Ninth Grade Mathematics: Raw, Ungrouped & Grouped Frequency Tables
When data is collected from a population, it initially appears as an unstructured list of numbers known as raw data. In this raw state, patterns are obscured by chaos. In this chapter, you will master the methods of transforming raw data into structured frequency distributions, constructing ungrouped and grouped frequency tables, and understanding class intervals, boundaries, and class marks.
Raw Data and Array Formation
Raw data is the original, unprocessed collection of observations in the exact order in which they were recorded. The first step in organizing numerical raw data is arranging it in an array—a list ordered in ascending (lowest to highest) or descending order.
Raw Scores of 20 Students:
14, 18, 12, 15, 18, 14, 16, 15, 12, 18, 14, 15, 16, 14, 18, 12, 15, 16, 14, 15
Ordered Array:
12, 12, 12, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 18, 18, 18, 18
While arranging data into an array reveals the minimum (12) and maximum (18), handling large datasets containing hundreds or thousands of observations requires frequency tabulation.
Ungrouped Frequency Distribution Tables
The frequency (denoted by f) of a particular observation is the number of times that specific data value occurs in the dataset. An ungrouped frequency distribution lists each distinct value alongside its count, tracked using tally marks:
Ungrouped Frequency Table for Student Scores:
Value (x) Tally Marks Frequency (f)
-------------------------------------------------
12 ||| 3
14 |||| 5
15 |||| 5
16 ||| 3
18 |||| 4
-------------------------------------------------
Total Sigma f = 20
The Greek capital letter Sigma denotes summation. The sum of all frequencies (Sigma f) must always equal the total sample size N (here, 20).
Grouped Frequency Distributions: Class Intervals
When a dataset spans a wide numerical range, an ungrouped table becomes unmanageably long. To condense the data, we group observations into non-overlapping numerical categories called class intervals.
Anatomy of a Class Interval: [ 20 - 30 )
- Lower Class Limit: 20
- Upper Class Limit: 30
- Class Width (Size): Upper Limit - Lower Limit = 30 - 20 = 10
- Class Mark (Midpoint): (Lower Limit + Upper Limit) / 2 = (20 + 30) / 2 = 25
Inclusive vs. Exclusive Methods of Grouping
There are two primary conventions for defining class intervals:
Method 1: Exclusive (Continuous) Method
Intervals: 0 - 10, 10 - 20, 20 - 30, 30 - 40
- A value equal to the upper limit belongs to the NEXT class interval.
For example, the number 20 is placed in [20 - 30), NOT in [10 - 20).
- This creates continuous boundaries without gaps: essential for histograms!
Method 2: Inclusive (Discrete) Method
Intervals: 1 - 10, 11 - 20, 21 - 30, 31 - 40
- Both limits are included in the interval: 10 belongs to the first class,
and 11 belongs to the second.
- Creates an apparent gap of 1 unit between consecutive classes.
To convert inclusive intervals into continuous true class boundaries, find the gap between classes (typically 1 unit), divide it by 2 (0.5), subtract 0.5 from each lower limit, and add 0.5 to each upper limit: Class 11 - 20 becomes true class boundary: 10.5 - 20.5.
Cumulative Frequency Distributions
A cumulative frequency distribution tracks running totals, showing how many observations fall below or above a specific boundary:
Grouped Cumulative Frequency Table:
Class Interval Frequency (f) Cumulative Frequency (Less than)
------------------------------------------------------------------------
0 - 10 4 4
10 - 20 7 4 + 7 = 11
20 - 30 12 11 + 12 = 23
30 - 40 8 23 + 8 = 31
40 - 50 5 31 + 5 = 36
------------------------------------------------------------------------
Total Sigma f = 36
Cumulative frequencies are indispensable for calculating medians, quartiles, percentiles, and plotting ogive curves.
Chapter Exercises
Exercise 1: The marks obtained by 30 students in a mathematics test (out of 50) are given below. Construct a grouped frequency distribution table using exclusive class intervals with a uniform class size of 10 (starting from 0 - 10): 12, 25, 38, 42, 19, 8, 34, 27, 45, 14, 22, 31, 16, 29, 39, 48, 7, 24, 33, 41, 18, 26, 35, 47, 11, 23, 30, 44, 15, 28.
Exercise 2: Convert the inclusive class intervals 21 - 30, 31 - 40, 41 - 50, and 51 - 60 into continuous true class boundaries. Calculate the class mark (midpoint) of each interval.
Exercise 3: Explain the difference between an individual frequency and a cumulative frequency, illustrating how cumulative frequency reveals the number of students scoring below a threshold.
Solutions and Explanations
Solution 1: Tally the 30 test marks into exclusive class intervals of width 10: Class Interval 0 - 10 (scores from 0 to less than 10): scores are 8, 7. Frequency = 2. Class Interval 10 - 20 (scores from 10 to less than 20): scores are 12, 19, 14, 16, 18, 11, 15. Frequency = 7. Class Interval 20 - 30 (scores from 20 to less than 30): scores are 25, 27, 22, 29, 24, 26, 23, 28. Frequency = 8. Class Interval 30 - 40 (scores from 30 to less than 40): scores are 38, 34, 31, 39, 33, 35, 30. Frequency = 7. Class Interval 40 - 50 (scores from 40 to 50): scores are 42, 45, 48, 41, 47, 44. Frequency = 6. Sum of frequencies: 2 + 7 + 8 + 7 + 6 = 30. The grouped frequency table is complete.
Solution 2: The gap between consecutive inclusive intervals is 31 - 30 = 1 unit. Half the gap is 1 / 2 = 0.5. Subtract 0.5 from each lower limit and add 0.5 to each upper limit to produce true class boundaries: Interval 21 - 30 becomes 20.5 - 30.5; Class Mark = (20.5 + 30.5) / 2 = 51 / 2 = 25.5. Interval 31 - 40 becomes 30.5 - 40.5; Class Mark = (30.5 + 40.5) / 2 = 71 / 2 = 35.5. Interval 41 - 50 becomes 40.5 - 50.5; Class Mark = (40.5 + 50.5) / 2 = 91 / 2 = 45.5. Interval 51 - 60 becomes 50.5 - 60.5; Class Mark = (50.5 + 60.5) / 2 = 111 / 2 = 55.5. Notice that the class mark of the original interval (21 + 30)/2 = 25.5 is identical to the class mark of the true boundaries.
Solution 3: An individual frequency f represents the count of observations occurring exclusively within a single specified category or interval. For instance, in the table above, the frequency of the 20 - 30 interval is 8, meaning exactly 8 students scored between 20 and 30. By contrast, cumulative frequency represents a running accumulation of frequencies from the beginning of the distribution up to the upper boundary of a given class. The cumulative frequency for the 20 - 30 class is 2 + 7 + 8 = 17. This immediately reveals that exactly 17 students scored less than 30 marks, without requiring the investigator to sum individual rows manually.