Measure of Central tendancy psm topic

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Here is a complete PSM / Biostatistics note on Measures of Central Tendency:

Measures of Central Tendency - PSM / Biostatistics

A measure of central tendency is a single summary value that represents the center or typical value of a dataset. The three classical measures are: Mean, Median, and Mode.

1. MEAN (Arithmetic Mean)

Definition: Sum of all observations divided by the total number of observations.

$$\bar{x} = \frac{\sum x_i}{n}$$

It is a parametric method - derived from a defined formula.
Uses every value in the dataset, so it is the most sensitive to extreme values (outliers).
Best used when data are normally (symmetrically) distributed.

Types of Mean

Type	Formula	When Used
Arithmetic Mean	Σx / n	Routine data; symmetric distributions
Geometric Mean	ⁿ√(x₁ × x₂ × ... × xₙ)	Log-normal distributions; minimizes effect of extreme values. Log GM = Σ(log xᵢ)/n
Weighted Mean	Σ(wᵢxᵢ) / Σwᵢ	When observations have different importance/weight
Harmonic Mean	n / Σ(1/xᵢ)	Rates and ratios (e.g., speed, dilution titers)

Example of Geometric Mean vs Arithmetic Mean: For values: 3, 3, 4, 4, 5, 5, 5, 6, 6, 8, 9, 10, 15, 21

Arithmetic Mean = 7.2
Geometric Mean = 6.09 (better reflects the cluster of lower values)
Henry's Clinical Diagnosis and Management by Laboratory Methods

2. MEDIAN

Definition: The middle value that divides the distribution exactly in half - one half above, one half below. Also called the 50th percentile.

It is a non-parametric method - derived from a count, not a formula.
Not affected by extreme values (outliers) - robust measure.
Best used when data are skewed or when outliers are present.
For ordinal data (e.g., pain score 0-5), the median is the preferred summary.

How to find:

Arrange all values in ascending order.
If n is odd: median = middle value [(n+1)/2 th term]
If n is even: median = average of the two middle values

3. MODE

Definition: The most frequently occurring value in a dataset.

Not very useful for comparing datasets, but helpful for understanding the shape of a distribution.
A dataset can be unimodal (one mode), bimodal (two modes - suggests two subpopulations), or multimodal.
Skewness does not affect the mode.
Henry's Clinical Diagnosis and Management by Laboratory Methods, p. 150

Relationship in Different Distributions

Normal (Gaussian / Bell-Shaped) Distribution

In a perfect Gaussian distribution:

Mean = Median = Mode

Gaussian (Normal) Distribution showing SD ranges

Figure: Idealized Gaussian distribution - areas under the curve at ±1, 2, and 3 SD from the mean - Henry's Clinical Diagnosis and Management

The SD properties of a normal distribution:

Mean ± 1 SD covers 68.2% of values
Mean ± 2 SD covers 95.5% of values
Mean ± 2 SD covers 99.7% of values

Skewed Distributions

Skewness shifts the mean but the mode stays fixed.

Distribution	Relationship	Tail Direction
Positively skewed (right skewed)	Mode < Median < Mean	Tail toward right (high values)
Negatively skewed (left skewed)	Mean < Median < Mode	Tail toward left (low values)

Key: Mean is always pulled toward the long tail. In a positively skewed dataset, mean > median > mode.

Quick Compendium of Clinical Pathology 5th Edition

Which Measure to Use?

Situation	Best Measure
Normally distributed data	Mean
Skewed data / outliers present	Median
Nominal / categorical data	Mode
Log-normal distribution (e.g., antibody titers, bacterial counts)	Geometric Mean
Ordinal variables (e.g., pain scale, Likert)	Median
Bimodal distribution	Mode (or report both peaks)

Quick Comparison Table

Feature	Mean	Median	Mode
Formula-based	Yes (parametric)	No (non-parametric)	No
Uses all values	Yes	No	No
Affected by outliers	Most affected	Least affected	Not affected
Affected by skewness	Yes	Moderate	No
Best for	Normal distribution	Skewed data	Nominal/bimodal data
Position in +ve skew	Highest	Middle	Lowest

PSM Exam High-Yield Points

"Which is not a measure of central tendency?" - Standard Deviation (SD), Variance, Range - these are measures of dispersion, not central tendency.
In a positively skewed distribution: Mean > Median > Mode (tail is on the right, mean is pulled right).
In a negatively skewed distribution: Mean < Median < Mode.
Geometric mean is used for antibody titer calculations and log-normal data.
The median is the measure of choice for income data (highly skewed due to extreme earners).
Regression to the mean: Patients selected because of an unusually high/low test result tend to have values closer to the mean on repeat testing - important concept in clinical trial design.

Sources: Henry's Clinical Diagnosis and Management by Laboratory Methods; Quick Compendium of Clinical Pathology 5th Ed.; Rockwood and Green's Fractures in Adults 10th Ed.; Swanson's Family Medicine Review

Inshort pls

Measures of Central Tendency - Quick Notes

3 Main Measures

Measure	Definition	Formula
Mean	Sum ÷ total observations	x̄ = Σx / n
Median	Middle value (50th percentile)	Count-based
Mode	Most frequent value	Count-based

Skewed Distribution - KEY RULE

Distribution	Order
Normal	Mean = Median = Mode
Positive skew (right tail)	Mode < Median < Mean
Negative skew (left tail)	Mean < Median < Mode

Mean is always pulled toward the tail.

Which to Use?

Data Type	Use
Normal data	Mean
Skewed / outliers	Median
Categorical	Mode
Log-normal (antibody titers)	Geometric Mean

Exam Points

SD, Variance, Range = measures of dispersion, NOT central tendency
Geometric Mean minimizes effect of extreme values
Median is best for income data (skewed)
Mean is most affected by outliers; Mode is least affected

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