Statistics — Mean, Median, Mode
Introduction
Statistics deals with collection, organisation, analysis, and interpretation of data. In ICSE Class 10, you learn to compute measures of central tendency — mean, median, and mode — for both grouped and ungrouped data, and to represent data graphically through histograms and ogives.
Mean for Ungrouped Data
Mean = (Sum of all observations) / (Number of observations) = Σx / n
Mean for Grouped Data
Direct Method
Mean = Σ(fᵢ × xᵢ) / Σfᵢ
Where xᵢ = class mark = (lower limit + upper limit) / 2, fᵢ = frequency.
Assumed Mean Method
Mean = A + Σ(fᵢ × dᵢ) / Σfᵢ
Where A = assumed mean, dᵢ = xᵢ − A, xᵢ = class mark.
Step Deviation Method
Mean = A + [Σ(fᵢ × uᵢ) / Σfᵢ] × h
Where uᵢ = (xᵢ − A) / h, h = class size.
Median
For Ungrouped Data
Arrange data in ascending order. If n is odd: median = (n+1)/2 th term. If n is even: median = average of n/2 th and (n/2 + 1) th terms.
For Grouped Data
Median = l + [(n/2 − cf) / f] × h
Where l = lower limit of median class, n = total frequency, cf = cumulative frequency of class preceding median class, f = frequency of median class, h = class size.
Mode
For Ungrouped Data
The value that occurs most frequently.
For Grouped Data
Mode = l + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h
Where l = lower limit of modal class, f₁ = frequency of modal class, f₀ = frequency of class preceding modal class, f₂ = frequency of class succeeding modal class, h = class size.
Empirical Relationship
3 Median = Mode + 2 Mean
Cumulative Frequency Curve (Ogive)
- Less than ogive — Plot less than cumulative frequencies against upper class boundaries.
- More than ogive — Plot more than cumulative frequencies against lower class boundaries.
- The intersection of the two ogives gives the median.
Quartiles
Q₁ = l + [(n/4 − cf) / f] × h (First quartile) Q₂ = Median = l + [(n/2 − cf) / f] × h (Second quartile) Q₃ = l + [(3n/4 − cf) / f] × h (Third quartile)
Worked Examples
Example 1: Mean (Direct Method)
| Class | 0−10 | 10−20 | 20−30 | 30−40 | 40−50 |
|---|---|---|---|---|---|
| Frequency | 5 | 8 | 12 | 7 | 3 |
Find the mean.
Solution: Class marks: 5, 15, 25, 35, 45 Σf = 5 + 8 + 12 + 7 + 3 = 35 Σfx = 5×5 + 8×15 + 12×25 + 7×35 + 3×45 = 25 + 120 + 300 + 245 + 135 = 825 Mean = 825 / 35 = 23.57
Example 2: Median (Grouped Data)
Using the same data, find the median.
Solution: n/2 = 35/2 = 17.5 Cumulative frequencies: 5, 13, 25, 32, 35 Median class: 20−30 (cf of preceding class = 13, f = 12) Median = 20 + (17.5 − 13) / 12 × 10 = 20 + 4.5/12 × 10 = 20 + 3.75 = 23.75
Example 3: Mode (Grouped Data)
Using the same data, find the mode.
Solution: Modal class: 20−30 (highest frequency = 12) l = 20, f₁ = 12, f₀ = 8, f₂ = 7, h = 10 Mode = 20 + (12 − 8) / (24 − 8 − 7) × 10 = 20 + 4/9 × 10 = 20 + 4.44 = 24.44
Example 4: Empirical Relationship
If mean = 25 and mode = 22, find the median.
Solution: Using 3 Median = Mode + 2 Mean 3 Median = 22 + 2 × 25 = 22 + 50 = 72 Median = 72 / 3 = 24
Histogram vs Bar Graph
| Feature | Histogram | Bar Graph |
|---|---|---|
| Data type | Continuous (grouped) | Discrete/categorical |
| Bar width | Proportional to class size | Uniform width |
| Bars touching | Yes | No |
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| Using class limits instead of class marks | Class mark = (L + U) / 2 |
| Confusing cumulative frequency with frequency | cf = sum of all previous frequencies |
| Wrong median class identification | Find n/2 first, then locate in cumulative frequency column |
| Including gaps in histogram | Bars must touch (no gaps between bars) |
ICSE Exam Focus
Statistics carries 10–14 marks in ICSE exams. Questions include:
- Computing mean (direct/assumed mean/step deviation).
- Computing median (ungrouped and grouped).
- Computing mode (grouped data).
- Drawing ogives and reading median from graph.
- Histogram-based problems.
Marks Blueprint:
| Topic | Marks |
|---|---|
| Mean (any method) | 3 |
| Median (grouped data) | 3 |
| Mode (grouped data) | 3 |
| Ogive / Histogram | 4 |
| Quartiles | 2 |
Self-Test Questions
- Compute the mean of the following data using the step deviation method:
| Class | 10−20 | 20−30 | 30−40 | 40−50 | 50−60 |
|---|---|---|---|---|---|
| f | 4 | 6 | 10 | 5 | 5 |
-
Find the median for the data in Q1.
-
Find the mode for the data in Q1.
-
If median = 30 and mean = 28, find the mode using the empirical relationship.
-
Explain how to draw a 'less than' ogive and how the median can be found from it.
-
Draw a histogram and estimate the mode for the frequency distribution:
| Marks | 0−10 | 10−20 | 20−30 | 30−40 | 40−50 |
|---|---|---|---|---|---|
| Students | 3 | 7 | 10 | 6 | 4 |
In ICSE, statistical computations carry method marks. Write each step clearly — Σfx, Σf, and the formula — even if your final numeric answer is off by rounding.
