Chapter 7.1 — Introduction — Central Tendency

The Arithmetic Mean

A trader's sales across six days — ₹5000, ₹4200, ₹4800, ₹6600, ₹5400, ₹5200 — total ₹31,200, and dividing by 6 gives a mean of ₹5,200. The relationship also runs in reverse: knowing the mean and the count tells you the total, without needing the individual values at all — sum = mean × number of observations, so a trader whose average sales over 6 days is known to be ₹5,200 must have taken in ₹31,200 in total that week.

What Makes the Mean Behave Predictably

Two properties make the mean easy to reason about without recomputing it from scratch:

• Shift every observation by the same amount, and the mean shifts by that same amount. The observations 7, 10, 15, 21, 27 have mean 16; adding 4 to every single one (11, 14, 19, 25, 31) gives a new mean of exactly 20 — 4 more, matching the shift applied.
• Scale every observation by the same factor, and the mean scales by that same factor. Multiplying those same original observations by 3 (21, 30, 45, 63, 81) gives a new mean of 48 — exactly 3 times the original 16.

Both properties follow from how the mean is built: since every term inside the sum shifts or scales identically, the whole sum shifts or scales identically too, and dividing by the same unchanged count carries that shift or scale straight through to the final mean.

Why the Mean Formula Uses Σ Notation

Written formally, the mean of observations x₁, x₂, ..., xₙ is x̄ = Σxᵢ/n, where the Greek letter sigma (Σ) simply means "add up everything that follows it." This is nothing more than shorthand for "sum of observations divided by number of observations" — the same formula already used above, just written in a form that scales cleanly to a data set of any size without needing to write out every single term by name. Getting comfortable reading Σxᵢ as "add up all the x-values" rather than treating it as a new, separate concept makes the notation far less intimidating than it first appears, and it's notation that reappears constantly in statistics from this point onward, in Class 9 and beyond, always meaning exactly the same thing: sum everything the sigma symbol sits in front of, nothing more mysterious than that, however unfamiliar or oddly strange the symbol itself might look at first glance.

The Median — The Middle, Not the Average

Arrange the data in order (ascending or descending, either works), and the median is whatever value sits in the middle.

For 12, 21, 15, 19, 11 (5 values): ordered, this is 11, 12, 15, 19, 21, and since n=5 is odd, the median is the 3rd value: 15. For 12, 21, 15, 19, 11, 17 (6 values): ordered, this is 11, 12, 15, 17, 19, 21, and since n=6 is even, the median is the average of the 3rd and 4th values, (15+17)/2 = 16.

Unlike the mean, the median doesn't care about exactly how large or small the extreme values are — only about their position once sorted. A single very large or very small observation can pull the mean noticeably, while barely affecting the median at all.

Why the Median Needs Two Different Formulas

The split into an odd-count formula and an even-count formula isn't arbitrary — it comes directly from what "the middle" means for a list with no exact centre. An odd-length list has one value sitting precisely in the middle, with equal counts on either side, so that single value is the median outright. An even-length list has no single middle value — instead, two values sit closest to the centre, one on each side — so the median is defined as their average, splitting the difference between the two candidates for "most central" as evenly as possible.

The Mode — Whatever Shows Up Most

The mode is simply the most frequently occurring value. In 10, 8, 6, 8, 5, 7, 7, 8, the value 8 appears three times — more than anything else — so the mode is 8. A data set can have more than one mode (7, 6, 5, 6, 7, 4, 6, 7 has both 6 and 7 tied for most frequent), or none at all if every value appears exactly once, as in 3, 5, 6, 7, 9, 11.

What Happens When One Number Is Way Off From the Rest

It's worth seeing directly how differently the mean and median respond to an unusually extreme value. Take the five ages 20, 22, 24, 26, 78 — the last one far larger than the rest. The mean is (20+22+24+26+78)/5 = 34, pulled noticeably upward by the single large value, even though four of the five ages sit well below it. The median, however, is simply the middle value once sorted — 24 — completely unaffected by exactly how large that fifth value happens to be; it would give the same median of 24 whether the largest age were 78, 780, or 7,800 — the median only sees where a value ranks, never how far it sits from its neighbours. This is precisely why the median is often preferred over the mean when a data set might contain a few unusually extreme values — income data and property prices are classic examples where a handful of very large values would otherwise distort the mean far more than they distort the median.

Three Different Ways of Answering the Same Question

Mean, median and mode all answer "what's typical here?", but they can give genuinely different answers on the same data, and each is more informative in different situations. The mean is sensitive to every value and every extreme; the median only cares about rank order; the mode only cares about repetition. A trader tracking average daily earnings wants the mean; a report on "typical" house prices in an area with a few extremely expensive outliers might prefer the median; a shop restocking the most commonly requested shoe size wants the mode. Choosing the right one for a given question, not just calculating all three by rote, is the real skill this chapter builds toward — starting with Exercise 7.1, and continuing into how these same three ideas are computed once data is grouped into class intervals in Exercise 7.2 — where the individual observations are no longer listed one by one, but the same underlying questions about "what's typical" still need answering.