Chapter 14.3 — Exercise 14.2 — Mode of Grouped Data
Mode of the grouped data. This is Lesson 3 of 5 in Chapter 14: Statistics.
From Average to Most Common
The ordinary mean answers "what does this data average out to." The mode answers a different question entirely — which single value shows up more often than any other. A dataset can have one mode (unimodal), two (bimodal), or more, and for grouped data it needs its own formula, built from the class with the highest frequency and its two immediate neighbours.
The Mode Formula for Grouped Data
Mode = l + [(f₁−f₀) / (2f₁−f₀−f₂)] × h
l = lower boundary of the modal class
f₁ = frequency of the modal class (the class with maximum frequency)
f₀ = frequency of the class immediately before the modal class
f₂ = frequency of the class immediately after the modal class
h = size of the modal classTwenty households' family sizes, sorted into five classes of size 2, illustrate every piece of this formula at once:
| Class (family size) | fᵢ (families) |
|---|---|
| 1–3 | 7 |
| 3–5 | 8 ← modal class |
| 5–7 | 2 |
| 7–9 | 2 |
| 9–11 | 1 |
Modal class = 3–5 (highest frequency, 8)
l=3, f₁=8, f₀=7, f₂=2, h=2
Mode = 3 + [(8−7)/(16−7−2)]×2 = 3 + (1/7)×2 ≈ 3.29Only three numbers from the entire table ever enter the formula — the modal class's own frequency and its two direct neighbours. The classes further away (5–7, 7–9, 9–11 here) play no role in the calculation at all, which is worth noticing: unlike the mean, which genuinely weighs every single class, the mode formula is a strictly local calculation around wherever the data happens to peak.
Hospital Patients: Mode Against Mean
Eighty patients' ages, grouped into six 10-year classes from 5–65, ask for both the mode and the mean side by side — a direct comparison of what each measure actually reports about the same dataset.
| Class (years) | fᵢ | xᵢ | fᵢxᵢ |
|---|---|---|---|
| 5–15 | 6 | 10 | 60 |
| 15–25 | 11 | 20 | 220 |
| 25–35 | 21 | 30 | 630 |
| 35–45 | 23 ← modal | 40 | 920 |
| 45–55 | 14 | 50 | 700 |
| 55–65 | 5 | 60 | 300 |
| Total | 80 | 2830 |
Mode: l=35, f₁=23, f₀=21, f₂=14, h=10
Mode = 35 + [(23−21)/(46−21−14)]×10 = 35 + (2/11)×10 ≈ 36.82 years
Mean (direct method): 2830/80 = 35.375 ≈ 35.38 yearsThe mode (36.82) sits noticeably above the mean (35.38) here, and the reason is visible directly in the frequency table: the two classes right around the peak (25–35 and 45–55, with 21 and 14 patients) are heavier than the classes at either extreme, pulling the mode toward where patients actually cluster most densely, while the mean keeps averaging in the thinner, older tail (55–65) that drags it back down slightly. Neither number is "wrong" — they're simply measuring different things about the same 80 patients.
Electrical Components' Lifetimes
Two hundred twenty-five electrical components, grouped by lifetime into six 20-hour classes, ask for the mode alone — a single-formula application once the modal class is identified.
| Class (hours) | Frequency |
|---|---|
| 0–20 | 10 |
| 20–40 | 35 |
| 40–60 | 52 |
| 60–80 | 61 ← modal |
| 80–100 | 38 |
| 100–120 | 29 |
l=60, f₁=61, f₀=52, f₂=38, h=20
Mode = 60 + [(61−52)/(122−52−38)]×20 = 60 + (9/32)×20 = 60+5.625 = 65.625 hoursNotice that the modal class here, 60–80, isn't flanked by two roughly equal neighbours — 52 on one side, only 38 on the other. That asymmetry is exactly what pulls the mode toward 65.6 rather than landing near the modal class's own midpoint of 70; a modal class with a much heavier neighbour on one side always pulls the mode away from that heavier side.
Household Expenditure: Step Deviation Meets Mode
Two hundred families' monthly household expenditure in Gummadidala village, grouped into eight Rs. 500-wide classes, asks for the mode alongside a mean computed via the step deviation method — the two formulas run independently, on the same table.
| Class (Rs.) | fᵢ | xᵢ | uᵢ=(xᵢ−2750)/500 | fᵢuᵢ |
|---|---|---|---|---|
| 1000–1500 | 24 | 1250 | −3 | −72 |
| 1500–2000 | 40 ← modal | 1750 | −2 | −80 |
| 2000–2500 | 33 | 2250 | −1 | −33 |
| 2500–3000 | 28 | 2750 | 0 | 0 |
| 3000–3500 | 30 | 3250 | 1 | 30 |
| 3500–4000 | 22 | 3750 | 2 | 44 |
| 4000–4500 | 16 | 4250 | 3 | 48 |
| 4500–5000 | 7 | 4750 | 4 | 28 |
| Total | 200 | −35 |
Modal class = 1500–2000 (highest frequency, 40): l=1500, f₁=40, f₀=24, f₂=33, h=500
Mode = 1500 + [(40−24)/(80−24−33)]×500 = 1500 + (16/23)×500 ≈ 1847.83
Mean (step deviation): 2750 + (−35/200)×500 = 2750 − 87.5 = Rs. 2662.50It's worth double-checking which class actually has the maximum frequency before applying the formula, rather than assuming it's the first or most prominent-looking row — here it's the second class, 1500–2000 with 40 families, not the first class, even though 24 is still a fairly large number in its own right. The mode, Rs. 1847.83, and the mean, Rs. 2662.50, land noticeably far apart this time — a wider gap than the hospital-patients example above, reflecting a more heavily skewed spending distribution among these 200 families.
Teacher-Student Ratio
Thirty-five Indian states' pupil-teacher ratios, grouped into eight classes of size 5 (with two classes recording zero states), ask for the mode and mean together, then an interpretation of what the two values suggest about the ratios.
| Class (students per teacher) | fᵢ (states) | xᵢ | dᵢ=xᵢ−32.5 | fᵢdᵢ |
|---|---|---|---|---|
| 15–20 | 3 | 17.5 | −15 | −45 |
| 20–25 | 8 | 22.5 | −10 | −80 |
| 25–30 | 9 | 27.5 | −5 | −45 |
| 30–35 | 10 ← modal | 32.5 | 0 | 0 |
| 35–40 | 3 | 37.5 | 5 | 15 |
| 40–45 | 0 | 42.5 | 10 | 0 |
| 45–50 | 0 | 47.5 | 15 | 0 |
| 50–55 | 2 | 52.5 | 20 | 40 |
| Total | 35 | −115 |
Mode: l=30, f₁=10, f₀=9, f₂=3, h=5
Mode = 30 + [(10−9)/(20−9−3)]×5 = 30 + (1/8)×5 = 30.625
Mean: 32.5 + (−115/35) = 32.5 − 3.29 ≈ 29.21The two zero-frequency classes (40–45, 45–50) don't break the calculation at all — they simply contribute nothing to either Σfᵢdᵢ or the mode formula, since neither one is the modal class or its immediate neighbour. Mode (30.6) and mean (29.2) land close together this time, both suggesting most states cluster near a ratio of roughly 30 students per teacher, unlike the more widely separated mode and mean seen in the household-expenditure problem just above.
Two More, Mode Only
| Dataset | Modal class | Mode |
|---|---|---|
| Runs scored by top ODI batsmen (8 classes, 3000–11000) | 4000–5000 (18 batsmen) | ≈4608.7 runs |
| Cars passing a spot in a time interval (8 classes, 0–80) | 40–50 (20 intervals) | ≈44.7 cars |
Both problems here skip the mean entirely and ask for the mode alone — a reminder that the mode is a genuinely standalone measure, not something that only makes sense alongside a mean for comparison. Sometimes "which value occurs most often" is exactly the question being asked, with no need for any other measure of the data at all. A sports statistic like "most common run-total among top batsmen" or a traffic count like "busiest ten-minute window for cars" are both naturally mode questions from the outset, not mean questions dressed up differently.
Onward to the Median
Every calculation in this exercise located a dataset's single most frequent value using the modal class and its two immediate neighbours. Exercise 14.3 introduces the third and final measure of central tendency, the median — the exact middle value once a dataset is sorted — built from a genuinely new column, cumulative frequency, that neither the mean nor the mode formulas ever needed. Revisit Exercise 14.1 for the mean methods this exercise repeatedly compared its mode calculations against.