I now have all the content from Park's PSM to compile a thorough biostatistics guide. Here it is:
Complete Biostatistics for PSM Exam
(Based on Park's Textbook of Preventive and Social Medicine)
CHAPTER OUTLINE
- Definition and Scope
- Types of Data
- Presentation of Data
- Statistical Averages (Measures of Central Tendency)
- Measures of Dispersion
- Normal Distribution
- Sampling
- Tests of Significance
- Correlation and Regression
- Key Formulas at a Glance
1. DEFINITION AND SCOPE
Biostatistics = the application of statistical methods to biological and health data.
Statistics involves:
- Collection of data
- Organization and presentation of data
- Analysis and interpretation of data
- Drawing conclusions
Why biostatistics in PSM?
- To plan and evaluate health programmes
- To measure health status of populations
- To compare disease occurrence over time and place
- To test whether an observed difference is real or due to chance
2. TYPES OF DATA
A. Qualitative (Categorical) Data
Observations that can be sorted into categories - NOT measured numerically.
- Nominal (no order): Blood group (A, B, O, AB), sex, religion
- Ordinal (ordered categories): Severity (mild/moderate/severe), socioeconomic status (low/middle/high)
- Dichotomous/Binary: Only two categories - diseased/not diseased, alive/dead
B. Quantitative (Numerical) Data
Observations expressed in numbers.
- Discrete: Only whole number values possible - number of children, number of hospital admissions
- Continuous: Can take any value within a range - height, weight, blood pressure, temperature
Key Distinction for Tests:
| Data Type | Measure of Average | Test of Significance |
|---|
| Qualitative/Categorical | Mode, proportion | Chi-square test |
| Quantitative (normal distribution) | Mean | t-test, z-test |
| Quantitative (non-normal / skewed) | Median | Non-parametric tests |
3. PRESENTATION OF DATA
A. Tabular Presentation
- Simple/frequency tables
- Contingency tables (cross-tabulation, used in chi-square)
B. Diagrams and Charts
| Type | Best Used For |
|---|
| Bar chart (simple) | Comparing discrete categories |
| Multiple bar chart | Comparing two or more groups side-by-side |
| Component (stacked) bar chart | Showing parts of a whole within bars |
| Histogram | Continuous data (frequency distribution) - bars touch each other |
| Frequency polygon | Continuous data - midpoints of histogram bars joined by lines |
| Pie chart | Proportional/percentage data (360° = 100%) |
| Scatter diagram (dot plot) | Showing relationship between two continuous variables |
| Line graph | Trends over time |
Histogram vs. Bar chart:
- In a histogram, bars touch each other (continuous data)
- In a bar chart, bars are separated (discrete/categorical data)
4. STATISTICAL AVERAGES (MEASURES OF CENTRAL TENDENCY)
These summarize the "centre" of a distribution. Three main types:
(a) Arithmetic Mean (x̄)
Definition: Sum of all observations divided by the number of observations.
Formula:
x̄ = Σx / n
Example: Diastolic BP of 10 individuals: 83, 75, 81, 79, 71, 95, 75, 77, 84, 90
- Total = 810
- Mean = 810/10 = 81
Advantages:
- Easy to calculate and understand
- Uses all values in the data
- Best measure for normally distributed data
Disadvantages:
- Unduly influenced by extreme (outlier) values
- May give an unrealistic value (e.g., 4.76 children)
(b) Median
Definition: The middle value when data is arranged in ascending or descending order.
Rule:
- Odd number of observations → middle value
- Even number of observations → average of the two middle values
Example: 9 values: 71, 75, 75, 77, 79, 81, 83, 84, 95 → Median = 79
Example: 10 values: 71, 75, 75, 77, 79, 81, 83, 84, 90, 95 → Median = (79+81)/2 = 80
Advantages:
- Not affected by extreme values
- Best for skewed distributions
When to use median over mean:
- When one extreme value distorts the mean
- Example: Income of 7 people: 5, 5, 5, 7, 10, 20, 102
- Mean = 154/7 = 22 (distorted by 102)
- Median = 7 (far more representative)
(c) Mode
Definition: The most frequently occurring value in a distribution.
Example: BP values: 85, 75, 81, 79, 71, 95, 75, 77, 75, 90, 81, 71, 75, 79, 95, 75, 77, 84, 75, 81, 75
- Mode = 75 (appears most frequently)
Advantages:
- Easy to understand
- Not affected by extreme values
Disadvantages:
- Exact location is often uncertain
- Not often used in biological statistics
Relationship Between Mean, Median, and Mode:
| Distribution | Relationship |
|---|
| Normal (symmetrical) | Mean = Median = Mode |
| Positively skewed (tail to right) | Mode < Median < Mean |
| Negatively skewed (tail to left) | Mean < Median < Mode |
Tip: In a skewed distribution, the median is always between the mean and mode.
5. MEASURES OF DISPERSION
Dispersion = how spread out the data values are around the central value.
(a) Range
Definition: Difference between the highest and lowest values in a sample.
Example: BP values: 83, 75, 81, 79, 71, 90, 75, 95, 77, 94
- Highest = 95, Lowest = 71
- Range = 71 to 95 (or difference = 24)
Limitation: Only uses the two extreme values; tells nothing about the distribution of values in between.
(b) Mean Deviation (MD)
Definition: Average of the absolute deviations from the arithmetic mean.
Formula:
MD = Σ|x - x̄| / n
Steps:
- Calculate the mean
- Find the deviation of each value from the mean
- Ignore the sign (take absolute values)
- Average those deviations
Example: BP values: 83, 75, 81, 79, 71, 95, 75, 77, 84, 90 (Mean = 81)
- Sum of deviations = 2+6+0+2+10+14+6+4+3+9 = 56 (ignoring signs)
- MD = 56/10 = 5.6
(c) Standard Deviation (SD or σ)
Definition: "Root-Mean-Square Deviation" - the most commonly used measure of dispersion.
Formula:
- For large samples (n > 30): SD = √(Σ(x - x̄)² / n)
- For small samples (n < 30): SD = √(Σ(x - x̄)² / (n-1)) ← use (n-1) to avoid underestimation
Steps to calculate SD:
- Find the mean (x̄)
- Find deviation of each value from mean (x - x̄)
- Square each deviation: (x - x̄)²
- Sum the squared deviations: Σ(x - x̄)²
- Divide by n (or n-1 for small samples)
- Take the square root
Notation:
- Population parameters: Greek letters → mean = μ, SD = σ
- Sample statistics: Roman letters → mean = x̄, SD = s
(d) Variance
= SD² (the square of standard deviation)
(e) Coefficient of Variation (CV)
Used to compare variability between two groups with different units or means:
CV = (SD / Mean) × 100
6. NORMAL DISTRIBUTION
The Normal Curve (Gaussian Distribution)
- Bell-shaped, perfectly symmetrical curve
- Mean = Median = Mode (all coincide at the centre)
- Total area under the curve = 1 (or 100%)
- Based on an infinitely large number of observations
Properties - The 68-95-99.7 Rule (Critical for Exams!)
| Range | Percentage of observations included |
|---|
| Mean ± 1 SD | 68.3% of observations |
| Mean ± 2 SD | 95.4% (~95%) of observations |
| Mean ± 3 SD | 99.7% (~almost all) of observations |
Standard Normal Curve
A special form of the normal curve with:
- Mean = 0
- SD = 1
- Total area = 1
Standard Normal Deviate (Z-score)
Used to find the position of any value relative to the mean, in units of SD:
Z = (x - x̄) / σ
Example: Pulse of a group: mean = 72, SD = 2. What is the probability that a randomly chosen male has pulse ≥ 80?
- Z = (80 - 72) / 2 = 4
- Area corresponding to Z=4 = 0.49997 (from table)
- Area beyond Z=4 = 0.5 - 0.49997 = 0.00003
- Probability = only 3 in 100,000 individuals would have pulse ≥ 80
7. SAMPLING
Why Sample?
When the population is too large to study completely, a representative sample is selected. The sample must be representative of the whole population.
Sampling Frame
A complete listing of all members of the universe (population) from which the sample is drawn. Its accuracy is critical.
Types of Sampling
| Method | Description | Key Feature |
|---|
| Simple Random Sampling | Each unit is assigned a number; random number table used to select | Every unit has an equal chance; maximum number of possible samples |
| Systematic Random Sampling | Every k-th unit is selected (e.g., every 10th house) | Each unit still has equal probability but fewer possible samples |
| Stratified Random Sampling | Population divided into strata (subgroups); random sample taken from each | Ensures representation of all subgroups |
| Cluster Sampling | Population divided into clusters (e.g., villages); entire clusters randomly selected | Used when listing individuals is impractical |
| Multistage Sampling | Two or more stages of random sampling | Used in large national surveys (e.g., NFHS) |
| Quota Sampling | Non-random; investigator fills fixed quotas from strata | Prone to bias; not truly random |
| Purposive Sampling | Non-random; investigator deliberately selects | Prone to bias |
Sampling Errors
Occur because data are gathered from a sample, not the entire population. Factors:
- Larger sample size → smaller sampling error
- Greater variability in population → larger sampling error
Non-Sampling Errors
Due to:
- Poorly calibrated instruments
- Observer variation
- Incomplete coverage of selected subjects
- Conceptual errors
- Often MORE important than sampling errors
Standard Error (SE)
The standard deviation of the sampling distribution of the mean.
SE of Mean = σ / √n (or S / √n from sample)
Key concepts:
- 95% of sample means lie within μ ± 2 SE of the population mean
- SE decreases as sample size (n) increases
- SE is the key measure used in tests of significance
8. TESTS OF SIGNIFICANCE (HYPOTHESIS TESTING)
The Null Hypothesis (H₀)
Before testing, we set up a null hypothesis - the assumption that there is no real difference between the groups being compared. The test then tries to disprove this.
P-value (Probability Value)
- P = probability that the observed difference is due to chance alone
- P < 0.05 → difference is statistically significant (1 in 20 chance it's due to chance)
- P < 0.01 → difference is highly significant (1 in 100 chance)
- P > 0.05 → difference is NOT significant (likely due to chance)
(a) Standard Error of the Mean
Use: To determine if a sample mean is significantly different from the population mean.
Confidence Intervals:
- 95% CI = x̄ ± 2 SE (limits within which the true population mean lies in 95% of samples)
- 99% CI = x̄ ± 2.58 SE
Example: Sample of 25 males, mean temp = 98.14°F, SD = 0.6
- SE = 0.6/√25 = 0.12
- 95% CI = 98.14 ± 2(0.12) = 97.90 to 98.38°F
- Probability only 1 in 20 (P=0.05) that population mean is outside these limits
(b) Standard Error of Proportion
Use: For qualitative data when dealing with proportions (e.g., percentage of males).
Formula:
SE of Proportion = √(pq/n)
where p = proportion of one category, q = 1-p, n = sample size
Decision: If relative deviate (difference/SE) > 2, the difference is significant (P < 0.05)
(c) Standard Error of Difference Between Two Means
Use: To compare means of two groups (e.g., drug vs. control group).
Formula:
SE of difference = √(SD₁²/n₁ + SD₂²/n₂)
Decision: Calculate t = difference of means / SE of difference
- Compare with t-table at appropriate degrees of freedom
- If t > critical value → significant difference
(d) Chi-Square Test (χ²)
Use: The most important test for categorical (qualitative) data. Tests whether observed frequencies differ from expected frequencies.
Formula:
χ² = Σ(O - E)² / E
where O = Observed frequency, E = Expected frequency
Steps:
- State the null hypothesis
- Calculate expected values: E = (row total × column total) / grand total
- Calculate χ² = Σ(O-E)²/E
- Find degrees of freedom (df) = (rows-1)(columns-1)
- For a 2×2 table: df = (2-1)(2-1) = 1
- Compare calculated χ² with critical value from chi-square table
- For df=1: χ² critical value at P=0.05 is 3.84
- If calculated χ² < 3.84 → NOT significant (accept H₀)
- If calculated χ² > 3.84 → Significant (reject H₀)
Important limitation: The test is valid only if the expected number in each cell is ≥ 2. If expected frequencies are very small, use Fisher's Exact Test instead.
Exam-ready Chi-square table (df=1):
| P value | χ² critical value |
|---|
| 0.10 | 2.71 |
| 0.05 | 3.84 |
| 0.02 | 5.41 |
| 0.01 | 6.64 |
| 0.001 | 10.83 |
Applications of chi-square:
- Testing vaccine efficacy (attacked vs. not attacked in two vaccine groups)
- Testing association between two categorical variables
- Testing goodness of fit
Summary: Which Test to Use?
| Situation | Test |
|---|
| Compare a sample mean to a known population mean (large sample n>30) | Z-test |
| Compare means of two groups (small sample n<30) | t-test (Student's t-test) |
| Compare proportions in two groups | SE of difference between proportions |
| Two or more groups, categorical data | Chi-square test |
| Small expected frequencies in categorical data | Fisher's Exact Test |
| Compare two means in paired samples | Paired t-test |
| Non-parametric (non-normal data) | Mann-Whitney U, Wilcoxon, Kruskal-Wallis |
9. CORRELATION AND REGRESSION
Correlation
Definition: A statistical method to determine whether there is a linear relationship between two variables (e.g., height and weight, temperature and pulse, age and vital capacity).
Correlation Coefficient (r)
- Represented by the symbol r (Pearson's correlation coefficient)
- Ranges from -1 to +1
Formula:
r = Σ(x - x̄)(y - ȳ) / √[Σ(x - x̄)² × Σ(y - ȳ)²]
Interpretation of r:
| Value of r | Meaning |
|---|
| r = +1 | Perfect positive correlation |
| r = 0 | No correlation |
| r = -1 | Perfect negative correlation |
| r = +0.7 to +1 | Strong positive correlation |
| r = +0.4 to +0.7 | Moderate positive correlation |
| r = 0 to +0.4 | Weak positive correlation |
Scatter Diagram:
- Used to visually assess correlation before calculating r
- Dots clustering around a straight line = linear relationship
- No clustering pattern = no relationship
Positive correlation: Both variables increase together (e.g., height and weight)
Negative correlation: As one increases, the other decreases (e.g., age and FEV₁)
Important: Correlation does NOT imply causation!
Regression
Definition: A statistical method that describes the relationship between variables and allows prediction of one variable from another.
Regression line (line of best fit):
Y = a + bX
Where:
- Y = dependent variable (outcome)
- X = independent variable (predictor)
- a = Y-intercept (value of Y when X = 0)
- b = regression coefficient (slope - how much Y changes for each unit change in X)
Difference from correlation:
- Correlation: measures the strength of association between two variables
- Regression: measures the direction and magnitude of the relationship; allows prediction
10. KEY FORMULAS AT A GLANCE
| Formula | What it calculates |
|---|
| x̄ = Σx/n | Arithmetic mean |
| SD = √[Σ(x-x̄)²/(n-1)] | Standard deviation (small sample) |
| SE = SD/√n | Standard error of the mean |
| Z = (x - x̄)/σ | Standard normal deviate |
| χ² = Σ(O-E)²/E | Chi-square statistic |
| df = (r-1)(c-1) | Degrees of freedom |
| SE (proportion) = √(pq/n) | Standard error of proportion |
| 95% CI = x̄ ± 2 SE | Confidence interval |
11. QUICK REVISION TABLE - HIGH-YIELD EXAM POINTS
| Topic | Key Fact |
|---|
| Mean = Median = Mode | Only in a normal distribution |
| Median best used for | Skewed distributions (not affected by outliers) |
| Mode | Most frequently occurring value; rarely used in medical statistics |
| SD measures | Spread of individual observations around the mean |
| SE measures | Reliability/precision of the sample mean |
| Mean ± 2 SD | Contains 95.4% of observations in normal distribution |
| Mean ± 1 SD | Contains 68.3% of observations |
| P < 0.05 | Statistically significant (1 in 20) |
| P < 0.01 | Highly significant (1 in 100) |
| Chi-square valid when | Expected cell frequency ≥ 2 |
| Chi-square for 2×2 table | Degrees of freedom = 1; critical value at P=0.05 = 3.84 |
| r = +1 | Perfect positive correlation |
| r = 0 | No correlation |
| Scatter diagram | Used to visualize correlation between two continuous variables |
| Null hypothesis | Assumes no difference between compared groups |
| Confidence interval | Range within which the true population mean is likely to fall |
*(Park's Textbook of Preventive and Social Medicine, Biostatistics Chapter)*This is a complete ground-up guide to Biostatistics as covered in PSM. Here is a quick orientation to what was covered:
What you just learned - Summary Map:
BIOSTATISTICS
│
├── TYPES OF DATA
│ ├── Qualitative (nominal, ordinal, binary)
│ └── Quantitative (discrete, continuous)
│
├── PRESENTATION
│ ├── Tables, Bar charts, Histogram, Pie chart, Scatter diagram
│ └── Histogram (bars touch) vs Bar chart (bars separate)
│
├── CENTRAL TENDENCY
│ ├── Mean → use for normal data; affected by outliers
│ ├── Median → use for skewed data; NOT affected by outliers
│ └── Mode → most frequent value
│
├── DISPERSION
│ ├── Range → simplest; only uses extremes
│ ├── Mean Deviation → average of absolute deviations
│ └── Standard Deviation → most useful; "root-mean-square deviation"
│
├── NORMAL DISTRIBUTION
│ ├── Mean = Median = Mode
│ ├── ±1 SD = 68.3%
│ ├── ±2 SD = 95.4%
│ └── Z-score = (x - mean)/SD
│
├── SAMPLING
│ ├── Simple random, Systematic, Stratified, Cluster, Multistage
│ └── SE = SD/√n (decreases as n increases)
│
├── TESTS OF SIGNIFICANCE
│ ├── P < 0.05 = significant
│ ├── Z-test / t-test → for means (quantitative data)
│ └── Chi-square → for proportions (categorical data); df=(r-1)(c-1)
│
└── CORRELATION & REGRESSION
├── r = -1 to +1; r=0 means no correlation
└── Regression: Y = a + bX; used for prediction