Chapter 14.1 — Introduction to Statistics
Methods to find the mean of grouped data. This is Lesson 1 of 5 in Chapter 14: Statistics.
What Statistics Adds to a Single Number
A single observation rarely tells the full story — nine days of movie attendance, thirty patients' ages, two hundred families' monthly spending. Statistics takes an entire collection of numbers like this and boils it down to a small set of representative values: the mean, the mode, the median. This chapter builds all three, starting with the one most people already call "the average."
The Mean of Raw Data
The mean — the ordinary average — is the sum of every observation divided by how many observations there are. A movie theatre manager tracking attendance across nine days recorded 81, 89, 92, 85, 93, 62, 85, 105, and 90 people:
Mean = (sum of observations) / (number of observations)
= (81+89+92+85+93+62+85+105+90) / 9
= 782 / 9
= 86.9 (approx.)A second dataset — Virat Kohli's test-century count across eight years, 3, 2, 4, 2, 4, 5, 5, 2 — shows a shortcut worth noticing before moving to larger datasets. Rather than adding eight separate numbers, the repeated values can be grouped by how often each occurs: 3 appears once, 2 appears three times, 4 appears twice, and 5 appears twice.
Sum = 1(3) + 3(2) + 2(4) + 2(5) = 3+6+8+10 = 27
Mean = 27/8 = 3.375 centuries per yearThis regrouped version generalises directly: if observations x₁, x₂, ..., xₙ occur with respective frequencies f₁, f₂, ..., fₙ, the mean becomes (f₁x₁+f₂x₂+...+fₙxₙ)/(f₁+f₂+...+fₙ), usually written Σfᵢxᵢ/Σfᵢ. That single formula is the one every method in the rest of this chapter builds from.
Grouped Data Needs a Class Mark First
Once data gets sorted into class intervals — "470–480," "480–490," and so on — there's no longer a single number to plug into xᵢ for each class, only a range. The fix is the class mark, the midpoint of each interval, standing in for every value that interval actually contains:
class mark = (upper class limit + lower class limit) / 2This is an approximation, not a precise recovery of the original data — a class mark of 475 for the interval "470–480" assumes the values inside that interval average out to roughly 475, which is close enough for grouped-data purposes but not identical to knowing every individual wage. Every method that follows in this chapter accepts that same small trade-off in exchange for being able to work with grouped data at all.
One more detail matters before any class mark can even be computed correctly: whether the class intervals given are continuous (one class's upper limit is the very next class's lower limit, like "470–480" followed by "480–490") or inclusive (a small gap sits between classes, like "471–480" followed by "481–490"). The class mark formula above assumes continuous boundaries. Inclusive classes need converting first — extending every upper limit up and every lower limit down by half the gap size — before the midpoint calculation gives a meaningful class mark at all. Every worked example and problem in this chapter's exercises uses continuous classes directly, except one leaf-measurement problem later on that starts inclusive and needs exactly this conversion before its median can be found.
Three Methods, One Answer: Direct, Assumed Mean, Step Deviation
Once every class has its class mark, the grouped mean can be found three genuinely different ways — and all three are guaranteed to land on the identical answer, since they're really the same formula rearranged rather than three separate ideas.
Direct Method: Mean = Σfᵢxᵢ / Σfᵢ
Assumed Mean Method: Mean = a + Σfᵢdᵢ / Σfᵢ, where dᵢ = xᵢ − a
Step Deviation Method: Mean = a + (Σfᵢuᵢ / Σfᵢ) × h, where uᵢ = (xᵢ − a)/h
a = an assumed mean (any convenient class mark)
h = the class size (assumed equal across all classes)Daily wages for 20 workers, sorted into six Rs. 10-wide classes from 470–480 up to 520–530 with frequencies 2, 3, 4, 2, 5, and 4, demonstrate all three methods landing on the same figure:
| Class (Rs.) | fᵢ | xᵢ | fᵢxᵢ | dᵢ = xᵢ−495 | fᵢdᵢ |
|---|---|---|---|---|---|
| 470–480 | 2 | 475 | 950 | −20 | −40 |
| 480–490 | 3 | 485 | 1455 | −10 | −30 |
| 490–500 | 4 | 495 | 1980 | 0 | 0 |
| 500–510 | 2 | 505 | 1010 | 10 | 20 |
| 510–520 | 5 | 515 | 2575 | 20 | 100 |
| 520–530 | 4 | 525 | 2100 | 30 | 120 |
| Total | 20 | 10070 | 170 |
Direct: 10070 / 20 = 503.50
Assumed Mean (a=495): 495 + 170/20 = 495 + 8.5 = 503.50
Step Deviation (a=495, h=10): 495 + (17/20)×10 = 495 + 8.5 = 503.50All three converge on Rs. 503.50 — not a coincidence, but a guarantee, since the assumed-mean and step-deviation formulas are algebraically just the direct formula rewritten around a chosen reference point a. Choosing a as one of the actual class marks, as done here (a=495, the fourth class), is what makes the middle dᵢ value come out to exactly 0, keeping the rest of the arithmetic noticeably smaller than working with the raw class marks directly.
Choosing Which Method to Use
The direct method is the most transparent but becomes the most arithmetically heavy once class marks are large, awkward numbers — multiplying every fᵢ by a four-digit xᵢ adds up fast across many classes. The assumed mean method trims that load by working with the smaller deviations dᵢ instead. The step deviation method trims it further still, dividing every deviation by the class size h before multiplying — genuinely useful when h is large, since it keeps every uᵢ value down to a small integer. None of the three is "more correct" than the others; the choice is purely about which one keeps the arithmetic most manageable for a given dataset.
A useful habit when facing a new grouped dataset: glance at the class marks before picking a method. Small class marks, generally single or double digits, rarely benefit from anything beyond the direct method. Large class marks in the hundreds or thousands are where the assumed mean method starts paying off. A large, evenly-sized class width on top of that is the specific signal that step deviation is worth the extra uᵢ column — without an evenly-sized class width, dividing by a single h wouldn't even be valid, since h has to be the same across every class for the step deviation formula to apply at all.
Where This Chapter Goes
Exercise 14.1 applies all three mean methods across nine genuinely different datasets — daily wages, plant counts, pollution readings, literacy rates — including one problem that runs the mean formula backward to recover a missing frequency. Exercise 14.2 introduces the mode, the single most frequently occurring value, with its own dedicated formula for grouped data. Exercise 14.3 introduces the median, the exact middle value once everything is sorted, built from a new column — cumulative frequency — that none of the mean methods needed. Exercise 14.4 closes the chapter by turning that same cumulative-frequency idea into a graph, reading the median directly off a curve instead of a formula.
Mean, mode, and median all answer the same underlying question — what single number best represents an entire dataset — but they answer it differently enough that a real dataset's three values rarely land on exactly the same number, and comparing them against each other, as several problems ahead do directly, often reveals more about the data's actual shape than any one of the three could alone.