPIDEMIOLOGY & RESEARCH METHODS
Median & Mode: Completing the Picture of Central Tendency Median The median is the middle value when observations are arranged in ascending or descending order. If n is odd, the middle observation is the median. If n is even, the average of the two middle observations gives the median. Example (DMFT of five children): Raw data: 4, 2, 4, 3, 1 Arranged: 1, 2, 3, 4, 4 Median = 3 Mode The mode is the value that occurs most frequently in a distribution. It is the most representative and easiest to find among the measures of central tendency. Example (pulse rate per min): 71, 72, 73, 68, 71, 71 Mode = 71/min Mode is not unique — it may yield two or more values (bimodal/multimodal distributions). It is also not frequently used in formal statistical analysis.
Measures of Dispersion: Why Averages Alone Are Not Enough Measures of central tendency alone do not completely summarize data. To fully describe a distribution, we must also understand the variation within the data. Measuring this variation is called a measure of dispersion — it tells us how individual observations are scattered or dispersed from the mean. Consider two sleep-producing drugs given to two groups: Drug A: 6, 2, 4, 3, 5, 2 Mean = 3.7 hours → Drug B: 1, 6, 7, 1, 2, 6 Mean = 3.7 hours → Though averages are identical, the variation within each group is very different. This illustrates why dispersion must be measured alongside central tendency — especially when comparing two or more groups. Range Difference between highest and lowest values. Simple but based only on two extreme values; does not consider all observations. Mean Deviation Sum of absolute differences from the mean divided by n. Simple and easy, but cannot be used in further mathematical expressions. Standard Deviation Square root of mean squared deviations. Most commonly used in statistical analysis. An improvement over mean deviation. Co-efficient of Variation Expressed as a percentage (SD/Mean × 100). Used to compare relative variability between two characteristics or groups.
Standard Deviation & Co-efficient of Variation Standard Deviation (SD) SD is defined as the square root of mean squared deviations from the mean. It is the most commonly used measure of dispersion in statistical analysis. Uses of Standard Deviation • Shows how data are scattered from the mean • Summarises deviation of a large distribution in one figure used as a unit of variation • Helps indicate whether variation of an individual from the mean is by chance • Helps find the suitable sample size in sampling techniques Mean Deviation Though simple and easy to calculate, mean deviation is not used in statistical analysis as it cannot be used in further mathematical expressions. Co-efficient of Variation (CV) CV is used to compare relative variability between two characteristics or groups — especially when units of measurement differ (e.g., weight in spleen vs. heart; pulse rate in young vs. old). Expressed as a percentage:
Normal Distribution (Gaussian Curve) Understanding how much variation is considered normal or due to biological variation is critical in health sciences. The normal distribution provides the basis for all conclusions drawn from a sample. When a large number of observations of any variable (height, blood pressure, pulse rate) are plotted as a frequency distribution curve with small class intervals, a bell-shaped normal curve is obtained. Bell-Shaped The curve is symmetrical and bell-shaped, with the peak at the center. Mean = Median = Mode All three measures of central tendency coincide at the center of the distribution. Tails Never Touch Theoretically, the tails of the curve never touch the baseline — they extend to infinity. 68.26% Mean ± 1 SD Observations covered within this limit 95.44% Mean ± 2 SD ~95% confidence level; 5% error (p = 0.05) 99.74% Mean ± 3 SD ~99% confidence level; 1% error Example: From a sample of 100 normal babies, mean birth weight = 2.5 kg, SD = 0.25 kg. Normal birth weight at 95% confidence = Mean ± 2SD = 2.5 ± 0.50 = 2.00 to 3.00 kg.
Standard Normal Curve & Probability Estimation Standard Normal Curve Although the normal curve differs for different characteristics and sample sizes, its properties remain the same. To estimate the area under the curve between any two ordinates, a single standardised normal curve has been devised with: • Total area = 1 • Mean = 0 • SD = 1 The distance of a value (X) from the mean in units of SD is called the standard normal deviate (Z): The probability of a reading falling outside the 95% confidence limits (Z > 2) is 1 in 20 (p = 0.05). Probability Estimation — Worked Example Pulse rate of normal healthy males: Mean = 72, SD = 2. What is the probability that a randomly chosen male has a pulse of 78 or more? Area of normal curve corresponding to deviate 3 = 0.4987. Area beyond 0.4987 = 0.5 0.4987 = − 0.0013 Therefore, only 13 out of 10,000 individuals would likely have a pulse rate of 78 or higher — a very rare occurrence.
Statistical Inference: Estimation of Population Values Statistical inference is the procedure of drawing conclusions from sample values. It consists of two aspects: (i) estimation of a population value and (ii) testing of hypothesis. When many samples are drawn from a population, each gives a slightly different result — this is called sampling variation. The measure of this variation is the Standard Error (SE). Standard Error Formulas Single sample mean: Two sample means: Single proportion: Two proportions: Worked Example From a sample of 100 normal babies: Mean birth weight = 2.5 kg, SD = 0.25 kg. At 95% confidence level, population mean birth weight = Mean ± 2SE: We interpret this as: with 95% confidence, the population mean birth weight lies between 2.45 and 2.55 kg.
Testing of Hypothesis: A Step-by-Step Framework A hypothesis is an assumption made before investigation regarding the outcome under study. A test procedure used to decide whether to reject or accept a hypothesis is called testing of hypothesis. Find Table Value Compute Ratio Choose α Set Hypotheses Types of Error Decision H₀ True H₀ False Accept H₀ Correct Type II Error (β) Reject H₀ Type I Error (α) Correct At 5% level of significance, the experimenter risks a wrong decision in only 5 out of 100 cases — i.e., 95% confidence in the correct decision. Worked Example: Anxiety Scores Hypertensive: bar{X}_1 = 5.91, SD₁ = 1.2, n₁ = 100 Normal: bar{X}_2 = 3.90, SD₂ = 1.3, n₂ = 100 Z table value at 5% l.o.s. = 1.96 Since 11.36 > 1.96, reject H₀. There is a statistically significant difference in mean anxiety scores between hypertensives and normals.
Tests of Significance & Correlation Parametric vs. Non-Parametric Tests Parametric tests use parameters like mean, SD, or proportion. They require: (a) normally distributed population, (b) comparable group SDs not significantly different, (c) interval or ratio scale data. If any assumption fails, use the equivalent non-parametric test. Comparison Parametric Non- Parametric Binomial Two different groups Independent t-test Mann- Whitney U Chi- square Two repeated (same group) Paired t-test Wilcoxon signed rank McNemar' s More than two groups ANOVA Kruskal-Wallis Chi- square More than two repeated Repeated ANOVA Friedman test — Correlation Correlation measures the relationship between two quantitatively measured variables. The correlation coefficient (r) ranges from 1 to − +1: • r = +1: Perfect positive correlation • r = 1 − : Perfect negative correlation • 0 < r < 1: Moderate positive correlation • −1 < r < 0: Moderate negative correlation • r = 0: No correlation
Regression Analysis & Conclusion Regression Regression analysis enables prediction of the value of one variable from the knowledge of another, provided they are linearly correlated. It helps identify risk factors associated with an outcome variable. The regression coefficient (β) quantifies how much the dependent variable changes per unit change in the independent variable: The regression equation to estimate the dependent variable (Y) from the independent variable (X) is: The scatter plot (Fig. 51.8) shows a positive linear correlation between age and systolic blood pressure (SBP). The regression line allows prediction of SBP for any given age. Conclusion Biostatistics is an inevitable part of the epidemiological toolkit. Health information is an integral part of the national health system and a basic tool of management for the progress of any society. The primary objective of a health information system is to provide reliable, relevant, up-to-date, adequate, and reasonably complete information for health managers at all levels. Most medical and dental students do not yet make adequate use of health information. However, all will become consumers of research and must understand the inferential statistical principles behind the reports they read. That is why every medical and dental professional should have a solid working knowledge of biostatistics. Key References • Dawson B, Trapp RG. Basic and Clinical Biostatistics (4th edn). McGraw-Hill, 2004. • Norman GR, Streiner DL. Biostatistics: The Bare Essentials (2nd edn). BC Decker, 2000.