Data Handling — Class 6 IMO

Tally Marks & Pictographs

Data can be recorded with tally marks in groups of five. A pictograph shows data with pictures, where one picture stands for a fixed number (the scale). If 1 picture = 10, then 4 pictures mean 40.

Example 1: On a pictograph 1 symbol = 10 books. How many books do 4 symbols show?

4 × 10 = 40 books.

Example 2: How many does one full tally group stand for?

Five.

Quick recap

Tally marks group counts in fives.
Pictograph: one picture stands for a fixed number (the scale).

✓ Quick check

On a pictograph 1 picture = 5 apples. How many apples do 6 pictures show?

6 × 5 = 30 apples.

Two full tally groups stand for ___ ?

5 + 5 = 10.

Bar Graphs & Double Bar Graphs

A bar graph shows data with bars whose heights give the values. A double bar graph shows two sets of data side by side so they can be compared — for example, the marks of two students.

Example 1: What does a double bar graph let us do?

Compare two sets of data side by side.

Example 2: If one bar is 20 and another is 15, what is the difference?

20 − 15 = 5.

Quick recap

Bar heights give the values; the tallest shows the most.
A double bar graph compares two sets of data.

✓ Quick check

Bar A is 20 and bar B is 15. What is the difference?

20 − 15 = 5.

A double bar graph is used to compare ___ ?

It compares two sets of data.

Mean (Average)

Four Measures of Central Tendency:

Measure	Definition	How to Find	Example: 2,4,6,8,10
Mean	Average	Sum ÷ Count	(2+4+6+8+10)/5 = 6
Median	Middle value	Order, find middle	2,4,6,8,10 → 6
Mode	Most frequent	Find most common	No mode (all appear once)
Range	Spread	Max - Min	10 - 2 = 8

When to Use Each:

Measure	Best Used When
Mean	Data is symmetrical, no outliers
Median	Data has outliers (very high/low values)
Mode	Data has repeating values (favorite colors)
Range	Want to know spread of data

MEAN AS "BALANCE POINT":

    Data: 2, 4, 6, 8, 10
    
    <----|----|----|----|----|---->
         2    4    6    8    10
              ↑    ↑
              ←──┼──→
         Left sum = Right sum
         (2+4=6, 8+10=18) Not equal? Actually mean=6 balances:
         Deviations: -4, -2, 0, +2, +4 → sum of deviations=0


MEDIAN VISUAL:

    Odd count (5 numbers):
    2, 4, 6, 8, 10 → median = 6 (3rd number)
    
    Even count (6 numbers):
    2, 4, 6, 8, 10, 12 → median = (6+8)/2 = 7


MODE VISUAL:

    Data: 3, 5, 5, 7, 7, 7, 9
    
    Frequency:
    9│        ■
    8│        
    7│     ■  ■  ■
    6│        
    5│  ■  ■  ■
    4│        
    3│  ■
    └──────────────
       3  5  7  9
        
    Mode = 7 (appears 3 times)


RANGE VISUAL:

    Data: 15, 22, 31, 18, 45
    
    Min = 15, Max = 45
    Range = 45 - 15 = 30
    
    <─── Range = 30 ───→
    ├─────┼─────────────┤
    15                  45

Example 1: Find the mean of 4, 8, 6.

(4 + 8 + 6) ÷ 3 = 18 ÷ 3 = 6.

Example 2: Find the mean of 10, 20, 30.

(10 + 20 + 30) ÷ 3 = 60 ÷ 3 = 20.

Example 3:

Find mean, median, mode, and range of: 5, 8, 12, 8, 7

Mean = (5+8+12+8+7)/5 = 40/5 = 8
Order: 5,7,8,8,12 → median = 8
Mode = 8 (appears twice)
Range = 12 - 5 = 7
Answer: Mean=8, Median=8, Mode=8, Range=7

Example 4:

Find median of: 22, 15, 30, 18, 25, 20

Order: 15,18,20,22,25,30
Even count: middle two are 20 and 22
Median = (20+22)/2 = 21
Answer: 21

Example 5:

Find mode(s): 1, 2, 2, 3, 3, 4

2 appears twice, 3 appears twice
Two modes = bimodal
Answer: 2 and 3

Quick recap

Mean = sum of values ÷ number of values.
The mean is also called the average.
Mean = sum ÷ count (average)
Median = middle value (order first!)
Mode = most frequent value(s)
Range = max - min (spread)
Outliers affect mean more than median

✓ Quick check

What is the mean of 2, 4, 6, 8?

(2 + 4 + 6 + 8) ÷ 4 = 20 ÷ 4 = 5.

What is the mean of 5, 5, 5?

(5 + 5 + 5) ÷ 3 = 5.