Statistics
Statistics
Collection and Organisation of Data
| Term | Definition | Example |
|---|---|---|
| Raw Data | Unprocessed collected values | Marks of 30 students: 45, 68, 72, 45, 91… |
| Tally Marks | Counting in groups of five | |||| //// || (12) |
| Frequency | Number of times a value appears | 45 appears 3 times → f = 3 |
| Ungrouped Data | Individual data values listed | 12, 15, 18, 20, 22 |
| Grouped Data | Data arranged in class intervals | 0–10, 10–20, 20–30… |
Class Interval: 0–10 (inclusive form), 0–9.5 (exclusive form)
Class Limits: Lower = 0, Upper = 10
Class Mark = (Lower + Upper)/2 = 5
Class Width = Upper – Lower = 10
Class Limits: Lower = 0, Upper = 10
Class Mark = (Lower + Upper)/2 = 5
Class Width = Upper – Lower = 10
In Indian board exams, continuous classes (no gap) are preferred for histograms.
Graphical Representation of Data
| Graph Type | Data Type | Key Feature |
|---|---|---|
| Pictograph | Discrete | Uses symbols (1 symbol = fixed quantity) |
| Bar Graph | Discrete/Categorical | Bars of equal width, height ∝ frequency |
| Double Bar Graph | Comparison of two data sets | Two bars side-by-side for each category |
| Histogram | Continuous grouped data | No gaps between bars |
| Frequency Polygon | Grouped data | Line joining mid-points of histogram tops |
| Pie Chart | Percentage distribution | Central angle = (Component/Total) × 360° |
India’s 2011 Census favourite graphs:
• Population by religion → Pie chart
• Literacy rate by states → Bar graph
• Rural-Urban population → Double bar graph
• Population by religion → Pie chart
• Literacy rate by states → Bar graph
• Rural-Urban population → Double bar graph
Reading and Interpreting Graphs
Most frequent exam traps:
• Misreading scale on Y-axis (especially when it starts from non-zero)
• Confusing percentage with actual value in pie charts
• Counting gaps in histogram as zero frequency
• Misreading scale on Y-axis (especially when it starts from non-zero)
• Confusing percentage with actual value in pie charts
• Counting gaps in histogram as zero frequency
| Graph | What examiners ask |
|---|---|
| Pictograph | If 1 symbol = 500 units, find total for 7 symbols |
| Double Bar | Which state showed maximum improvement? |
| Histogram | Frequency of class 30–40 when width varies |
| Pie Chart | Find central angle for 25% share → 90° |
Measures of Central Tendency – Ungrouped Data
| Measure | Formula | When to use |
|---|---|---|
| Mean (x̄) | Σx / n | Average marks, income |
| Median | (n+1)/2 th term (odd) Average of n/2 and n/2 + 1 (even) |
Skewed data, house prices |
| Mode | Most frequent value | Most popular shoe size |
For ungrouped data in exams:
Arrange in ascending order → Median position
Highest frequency → Mode (can be bimodal)
Arrange in ascending order → Median position
Highest frequency → Mode (can be bimodal)
Measures of Central Tendency – Grouped Data
Arithmetic Mean (Three Methods)
Direct: Σ(fᵢxᵢ) / Σfᵢ
Assumed Mean: x̄ = a + Σ(fᵢdᵢ)/Σfᵢ
Step Deviation: x̄ = a + h × [Σ(fᵢuᵢ)/Σfᵢ]
Direct: Σ(fᵢxᵢ) / Σfᵢ
Assumed Mean: x̄ = a + Σ(fᵢdᵢ)/Σfᵢ
Step Deviation: x̄ = a + h × [Σ(fᵢuᵢ)/Σfᵢ]
| Measure | Formula | Key Point for Exams |
|---|---|---|
| Median | l + [(N/2 – cf)/f] × h | l = lower limit of median class N = Σf |
| Mode | l + [(f₁ – f₀)/(2f₁ – f₀ – f₂)] × h | f₁ = frequency of modal class f₀, f₂ = previous & next |
| Empirical Relation | Mode = 3 Median – 2 Mean | Used in many CTET/UPTET questions |
Cumulative Frequency Distribution and Ogive
Less-than Ogive: Plot upper limit vs ≤ cumulative frequency
More-than Ogive: Plot lower limit vs ≥ cumulative frequency
Median ≈ intersection point of both ogives
More-than Ogive: Plot lower limit vs ≥ cumulative frequency
Median ≈ intersection point of both ogives
In board & competitive exams:
• Median from ogive is accepted even if slightly different from formula method
• Always draw both ogives when asked for “graphical location of median”
• Median from ogive is accepted even if slightly different from formula method
• Always draw both ogives when asked for “graphical location of median”
Applications and Problem Solving
| Real-Life India Examples | Favourite Exam Context |
|---|---|
| Runs scored by Virat Kohli in 20 innings | Find mean, median, mode |
| Monthly expenditure of 50 families | Missing frequency given mean |
| Literacy rate of states (pie chart) | Find angle for Uttar Pradesh |
| Rainfall in Delhi (histogram) | Days with rainfall > 30 mm |
Classic Missing Frequency Problem:
“If mean of 100 observations is 45 and mean of first 60 is 42, find mean of last 40”
→ (100×45 – 60×42)/40 = 49.5
“If mean of 100 observations is 45 and mean of first 60 is 42, find mean of last 40”
→ (100×45 – 60×42)/40 = 49.5