Hint: Class mark is the mid point of the class interval. Upper limit is the highest value of the class interval.
Similarly, the lower limit is the smallest value of the class interval. For finding the actual upper limits and actual lower limits , we need to make the upper limit of a certain class and lower limit of the next class to be equal and same for the lower limit.
\[d=\dfrac{\left( \text{upper limit - lower limit} \right)}{2}=\dfrac{2.1-2.0}{2}=0.05\]
\[\] Upper limit is the highest value of the class interval and the actual upper limit is obtained by adding 0.5 to the highest number if the number is represented as a whole number or add 0.05 to the highest number if the number is represented as decimal.
Similarly, the lower limit is the smallest value of the class interval and the actual lower limit is obtained by subtracting 0.5 to the smallest number if the number is the whole number or subtract 0.05 to the smallest number if the number is decimal.
Complete step by step answer:
Given, class intervals are 1.1−2.0,2.1−3.0,3.1−4.0
Since, the intervals are represented in decimal; we add and subtract 0.05 to get the actual limits.
For class interval 1.1−2.0,
Class mark is \[\dfrac{\left( 1.1+2.0 \right)}{2}=\dfrac{3.1}{2}=1.55\]
Actual upper limit is 2.0+0.05=2.05 and
actual lower limit is 1.1−0.05=1.05
Similarly for 2.1−3.0,
Class mark is \[\dfrac{\left( 2.1+3.0 \right)}{2}=\dfrac{5.1}{2}=2.55\]
Actual upper limit is 3.0+0.05 = 3.05 and
actual lower limit is 2.1−0.05 = 2.05
Similarly for 3.1−4.0,
Class mark is \[\dfrac{\left( 3.1+4.0 \right)}{2}=\dfrac{7.1}{2}=3.55\]
Actual upper limit is 4.0+0.05=4.05 and
actual lower limit is 3.1−0.05=3.05
Therefore, actual upper limits are 2.05,3.05,4.05
Actual lower limits are 1.05,2.05,3.05
Class marks are 1.55,2.55,3.55
Therefore the correct option is A.
Note:
Learn the differences between upper limit , lower limit, actual upper limit and actual lower limit. We should add 0.5 when it is a=whole number and 0.05 when it is a decimal for both lower limit and upper limit. Class mark is the midpoint of the class interval.
Understanding class intervals and their associated calculations is pivotal in statistics. The precise determination of upper and lower limits, along with class marks, is crucial for accurate data representation. Let's delve into the concepts encapsulated within the provided article.
Class Intervals
In statistics, data is organized into groups or intervals called class intervals. These intervals help in categorizing and summarizing data for easier analysis. Each interval includes a range of values and is represented by an upper and lower limit.
Class Mark
The class mark refers to the midpoint or center of a class interval. It's calculated by finding the average of the upper and lower limits of the interval.
Upper and Lower Limits
- Upper Limit: The highest value within a class interval.
- Lower Limit: The smallest value within a class interval.
Actual Upper and Lower Limits
To obtain more precise values for upper and lower limits:
- For whole numbers: Add 0.5 to the upper limit and subtract 0.5 from the lower limit.
- For decimal numbers: Add 0.05 to the upper limit and subtract 0.05 from the lower limit.
Application in the Provided Example
The article discusses class intervals such as 1.1−2.0, 2.1−3.0, and 3.1−4.0, with decimal representations.
For instance:
- For the interval 1.1−2.0:
- Class mark = (1.1 + 2.0) / 2 = 1.55
- Actual upper limit = 2.0 + 0.05 = 2.05
- Actual lower limit = 1.1 − 0.05 = 1.05
This process repeats for each interval, ensuring precise determination of actual upper and lower limits along with class marks.
Conclusion
Understanding these concepts—class intervals, class marks, upper and lower limits, as well as the calculations for obtaining actual upper and lower limits—is crucial for accurate statistical analysis. The distinction between these measures, particularly when dealing with decimal and whole number representations, ensures precise data representation and analysis.
