Statistics - Measures of Position In Data Science

1. What Are Measures of Position?

Measures of position describe where a particular data value stands relative to the rest of the dataset.

They help answer questions like:

Is this value high, low, or typical?

What proportion of data lies below (or above) a given value?

How extreme is a data point?

Unlike measures of central tendency (mean, median) or dispersion (variance, standard deviation), measures of position focus on relative standing.

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2. Why Measures of Position Matter in Data Science**

In data science, measures of position are crucial for:

Outlier detection (e.g., IQR method)

Feature scaling and normalization

Risk assessment (finance, insurance)

Model evaluation (percentile-based metrics)

Fair comparisons across populations

Example:

A test score of 85 means very different things depending on whether it is in the 60th percentile or the 95th percentile.

3. Types of Measures of Position

Main categories:

**Percentiles

Quartiles

Deciles

Z-scores (Standard Scores)

Ranks

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Each provides a different lens on relative position.

4. Percentiles

Definition

The p-th percentile is the value below which p% of the data falls.

Example:

90th percentile = value below which 90% of observations lie.

Properties

Percentiles range from 0 to 100

Not evenly spaced in value—depends on data distribution

How to Compute Percentiles

Given ordered data of size n:

Position of 𝑃𝑝 = 𝑝/100(𝑛+1)

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If the position is not an integer, interpolate.

Example :

Data:

10, 20, 30, 40, 50

Find the 60th percentile.

𝑃60=60/100(5+1)=3.6

Between 3rd and 4th values:

30+0.6(40−30)=36

So, P₆₀ = 36

Interpretation

60% of the data is ≤ 36

40% is ≥ 36

5. Quartiles

Quartiles divide data into four equal parts.

Quartile Meaning

Q₁ 25th percentile

Q₂ 50th percentile (Median)

Q₃ 75th percentile

Interquartile Range (IQR)

IQR=𝑄3−𝑄1

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Why important?

Measures spread of the middle 50%

Robust to outliers :

Used heavily in box plots and anomaly detection

Outlier Detection (IQR Rule)

Lower bound=𝑄1−1.5×𝐼𝑄𝑅

Upper bound=𝑄3+1.5×𝐼𝑄𝑅

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Values outside these bounds are considered outliers.

1. Deciles

Deciles split data into 10 equal parts.

Decile Percentile

D₁ 10th

D₅ 50th (Median)

D₉ 90th

Usage

Income distribution analysis

Population studies

Risk stratification

Example:

Top 10% income earners = above the 9th decile

1. Z-Scores (Standard Scores)
   Definition

A Z-score measures how many standard deviations a value is from the mean.

𝑍=𝑥−𝜇/𝜎

Where:

x = observation

μ = mean

σ = standard deviation

Interpretation

Z-score Meaning

0 Exactly at mean

+1 1 SD above mean

-2 2 SD below mean

Why Z-Scores Are Powerful

Standardize different scales

Enable comparison across datasets

Fundamental in machine learning pre-processing

Basis of normal distribution probabilities

Example

Mean = 70

SD = 10

Score = 85

𝑍=85−70/10=1.5

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Interpretation:

The score is 1.5 standard deviations above the mean

1. Relationship Between Z-Scores and Percentiles:

In a normal distribution:

Z Percentile

0 50%

1 ~84%

2 ~97.5%

-1 ~16%

This connection is vital in:

Hypothesis testing

Probability estimation

Statistical modelling:

1. Ranks
   Definition

Rank assigns an ordinal position to each observation.

Example:

Highest score → Rank 1

Next → Rank 2

Types of Ranking

Dense ranking (1,2,2,3)

Competition ranking (1,2,2,4)

Fractional ranking (2.5 for ties)

Limitations

Ignores magnitude differences

Not suitable for distance-based models

1. Measures of Position vs Measures of Central Tendency
   Aspect Central Tendency Position
   Focus Typical value Relative standing
   Examples Mean, Median Percentiles, Z
   Outliers Sensitive (mean) Often robust
   Use in ML Baseline Feature scaling, anomaly detection

11. Real-World Data Science Applications

1. Machine Learning

Feature normalization using Z-scores

Quantile transformation

1. Finance

Value-at-Risk (VaR) → percentile-based

Risk classification using deciles

1. Healthcare

Growth percentiles (BMI-for-age)

Lab result interpretation

1. Education

Standardized test scores

Admission cut-offs

1. Summary Table
   Measure Purpose Robust to Outliers
   Percentile Relative position Yes
   Quartile Spread & position Yes
   Decile Distribution segmentation Yes
   Z-score Standardized distance No
   Rank Order comparison Yes


2. Key Takeaways

Measures of position explain where a value lies, not just what it is.

Percentiles and quartiles are distribution-free.

Z-scores assume normality but allow deep comparisons.

In data science, they are foundational for scaling, anomaly detection, and interpretation.

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