Data Handling
1. Introduction to Data
Data is a collection of facts, observations, or information.
Types of data:
- Primary data: Collected FIRST-HAND by the investigator
- Secondary data: Collected by SOMEONE ELSE, used by the investigator
Raw data: Data as originally collected, before any organisation.
2. Frequency Distribution
A frequency distribution organises data by showing how MANY times each value occurs.
Worked Example: The marks of 20 students in a test (out of 10) are: 6, 4, 7, 9, 4, 6, 8, 7, 5, 6, 8, 7, 6, 5, 9, 7, 4, 6, 8, 7
Frequency Distribution Table:
| Marks (x) | Tally Marks | Frequency (f) |
|---|---|---|
| 4 | ||
| 5 | ||
| 6 | ||
| 7 | ||
| 8 | ||
| 9 | ||
| Total | 20 |
'Tally marks are grouped in FIVES — four vertical lines and one diagonal — for easy counting.'
3. Pie Chart (Circle Graph)
A pie chart represents data as SECTORS of a circle. The central angle of each sector is proportional to its frequency.
Central angle = (Frequency / Total) × 360°
Worked Example: Draw a pie chart for the favourite colours of 36 students: Blue: 12, Red: 9, Green: 6, Yellow: 5, Others: 4
| Colour | Frequency | Central Angle |
|---|---|---|
| Blue | 12 | 12/36 × 360 = 120° |
| Red | 9 | 9/36 × 360 = 90° |
| Green | 6 | 6/36 × 360 = 60° |
| Yellow | 5 | 5/36 × 360 = 50° |
| Others | 4 | 4/36 × 360 = 40° |
| Total | 36 | 360° |
4. Histogram
A histogram is a BAR GRAPH for grouped (continuous) data. Bars are TOUCHING (no gaps).
Differences: Histogram vs Bar Graph
| Histogram | Bar Graph |
|---|---|
| Data is NUMERICAL and CONTINUOUS | Data is CATEGORICAL |
| Bars TOUCH each other | Bars have GAPS |
| Width of bars represents class size | All bars have SAME width |
| No gaps between bars | Gaps between categories |
Worked Example: Draw a histogram for the following data:
| Marks | 0–10 | 10–20 | 20–30 | 30–40 | 40–50 |
|---|---|---|---|---|---|
| Frequency | 4 | 8 | 12 | 6 | 2 |
Bars of width 10 with heights 4, 8, 12, 6, 2 respectively, touching each other.
5. Mean (Average)
Mean = Sum of all observations / Number of observations
Formula: x̄ = (Σxᵢ)/n
Worked Example: Find the mean of 12, 15, 18, 21, 24.
Mean = (12 + 15 + 18 + 21 + 24)/5 = 90/5 = 18
Worked Example (Frequency): Find the mean of the following data:
| x | 5 | 8 | 10 | 12 | 15 |
|---|---|---|---|---|---|
| f | 3 | 4 | 6 | 5 | 2 |
Σfx = 5×3 + 8×4 + 10×6 + 12×5 + 15×2 = 15 + 32 + 60 + 60 + 30 = 197 Σf = 3 + 4 + 6 + 5 + 2 = 20 Mean = 197/20 = 9.85
6. Median
The median is the MIDDLE value when data is ARRANGED IN ORDER.
Steps:
- Arrange data in ASCENDING or descending order.
- If n is ODD: Median = value at position (n+1)/2
- If n is EVEN: Median = AVERAGE of values at positions n/2 and n/2 + 1
Worked Example: Find the median of 12, 5, 8, 15, 9, 10, 7.
Arranging: 5, 7, 8, 9, 10, 12, 15 n = 7 (odd). Median position = (7+1)/2 = 4th Median = 9
Worked Example: Find the median of 3, 7, 5, 9, 11, 6.
Arranging: 3, 5, 6, 7, 9, 11 n = 6 (even). Median = (6/2-th + (6/2+1)-th)/2 = (3rd + 4th)/2 = (6 + 7)/2 = 6.5
7. Mode
The mode is the value that occurs MOST FREQUENTLY.
Worked Example: Find the mode of 2, 3, 5, 3, 4, 3, 6, 5, 3, 7.
3 occurs FOUR times (most frequent). Mode = 3.
'A dataset can have ONE mode (unimodal), TWO modes (bimodal), or NO mode (all values occur once).'
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'Mean is always one of the data values' | Mean is an AVERAGE — it may NOT be an actual data point |
| 'Median is the middle value of UNSORTED data' | ALWAYS sort data BEFORE finding the median |
| 'Using histogram for categorical data' | Use BAR GRAPH for categories. Use HISTOGRAM for continuous numerical data |
| 'Finding mode in a uniform dataset' | If all values appear the SAME number of times, there is NO mode |
ICSE Exam Focus (5–7 marks)
- 2-mark questions: Find mean, median, or mode of ungrouped data
- 3-mark questions: Construct frequency distribution table
- 4-mark questions: Draw and interpret pie charts
- 5-mark questions: Draw histogram for grouped data
- 6-mark questions: Combined problems — all three measures of central tendency
Self-Test
Q1. Find the mean of first five prime numbers. A1. First five primes: 2, 3, 5, 7, 11. Mean = (2+3+5+7+11)/5 = 28/5 = 5.6.
Q2. Find the median of 14, 9, 22, 5, 18, 11, 7. A2. Arranging: 5, 7, 9, 11, 14, 18, 22. n=7 (odd). Median (4th) = 11.
Q3. Find the mode of 4, 6, 8, 4, 7, 6, 4, 9, 6, 4. A3. 4 occurs 4 times, 6 occurs 3 times. Mode = 4.
Q4. In a pie chart, the central angle for a category with 40 out of 200 items is: A4. Angle = (40/200) × 360 = 72°.
Q5. The marks of 10 students are 34, 28, 45, 39, 42, 31, 37, 44, 40, 35. Find the mean. A5. Sum = 375. Mean = 375/10 = 37.5.
Q6. When is a histogram used instead of a bar graph? A6. A histogram is used for CONTINUOUS numerical data grouped into intervals (e.g., marks ranges, ages). A bar graph is used for CATEGORICAL or discrete data.
