SOF IMO · Bite-size lessons

Statistics

About 7 min · Mathematical Reasoning · English

Time worked out from the lesson's 469 words at 150 words a minute, plus a minute for each question.

Statistics is the branch of mathematics dealing with collection, organization, analysis and interpretation of numerical data. For SOF IMO, this topic focuses on measures of central tendency (mean, median, mode) for both ungrouped and grouped data, along with graphical representation of data through bar graphs, histograms, frequency polygons and pie charts.

Key ideas

• Ungrouped data consists of individual observations listed separately, while grouped data organizes observations into class intervals with their frequencies. Ungrouped data is easier to handle but becomes unwieldy for large datasets.
• Mean (arithmetic average) represents the sum of all observations divided by their count. For grouped data, we use class marks (midpoints) multiplied by frequencies, making it sensitive to extreme values.
• Median is the middle value when data is arranged in ascending or descending order, making it resistant to outliers. For even number of observations, median equals the average of the two middle values.
• Mode is the observation or class interval with the highest frequency. A dataset can be unimodal (one mode), bimodal (two modes) or multimodal, while some datasets have no mode if all frequencies are equal.
• Class mark (midpoint) for any class interval equals (lower limit + upper limit)/2 and serves as the representative value for that entire class in grouped data calculations.

Watch out for

• Confusing class mark with class boundaries → Class mark is the midpoint used for calculations, while boundaries (like 9.5, 19.5) are used for continuous data representation. For mean calculation, always use class marks.
• Forgetting to arrange data in order before finding median → Students often pick the middle position from unsorted data. Always sort first: median requires ordered data.
• Using mode instead of modal class in grouped data → For grouped data, we first identify the modal class (highest frequency interval) and then find the mode with Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h. Don't report a class mark as "the mode."

Quick reference

• Mean = Σ(f × x) / Σf for grouped data; sensitive to extreme values
• Median = middle value when ordered; use n/2 position for cumulative frequency in grouped data
• Mode = highest frequency observation or class
• Class mark = (Lower + Upper limit) / 2
• Always arrange ungrouped data in order before finding median

Check yourself: 3 questions

This lesson is cut from the topic's note, not rewritten. Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain. The questions are from Shishya's validated question bank.

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