1. Types of data
- Qualitative (categorical): e.g. colour, gender
- Quantitative: discrete (counts), continuous (measurements)
2. Data presentation
Tables: frequency table, grouped frequency table
Graphs: bar chart, pie chart, histogram, frequency polygon, ogive (cumulative frequency curve)
3. Measures of central tendency
Mean (ungrouped): x̄ = (Σx) / n
Median: middle value when data ordered (or average of two middle values)
Mode: most frequent value
Example 1: Data: 3, 7, 7, 10, 12 → mean = (3+7+7+10+12)/5 = 39/5 = 7.8; median = 7; mode = 7.
Grouped mean (shortcut): For grouped data with class midpoints mᵢ and frequencies fᵢ,
x̄ = (Σ fᵢ mᵢ) / (Σ fᵢ)
4. Measures of dispersion
Range = max − min
Variance (population): σ² = (Σ(x−μ)²)/N
Sample variance: s² = (Σ(x−x̄)²)/(n−1)
Standard deviation: σ = √σ², s = √s²
Example 2: Data: 2, 4, 4, 6 → mean = 4; variance = [(4+0+0+4)/4]=2; sd=√2≈1.414
5. Cumulative frequency & median from grouped data
Use linear interpolation inside the median class:
Median ≈ L + ((N/2 − CF)/f) × w
Where L=lower boundary of median class, CF=cumulative freq before class, f=frequency of class, w=class width
Example 3: Class intervals and frequencies: 0–9:2, 10–19:5, 20–29:8, 30–39:5 (N=20). Median class is 20–29.
L=19.5 (if using boundaries), CF before=7, f=8, w=10.
Median ≈ 19.5 + ((10 − 7)/8)×10 = 19.5 + (3/8)×10 = 19.5 + 3.75 = 23.25
6. Percentiles & Quartiles
Quartiles split data into 4 equal parts. Percentiles into 100 parts.
7. Examples & practice
Practice 1: Calculate mean, median, mode, range for: 5, 8, 12, 5, 9, 5.
Ans: mean=(44)/6≈7.33; median=(5+8)/2=6.5 (ordered:5,5,5,8,9,12); mode=5; range=12−5=7
Practice 2 (grouped): Classes 0–9:3, 10–19:7, 20–29:10, 30–39:5. Find N, cumulative frequencies, approximate median.
Ans: N=25; cumulative before median class (20–29) =10; median class f=10; L=19.5; median≈19.5+((12.5−10)/10)×10=21.0