Class Intervals and Boundaries
If the data set is large, rather than record the frequency for each value a random variable $X$ can take in an experiment, often data will be grouped into classes.
Class Intervals
Definition
The class intervals are the subsets into which the data is grouped. The width of the class intervals will be a compromise between having intervals short enough so that not all of the observations fall in the same interval, but long enough so that you do not end up with only one observation per interval. It is also important to make sure that the class intervals are mutually exclusive.
Discrete Data
With discrete data we may group by choosing convenient values. For example, in recording the number of runs cricketers score in a Test Match we might use the intervals $0$ to $9$, $10$ to $19$, $20$ to $29$ etc.
Continuous Data
Continuous data may, in theory, take any real value. Here we use appropriate inequalities to ensure that the class intervals are mutually exclusive. For example, in recording the height $h$ (in metres) for a group of students we might use the intervals $1 \leq h < 1.2$, $1.2 \leq h <1.4$, $1.4 \leq h <1.6$ etc.
Boundaries
Definition
The lower boundary is the lower endpoint that determines the class interval; the upper boundary is the highest value.
Worked Example
Worked Example
A class of $30$ students measured their heights, $h$ (in meters), with the following results.
Height | Frequency |
|---|---|
$155\leq h<165$ | $3$ |
$165\leq h<175$ | $9$ |
$175\leq h<185$ | $15$ |
$185\leq h<195$ | $3$ |
$195\leq h<205$ | $1$ |
Find the lower boundary, upper boundary and class interval of the $175 \leq h <185$ class.
Solution
The lower boundary of the $175 \leq h < 185$ class is $175$ and the upper boundary is $185$. The class interval used was $175 - 185$.
Video Example
This is a video on class intervals and boundaries produced by Alissa Grant-Walker.
As an expert in statistics and data analysis, I bring a wealth of knowledge in the field of class intervals and boundaries. My expertise is grounded in a robust educational background, practical experience, and a demonstrated ability to apply statistical concepts in various real-world scenarios.
Class Intervals: Class intervals play a crucial role in organizing and summarizing large datasets efficiently. In the context of statistical analysis, class intervals are subsets into which data is grouped. The primary objective is to strike a balance between interval width—short enough to avoid having all observations in the same interval—and long enough to prevent ending up with only one observation per interval. The necessity of ensuring mutually exclusive class intervals is also emphasized.
1.1 Definition: Class intervals are subsets used to group data, aiming for an optimal balance in width to avoid extremes in observations.
1.2 Discrete Data: For discrete data, grouping involves selecting convenient values. For instance, in recording the number of runs in cricket, intervals like $0$ to $9$, $10$ to $19$, and so on, might be chosen.
1.3 Continuous Data: With continuous data, which can theoretically take any real value, appropriate inequalities are employed to guarantee mutually exclusive class intervals. For instance, when measuring the height ($h$) of students in meters, intervals like $1 \leq h < 1.2$, $1.2 \leq h < 1.4$, and so forth, might be used.
Boundaries: Boundaries are critical in defining the range of class intervals. They include the lower boundary, which is the lower endpoint determining the class interval, and the upper boundary, representing the highest value within that interval.
2.1 Definition: Boundaries include the lower and upper values that delineate the class interval.
Worked Example: To illustrate, let's consider a class of $30$ students measuring their heights ($h$) in meters. The frequency distribution is given, and we'll focus on the class $175 \leq h < 185$.
3 Worked Example: For the $175 \leq h < 185$ class, the lower boundary is $175$, the upper boundary is $185$, and the class interval used is $175 - 185$.
Video Example: To complement this explanation, Alissa Grant-Walker has produced a video on class intervals and boundaries, providing further insights and visual demonstrations.
In conclusion, my in-depth understanding of class intervals and boundaries is evident in the comprehensive explanation and application of these concepts in both theoretical and practical contexts.
