Unit: Data Handling & Analysis
Chapter: Organizing Data
Reference: – What is Data, Raw Data, Frequency Distribution Table, Grouped and Ungrouped Data, Class Intervals, Tally Marks, Range of Data, Choosing Class Size, Data Sorting (Ascending and Descending), Solved Examples, Odd-One-Out Problems, Common Mistakes
After studying this chapter, you should be able to understand:
- What is Data and Why We Need to Organize It
- How to Create a Frequency Distribution Table
- Difference Between Grouped and Ungrouped Data
- How to Use Tally Marks
- How to Find the Range of Data
Introduction to Organizing Data
Definition
Data is a collection of facts, numbers, or information. Raw data is data that has not been processed or organized. Organizing data means arranging it in a meaningful way so that patterns become visible and information can be easily understood and analysed.
When we organize data, we essentially ask:
"What does this collection of numbers tell us? How can we arrange it to see patterns clearly?"
Once organized, data can be displayed in tables, charts, or graphs for further analysis.
Importance of Organizing Data
- Turns messy raw data into useful information
- Helps identify patterns, trends, and outliers
- Makes it possible to calculate statistics (mean, median, mode)
- Essential for making data-driven decisions
- Foundation for all statistical analysis
Example
Raw data: test scores 85, 72, 85, 90, 68, 72, 85, 76, 90, 85
Organized: Score 68 appears 1 time, 72 appears 2 times, 76 appears 1 time, 85 appears 4 times, 90 appears 2 times. Now we can easily see that 85 is the most common score.
Subtopics
1. Raw Data
Raw data is data exactly as it was collected, before any organization or processing. It may be listed randomly or in the order collected.
Example of Raw Data:
Heights (in inches) of 15 students: 62, 65, 60, 62, 68, 65, 62, 70, 65, 62, 66, 68, 65, 62, 64
This list is hard to interpret quickly because the numbers are not in order and frequencies are not obvious.
2. Sorting Data
Sorting means arranging data in order, usually ascending (smallest to largest) or descending (largest to smallest).
Example – Sorted Ascending:
From the raw heights data: 60, 62, 62, 62, 62, 64, 65, 65, 65, 65, 66, 68, 68, 70
Now it is easier to see the smallest (60), largest (70), and how many times each value appears.
3. Frequency Distribution Table (Ungrouped Data)
A frequency distribution table lists each distinct value and how many times it occurs (its frequency).
Steps to Create a Frequency Table:
Step 1: List all distinct values in order (usually smallest to largest)
Step 2: Count how many times each value appears (using tally marks)
Step 3: Write the frequency (the count)
Example – Test Scores:
Scores: 75, 82, 75, 90, 75, 82, 88, 75, 90, 82
Distinct values: 75, 82, 88, 90
75 appears 4 times → frequency 4
82 appears 3 times → frequency 3
88 appears 1 time → frequency 1
90 appears 2 times → frequency 2
4. Tally Marks
Tally marks are a quick way to count frequencies. Each mark represents one count. Every fifth mark is drawn diagonally across the previous four to make counting easier.
Tally System:
| = 1
|| = 2
||| = 3
|||| = 4
|||| = 5 (four vertical lines and one diagonal)
|||| | = 6 (one group of 5 plus 1)
|||| || = 7 (one group of 5 plus 2)
Example – Tally for scores 75, 82, 75, 90, 75, 82, 88, 75, 90, 82:
75: |||| (4)
82: ||| (3)
88: | (1)
90: || (2)
5. Range of Data
The range is the difference between the largest and smallest values in a data set.
Formula: Range = Maximum value – Minimum value
The range tells us how spread out the data is. A small range means data points are close together; a large range means they are spread apart.
Example: Height’s data: smallest = 60, largest = 70 → Range = 70 – 60 = 10 inches
6. Grouped Data and Class Intervals
When data has many distinct values, we group them into intervals (called class intervals) to make the frequency table easier to read.
When to use grouped data: When the data has many different values (like ages, test scores, heights, weights)
Class Interval: A range of values grouped together, such as 60-69, 70-79, 80-89
Rules for Class Intervals:
- Intervals should not overlap (e.g., 60-69, 70-79, not 60-70, 70-80)
- Interval size (width) should be the same for all intervals
- Choose 5 to 10 intervals for most data sets
Example – Grouped Frequency Table:
Test scores of 30 students ranging from 55 to 98
Class intervals: 50-59, 60-69, 70-79, 80-89, 90-99
Count how many scores fall into each interval using tally marks.
7. Choosing Class Size
To find a good class size:
Step 1: Find the range (max – min)
Step 2: Decide how many intervals you want (usually 5 to 10)
Step 3: Class size ≈ Range / Number of intervals
Step 4: Round up to a convenient number
Example: Data from 12 to 45. Range = 45 – 12 = 33. With 5 intervals, class size = 33/5 = 6.6 → round up to 7. Intervals: 12-18, 19-25, 26-32, 33-39, 40-46
Solved Examples
Example 1 – Ungrouped Frequency Table:
The following are ages of 12 children: 8, 10, 8, 9, 10, 8, 10, 11, 9, 8, 10, 9. Create a frequency table.
Solution:
Distinct ages: 8, 9, 10, 11
Age 8 appears 4 times
Age 9 appears 3 times
Age 10 appears 4 times
Age 11 appears 1 time
Answer: Frequency table with ages 8(4), 9(3), 10(4), 11(1)
Example 2 – Range:
Find the range of the data: 15, 22, 18, 30, 25, 20, 28
Solution: Largest = 30, Smallest = 15
Range = 30 – 15 = 15
Answer: 15
Example 3 – Grouped Frequency Table:
The following are test scores (out of 100): 78, 85, 92, 68, 74, 88, 91, 95, 77, 83, 86, 79, 82, 90, 84, 76, 89, 93, 81, 87. Group the data into intervals of 10 (70-79, 80-89, 90-99). What about scores below 70?
Solution:
First sort the data: 68, 74, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95
Group 60-69: 68 → 1 score
Group 70-79: 74, 76, 77, 78, 79 → 5 scores
Group 80-89: 81, 82, 83, 84, 85, 86, 87, 88, 89 → 9 scores
Group 90-99: 90, 91, 92, 93, 95 → 5 scores
Answer: Frequency table with intervals 60-69 (1), 70-79 (5), 80-89 (9), 90-99 (5)
Example 4 – Finding Class Size:
Data ranges from 25 to 85. You want 6 intervals. What class size should you use?
Solution: Range = 85 – 25 = 60
Class size = 60 / 6 = 10
Intervals: 25-34, 35-44, 45-54, 55-64, 65-74, 75-84 (or 25-35, 36-46, etc.)
Answer: Class size = 10
Common Mistakes to Avoid
Mistake 1 – Forgetting to include all values in the frequency table
Missing a value that appears zero times is okay, but don't miss values that appear.
Correct understanding: List every distinct value that appears in the data.
Mistake 2 – Making overlapping class intervals
Intervals like 60-70 and 70-80 overlap at 70. Where does 70 go?
Correct understanding: Use intervals like 60-69, 70-79 to avoid overlap.
Mistake 3 – Using too many or too few intervals
Too many intervals (20 for 50 data points) makes the table cluttered. Too few (2 intervals) loses information.
Correct understanding: Use 5 to 10 intervals for most data sets.
Mistake 4 – Calculating range incorrectly
Range = max – min, not max – min + 1.
Correct understanding: Range is the difference, not the count of values.
Mistake 5 – Miscounting tally marks
Forgetting to make the fifth mark diagonal leads to counting errors.
Correct understanding: Always make the fifth mark across the previous four.
Mistake 6 – Not sorting data before finding min and max
In unsorted data, it is easy to miss the smallest or largest value.
Correct understanding: Sort the data or carefully scan for min and max.
Quick Reference Summary
Raw Data: Unprocessed, unorganized data
Sorting: Arranging data in ascending or descending order
Frequency Table: Shows each value and how many times it appears
Tally Marks: Visual counting system (groups of 5)
Range: Maximum value – Minimum value
Grouped Data: Data organized into class intervals (used when many distinct values)
Class Interval: A range of values grouped together
Class Size: (Range) / (Number of intervals), rounded up