By Rishabh Kumar, IIT Guwahati + ISI Alumnus, Founder – Mathematics Elevate Academy

📌 Introduction: Why Descriptive Statistics Matter

Descriptive statistics are the first step in understanding data — whether you’re analyzing IB internal assessment results, interpreting A-Level exam trends, or preparing for university-level studies. Before diving into advanced topics like probability distributions or inferential statistics, you must master the tools that describe, summarize, and visualize data effectively.

In this post, we’ll cover all the essential concepts — mean, median, mode, standard deviation, quartiles, box plots, and more — with simple explanations, examples, and exam-specific tips for IB Math (AA & AI HL/SL) and A-Level Mathematics students.

🧠 What is Descriptive Statistics?

Descriptive statistics is a branch of statistics that deals with:

• Organizing data (tables, charts)
• Summarizing data (averages, variability)
• Interpreting key features of a dataset

Think of it as telling the “story” of a dataset before making any predictions or assumptions.

📊 1. Types of Data

Before summarizing, understand what you’re summarizing:

🔍 IB & A-Level exams often test this distinction directly.

➗ 2. Measures of Central Tendency

These values represent the “center” of a dataset.

🧮 Mean

Mean=∑xn\text{Mean} = \frac{\sum x}{n}Mean=n∑x​

Use for symmetrical data. Affected by outliers.

📍 Median

Middle value when data is ordered.

Use for skewed data or when outliers are present.

🔁 Mode

Most frequent value(s).

Useful for categorical or multimodal data.

📏 3. Measures of Spread (Dispersion)

Tells you how scattered the data is around the mean.

📉 Range

Range=Max−Min\text{Range} = \text{Max} – \text{Min}Range=Max−Min

📦 Interquartile Range (IQR)

IQR=Q3−Q1\text{IQR} = Q_3 – Q_1IQR=Q3​−Q1​

Helps in detecting outliers. Used in boxplots.

🔢 Variance and Standard Deviation

• Sample SD:

s=1n−1∑(xi−xˉ)2s = \sqrt{\frac{1}{n-1} \sum (x_i – \bar{x})^2}s=n−11​∑(xi​−xˉ)2​

• Population SD:

σ=1N∑(xi−μ)2\sigma = \sqrt{\frac{1}{N} \sum (x_i – \mu)^2}σ=N1​∑(xi​−μ)2​

IB and A-Level exams frequently require you to calculate SD using GDC/Excel and interpret its meaning.

📉 4. Quartiles, Percentiles & Z-Scores

These show the position of a value within the distribution.

🎯 Quartiles

• Q1Q_1Q1​: 25th percentile
• Q2Q_2Q2​: Median (50th percentile)
• Q3Q_3Q3​: 75th percentile

🧮 Z-Score

z=x−μσz = \frac{x – \mu}{\sigma}z=σx−μ​

Tells how far a value is from the mean in standard deviation units.

📊 5. Data Visualization Tools

📊 Histograms

• Visualizes frequency distribution of continuous data.
• No gaps between bars.

📌 Box and Whisker Plots

• Show minimum, Q1Q_1Q1​, median, Q3Q_3Q3​, and maximum.
• Detect outliers and compare distributions.

🔘 Cumulative Frequency Graphs (Ogives)

• Useful for estimating medians and quartiles.
• Common in IB Paper 2 questions.

📈 Scatter Plots

• Show correlation between two variables.
• Positive, negative, or no correlation.
• Used in regression lines and correlation coefficient rrr.

🔄 6. Correlation and Covariance (Intro)

🔗 Covariance

Shows direction of the relationship between two variables:

• Positive = move together
• Negative = move in opposite directions

📏 Correlation Coefficient (Pearson’s r)

r=Cov(X,Y)σXσYr = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y}r=σX​σY​Cov(X,Y)​

• Ranges from −1-1−1 to +1+1+1
• IB/A-level typically focus on linear correlation in scatter plots

📘 IB and A-Level Exam Tip Sheet

🎓 Final Thoughts

Descriptive statistics are essential for building statistical intuition. Whether you’re tackling an IA dataset, answering Paper 2 questions, or preparing for uni-level econometrics or ML, you must speak the language of data.

Master these concepts now, and you’ll find inferential statistics, probability distributions, and hypothesis testing far easier.

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