Frequency Distributions, Class Boundaries, Midpoints, and Relative Frequencies

Class Limits and Class Boundaries

• Class Limits:

  • The upper and lower bounds that define the range of data values contained inside a given class.

  • Limits are inclusive of the data values in the dataset.

  • Expressed in interval notation using square brackets: $[\text{Lower Limit}, \text{Upper Limit}]$.

• Class Boundaries:

  • The values that separate adjacent classes without leaving gaps.

  • Boundaries sit strictly outside the class limits and lie halfway between the upper limit of one class and the lower limit of the next class.

  • Expressed in interval notation using exclusive round parentheses: $(\text{Lower Boundary}, \text{Upper Boundary})$.

• Calculating Class Boundaries:

  • To calculate the boundary between two adjacent classes, compute the arithmetic mean (average) of the upper limit of the first class and the lower limit of the following class:     $\text{Class Boundary} = \frac{\text{Upper Limit of Class } i + \text{Lower Limit of Class } (i+1)}{2}$

  • Example calculation for adjacent classes where Class 1 upper limit is $54\,999$ and Class 2 lower limit is $55\,999$:     $\text{Boundary} = \frac{54\,999 + 55\,999}{2} = 54\,999.5$

  • The result $54\,999.5$ serves simultaneously as the upper boundary of Class 1 and the lower boundary of Class 2.

• Rule of Data Inclusion for Boundaries:

  • Data in datasets are rounded off (e.g., rounded up or to the nearest dollar).

  • Fundamental Constraint: No actual data point from a dataset is permitted to equal a class boundary.

  • If a data point equals a class boundary, the boundary system must be adjusted.

• Finding Subsequent Boundaries using Class Width:

  • Once a single class boundary is computed, all other lower and upper class boundaries are calculated by systematically adding or subtracting the class width ($w$).

  • Example using a class width of $w = 5\,000$:

  • Lower boundary of Class 1:       $54\,999.5 - 5\,000 = 49\,999.5$

  • Upper boundary of Class 1 / Lower boundary of Class 2:       $54\,999.5$

  • Upper boundary of Class 2 / Lower boundary of Class 3:       $54\,999.5 + 5\,000 = 59\,999.5$

  • Upper boundary of Class 3 / Lower boundary of Class 4:       $59\,999.5 + 5\,000 = 64\,999.5$

  • This process is repeated sequentially until the upper boundary of the highest class exceeds the maximum data value in the dataset.

Class Midpoints

• Definition:

  • The class midpoint represents the exact numerical center of a given class.

• Methods of Calculation:

  • Method 1 (Using Class Limits):     $\text{Class Midpoint} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2}$

  • Method 2 (Using Class Boundaries):     $\text{Class Midpoint} = \frac{\text{Lower Boundary} + \text{Upper Boundary}}{2}$

  • Both methods yield identical results. Calculating the midpoint using both formulas serves as an arithmetic verification tool. If the two methods yield different values, an error has occurred.

• Example Calculation:

  • For a class with a lower limit of $50\,000$ and an upper limit of $54\,999$:     $\text{Midpoint} = \frac{50\,000 + 54\,999}{2} = 52\,499.5$

• Properties and Use Cases:

  • A class midpoint does not need to be a valid or possible value within the raw dataset.

  • In a spatial visualization:

  • Class limits define the inner bounds.

  • Class boundaries sit outside the limits.

  • The midpoint sits dead center between the boundaries and limits.

  • Midpoints for remaining classes can be found rapidly by adding or subtracting the class width ($w$) to the calculated midpoint of an adjacent class.

  • Midpoints across adjacent classes with $w = 5\,000$:

  • Class 1 Midpoint: $52\,499.5$

  • Class 2 Midpoint: $52\,499.5 + 5\,000 = 57\,499.5$

  • Class 3 Midpoint: $57\,499.5 + 5\,000 = 62\,499.5$

  • Estimation of Class Average:

  • Midpoints serve as point estimates for the average value of data falling within each class.

  • Maximum error bound: The true mean of data inside a class can be at most half a class width ($\frac{w}{2}$) away from the class midpoint.

Absolute Frequencies and Relative Frequencies

• Absolute Frequency:

  • The total raw count of data points assigned to a specific class. Also referred to simply as class frequency ($f$).

• Sample Size ($n$):

  • The total count of all data points in the entire sample.

  • Defined mathematically using capital sigma ($\sum$) summation notation:     $n = \sum f$

• Relative Frequency:

  • The fraction, proportion, or percentage of the entire dataset that falls within a given class.

  • Mathematical Formula:     $\text{Relative Frequency} = \frac{f}{n} = \frac{f}{\sum f}$

Step-by-Step Conversion and Frequency Table Calculations

• Converting Relative Frequency Fractions to Percentages:

  1. Divide the class frequency $f$ by the total sample size $n$ to obtain a decimal value.

  2. Identify the first two digits following the decimal point; these digits form the integer percentage value.

  3. If only one decimal place exists (e.g., $0.4$), a trailing zero is implied (e.g., $0.40 = 40\%$).

• Exhaustive Dataset Calculations ($n = 25$):

  • Class 1:

  • Absolute Frequency $f_1 = 3$

  • Relative Frequency Calculation:       $\frac{3}{25} = 0.12$

  • Percentage Equivalent: $12\%$

  • Class 2:

  • Absolute Frequency $f_2 = 0$

  • Relative Frequency Calculation:       $\frac{0}{25} = 0.00$

  • Percentage Equivalent: $0\%$

  • Class 3:

  • Absolute Frequency $f_3 = 6$

  • Relative Frequency Calculation:       $\frac{6}{25} = 0.24$

  • Percentage Equivalent: $24\%$

  • Class 4:

  • Absolute Frequency $f_4 = 4$

  • Relative Frequency Calculation:       $\frac{4}{25} = 0.16$

  • Percentage Equivalent: $16\%$

  • Class 5:

  • Absolute Frequency $f_5 = 10$

  • Relative Frequency Calculation:       $\frac{10}{25} = 0.40$

  • Percentage Equivalent: $40\%$

  • Class 6:

  • Absolute Frequency $f_6 = 1$

  • Relative Frequency Calculation:       $\frac{1}{25} = 0.04$

  • Percentage Equivalent: $4\%$

  • Class 7:

  • Absolute Frequency $f_7 = 1$

  • Relative Frequency Calculation:       $\frac{1}{25} = 0.04$

  • Percentage Equivalent: $4\%$

Questions & Discussion

• Data Boundary Alignment:

  • Direct alignment between raw data values and class boundaries is prohibited. If raw data lands directly on a boundary, boundaries must be shifted or recalculated to maintain exclusiveness.

• Arithmetic Verification:

  • Midpoints can be cross-checked by computing both $\frac{\text{Lower Limit} + \text{Upper Limit}}{2}$ and $\frac{\text{Lower Boundary} + \text{Upper Boundary}}{2}$ to ensure full calculation accuracy.