Statistics: Finding the 'typical' value in a pile of numbers

🧮13.1 - 13.2 Mean of Grouped Data

Dump your whole class's test scores into one pile and a measure of central tendency is the friend who looks at that mess and says, in one number, "here's roughly what's going on." You already know three such friends from Class IX — mean, median, and mode. This chapter teaches them to work with grouped data, where individual scores are bundled into class intervals like 10-25, 25-40, and so on, instead of listed one by one.

①Method 1: The Direct Method

For ungrouped data, mean = sum of observations ÷ number of observations. For grouped data, the class mark xᵢ stands in for every observation inside that class, and gets weighted by its frequency fᵢ.

🚀13.2 Assumed Mean & Step-Deviation Methods

When xᵢ and fᵢ are large numbers, direct multiplication turns tedious fast — like doing long multiplication by hand during a timed test. 😩 So we shrink the numbers first, do easy arithmetic, then scale back at the end. The final answer never changes; only the path to it gets shorter.

②Method 2: Assumed Mean Method

Pick any class mark as an 'assumed mean' a — usually the middle one. Find how far every other xᵢ deviates from it: dᵢ = xᵢ − a. The true mean is then a plus the average deviation.

③Method 3: Step-Deviation Method

If the dᵢ values share a common factor (the class size h), shrink them further into uᵢ = (xᵢ − a) / h. It's like converting large rupee amounts into smaller unit-counts before adding — same total, friendlier numbers along the way.

🏆13.3 Mode of Grouped Data

If mean is the 'average performer', mode is the 'crowd favourite' — the value that shows up most often. Think of the most-streamed song of the week: not necessarily the best song by every measure, just the one everyone's playing the most. 🎵

In grouped data we can't spot one exact repeated value, so we locate the modal class — the class interval with the highest frequency — and estimate a mode that lies somewhere inside it.

• l = lower limit of the modal class
• h = size of the class interval (assumed equal)
• f₁ = frequency of the modal class
• f₀ = frequency of the class before the modal class
• f₂ = frequency of the class after the modal class

🔗13.4 Cumulative Frequency & Median of Grouped Data

Median is the 'middle person in the queue' — exactly half the data lies below it, half above. For grouped data, we first build a cumulative frequency table: a running total, like a scoreboard that keeps adding up as each new class interval joins in.

To find the median class, locate the class whose cumulative frequency is the first to reach or exceed n/2.

• l = lower limit of the median class
• n = total number of observations
• cf = cumulative frequency of the class before the median class
• f = frequency of the median class
• h = class size

🧭13.5 Choosing the Right Measure — Summary

Three friends, three personalities. 🤝 Mean accounts for every observation, making it ideal for comparing overall performance — but it's easily dragged around by extreme values, like one billionaire skewing a neighbourhood's 'average income.'

Median ignores exactly how extreme the extremes are, just cares about the middle — perfect for things like typical wages. Mode answers 'what's most popular?' — ideal for favourite TV shows, best-selling shoe sizes, or the most-used vehicle colour.

TOOL 1 Grouped Mean Calculator

Enter class intervals and frequencies; instantly see the mean computed by Direct, Assumed Mean, and Step-Deviation methods side by side, confirming all three agree.

TOOL 2 Mode & Median Finder

Enter class intervals and frequencies; the tool builds the cumulative frequency table, highlights the modal and median classes, and shows the mode/median formula substitution step by step.