A cumulative frequency distribution is given below Class 60-62 63-65 66-68 69-71 72-74 Cumulative frequency 3 20 36 48 50 Which one of the following class has maximum frequency?
63-65
Understanding Frequency Distributions
In statistics, a frequency distribution helps organize data by showing the frequency of each value or range of values. There are different ways to present this information, such as simple frequency distribution and cumulative frequency distribution.
Cumulative vs. Simple Frequency
A simple frequency for a class interval tells us exactly how many data points fall within that specific range. A cumulative frequency, on the other hand, tells us the total number of data points that are less than or equal to the upper limit of a given class interval. It is the sum of the frequencies of that class and all classes before it.
To find the simple frequency from a cumulative frequency distribution, we subtract the cumulative frequency of the previous class from the cumulative frequency of the current class.
Converting Cumulative Frequency to Simple Frequency
The formula to calculate the simple frequency ($F_i$) for the $i$-th class from its cumulative frequency ($C_i$) and the cumulative frequency of the previous class ($C_{i-1}$) is:
\(F_i = C_i - C_{i-1}\)
For the first class, the simple frequency is equal to its cumulative frequency ($F_1 = C_1$).
Step-by-Step Calculation of Simple Frequencies
We are given the following cumulative frequency distribution:
| Class | Cumulative Frequency |
|---|---|
| 60-62 | 3 |
| 63-65 | 20 |
| 66-68 | 36 |
| 69-71 | 48 |
| 72-74 | 50 |
Let's calculate the simple frequency for each class:
- Class 60-62: This is the first class. The simple frequency is equal to the cumulative frequency.
- \(F_{60-62} = 3\)
- Class 63-65: The simple frequency is the cumulative frequency of this class minus the cumulative frequency of the previous class (60-62).
- \(F_{63-65} = C_{63-65} - C_{60-62} = 20 - 3 = 17\)
- Class 66-68: The simple frequency is the cumulative frequency of this class minus the cumulative frequency of the previous class (63-65).
- \(F_{66-68} = C_{66-68} - C_{63-65} = 36 - 20 = 16\)
- Class 69-71: The simple frequency is the cumulative frequency of this class minus the cumulative frequency of the previous class (66-68).
- \(F_{69-71} = C_{69-71} - C_{66-68} = 48 - 36 = 12\)
- Class 72-74: The simple frequency is the cumulative frequency of this class minus the cumulative frequency of the previous class (69-71).
- \(F_{72-74} = C_{72-74} - C_{69-71} = 50 - 48 = 2\)
Analyzing Class Frequencies
Now we have the simple frequencies for each class interval:
| Class | Simple Frequency |
|---|---|
| 60-62 | 3 |
| 63-65 | 17 |
| 66-68 | 16 |
| 69-71 | 12 |
| 72-74 | 2 |
Identifying the Maximum Frequency Class
We compare the simple frequencies calculated for each class: 3, 17, 16, 12, and 2. The maximum value among these frequencies is 17. This maximum frequency corresponds to the class interval 63-65.
Conclusion
The class with the maximum frequency is 63-65, as it has a simple frequency of 17, which is the highest among all the given class intervals.
Revision Table: Frequency Concepts
| Concept | Description | Calculation (from previous class) |
|---|---|---|
| Simple Frequency | Number of observations in a specific class interval. | Cumulative Frequency of Current Class - Cumulative Frequency of Previous Class |
| Cumulative Frequency | Total number of observations less than or equal to the upper limit of a class interval. | Sum of Simple Frequencies up to the Current Class |
Additional Information: Types of Frequency Distributions
Frequency distributions can be presented in various ways:
- Ungrouped Frequency Distribution: Lists each distinct value in the data and its frequency. Suitable for discrete data with a small range of values.
- Grouped Frequency Distribution: Data is grouped into class intervals, and the frequency for each interval is listed. Suitable for continuous data or discrete data with a large range.
- Relative Frequency Distribution: Shows the proportion or percentage of observations in each class. Calculated as (Simple Frequency / Total Number of Observations).
- Cumulative Relative Frequency Distribution: Shows the proportion or percentage of observations less than or equal to the upper limit of each class.
Important Questions from Classification of Data
In an examination, 40% of candidates got second class. When the data are represented by a pie chart, what is the angle corresponding to second class?
Following histogram shows certain frequency distribution against class intervals.
The approximated mean of this distribution is:
The annual vehicles production (in lacs) in india is given in the pie chart.
If the annual production of motor cycle is 1.80 lacs, the annual production of Bicycle is
For determination of mode and median graphically, one considers:
The curve obtained by joining the points, whose x-coordinates are the upper limits of the class interval and y-coordinates are corresponding cumulative frequencies is called:
