Chapter 13: Statistics
This chapter deals with the three measures of central tendency for grouped data — mean, mode and median. For board exam 2027, you must know all three formulas by heart, the exact table layouts, and the empirical relationship connecting the three measures.
1. Basic Terms
- Class interval: a group such as 10–20; here 10 is the lower class limit and 20 is the upper class limit.
- Class mark (xi): the mid-point of a class interval.
xi = (upper limit + lower limit)/2. E.g. class mark of 10–25 is 17.5. - Class size (h): difference between upper and lower limits of a class (assumed equal for all classes in the formulas below).
- Frequency (fi): number of observations in a class.
- Cumulative frequency (cf): running total of frequencies up to (and including) a class. Needed only for the median.
2. Mean of Grouped Data
In grouped data we assume that the frequency of each class is centred at its class mark xi. There are three methods; all give (almost) the same answer.
(a) Direct Method
Mean (x̄) = Σfixi / Σfi
(b) Assumed Mean Method
Choose an assumed mean a (usually the middle class mark), take deviations di = xi − a.
Mean (x̄) = a + (Σfidi / Σfi)
(c) Step-Deviation Method
Divide each deviation by the class size h: ui = (xi − a)/h.
Mean (x̄) = a + h × (Σfiui / Σfi)
Worked table (same data, direct and step-deviation methods)
| Class | fi | xi | fixi | ui = (xi − 25)/10 | fiui |
|---|---|---|---|---|---|
| 0–10 | 5 | 5 | 25 | −2 | −10 |
| 10–20 | 8 | 15 | 120 | −1 | −8 |
| 20–30 | 12 | 25 | 300 | 0 | 0 |
| 30–40 | 10 | 35 | 350 | 1 | 10 |
| 40–50 | 5 | 45 | 225 | 2 | 10 |
| Total | 40 | 1020 | 2 |
Direct: Mean = 1020/40 = 25.5. Step-deviation (a = 25, h = 10): Mean = 25 + 10 × (2/40) = 25 + 0.5 = 25.5. Both agree.
Exam tip: Use the step-deviation method when class marks are large or the frequencies are big — it keeps the arithmetic small. If the question does not specify a method, any method earns full marks, but the table must be shown.
3. Mode of Grouped Data
The modal class is the class with the maximum frequency. Then:
Mode = l + ((f1 − f0) / (2f1 − f0 − f2)) × h
- l = lower limit of the modal class
- f1 = frequency of the modal class
- f0 = frequency of the class preceding the modal class
- f2 = frequency of the class succeeding the modal class
- h = class size
Example (data above): maximum frequency 12 ⇒ modal class 20–30, so l = 20, f1 = 12, f0 = 8, f2 = 10, h = 10.
Mode = 20 + ((12 − 8)/(24 − 8 − 10)) × 10 = 20 + (4/6) × 10 = 20 + 6.67 = 26.67 (approx.).
4. Median of Grouped Data
Steps: (1) build a cumulative frequency (cf) column; (2) find n = Σfi and n/2; (3) the median class is the class whose cumulative frequency is greater than or equal to n/2 for the first time; (4) apply:
Median = l + ((n/2 − cf) / f) × h
- l = lower limit of the median class
- n = number of observations (Σfi)
- cf = cumulative frequency of the class preceding the median class
- f = frequency of the median class
- h = class size
Worked table with cf column (same data)
| Class | fi | cf |
|---|---|---|
| 0–10 | 5 | 5 |
| 10–20 | 8 | 13 |
| 20–30 | 12 | 25 |
| 30–40 | 10 | 35 |
| 40–50 | 5 | 40 |
n = 40, n/2 = 20. The first cf ≥ 20 is 25, so the median class is 20–30; l = 20, cf (preceding) = 13, f = 12, h = 10.
Median = 20 + ((20 − 13)/12) × 10 = 20 + 70/12 = 20 + 5.83 = 25.83 (approx.).
Common mistake: In the median formula, cf is the cumulative frequency of the class before the median class — never of the median class itself. Also note that the modal class and the median class need not be the same class.
5. Empirical Relationship
3 Median = Mode + 2 Mean
This is an approximate relation for moderately skewed distributions. Check with our data: 3 × 25.83 = 77.5 and 26.67 + 2 × 25.5 = 77.67 — nearly equal, as expected. Use it in 1-mark questions to find the third measure when two are given, e.g. if Mean = 24 and Mode = 12, then Median = (12 + 48)/3 = 20.
6. Common Board Question Patterns
- Find the mean by a stated method (direct / assumed mean / step-deviation) — 3 marks.
- Find the missing frequency when the mean is given — form the equation Σfixi = Mean × Σfi and solve.
- Find the mode or median of a distribution — 3 marks; find two missing frequencies when the median and total frequency are given — 5 marks.
- 1-mark questions on class mark, modal class, median class, and the empirical relation.
- Case-study questions: a real-life table (daily wages, electricity bills, weights of students) followed by parts asking mean, modal class/mode and median.
Presentation tip: Always draw the full table (xi, fixi or di/ui, or cf) even for short data — marks are allotted to the table. State the modal/median class explicitly before substituting in the formula, and end with a clear statement line such as “Hence, the median marks are 25.83.”
Deleted from the rationalised syllabus (do NOT study)
- Cumulative frequency graphs (ogives) — “less than” and “more than” type curves.
- Graphical determination of the median (intersection of ogives).
Retained: mean by all three methods, mode of grouped data, median of grouped data (by formula only) and the empirical relation 3 Median = Mode + 2 Mean. Bimodal situations are avoided in the exam.