# Frequency Distribution and Statistical Data Analysis of Student Heights

## Statistical Data from Quantitative Variables

We use statistical methods to describe data derived from quantitative variables.

## Primitive Table of Student Heights

Table 5.1 shows the raw, unorganized height data (in cm) of a group of students. This type of table, with numerically unorganized elements, is called a primitive table: 166, 160, 150, 162, 160, 165, 160, 167, 164, 160, 162, 161, 168, 163, 156, 173, 160, 155, 164, 168, 155, 152, 163, 160, 155, 155, 159, 151, 170, 164, 154, 161, 156, 172, 153, 157, 156, 158, 158, 161.

It’s difficult to interpret this data effectively. Organizing the data, such as through ascending or descending sorting, creates a *roll*, which makes analysis easier.

Sorted Data (in cm): 150, 151, 152, 153, 154, 155, 155, 155, 156, 156, 156, 157, 158, 158, 159, 160, 160, 160, 160, 161, 161, 161, 162, 162, 163, 163, 164, 164, 164, 165, 166, 167, 168, 168, 170, 172, 173.

Now, we can easily identify the smallest height (150 cm) and the tallest (173 cm). The range of variation is 23 cm (173 – 150). We can also observe a concentration of heights between 160 cm and 165 cm.

## Frequency Distribution

A frequency distribution table further simplifies data analysis by listing each height value and the number of times it appears (its frequency).

### Table 5.3: Frequency Distribution of Student Heights

150 – 1; 151 – 1; 152 – 1; 153 – 1; 154 – 1; 155 – 4; 156 – 3; 157 – 1; 158 – 2; 160 – 5; 161 – 4; 162 – 2; 163 – 2; 164 – 3; 165 – 1; 166 – 1; 167 – 1; 168 – 2; 169 – 1; 170 – 1; 172 – 1; 173 – 1; Total: 40

To make this more concise, we can group height values into *class intervals* (or classes).

### Table 5.4: Frequency Distribution with Class Intervals

**Height of 40 Students**

- Class 1 (150-154 cm): 4
- Class 2 (154-158 cm): 9
- Class 3 (158-162 cm): 11
- Class 4 (162-166 cm): 8
- Class 5 (166-170 cm): 5
- Class 6 (170-174 cm): 3
- Total: 40

While grouping simplifies the table, we lose some detail. For example, we can no longer see the exact number of students with a height of 161 cm, but we know that 11 students have heights between 158 cm and 162 cm.

## Class Intervals and Frequency

### 5.3 Class Definitions

Classes are represented by ‘i’, where ‘i’ = 1, 2, 3,…K, and ‘K’ is the total number of classes.

*Example 1:* In Table 5.4, there are 6 classes. The range of the 3rd class is 158-162 cm.

### 5.3.2 Class Limits

Class boundaries are defined by the extremes of each class. The lower boundary is the lower limit (li), and the upper boundary is the upper limit (LI).

*Example 2:* In Table 5.4, for the 5th class, the lower limit (l5) is 166 cm, and the upper limit (L5) is 170 cm.

### 5.3.3 Class Interval Amplitude

The amplitude (Hi) of a class interval is the difference between its upper and lower limits: Hi = LI – li.

*Example 3:* For the 5th class in Table 5.4, the amplitude is h5 = 170 – 166 = 4 cm.

### 5.3.4 Total Distribution Range

The total distribution range (AT) is the difference between the upper limit of the last class (Lmax) and the lower limit of the first class (lmin): AT = Lmax – lmin.

*Example:* AT = 174 – 150 = 24 cm.

### 5.3.5 Sample Range

The sample range (AA) is the difference between the maximum and minimum values in the sample: AA = Xmax – Xmin.

*Example:* AA = 173 – 150 = 23 cm.

### 5.3.6 Class Midpoint

The midpoint (Xi) of a class is the average of its lower and upper limits: Xi = (li + LI) / 2.

*Example 5:* The midpoint of the 2nd class in Table 5.4 is X2 = (154 + 158) / 2 = 156 cm.

### 5.3.7 Simple Absolute Frequency

The simple absolute frequency (fi) is the number of observations in a class. In our example, f1 = 4, f2 = 9, f3 = 11, f4 = 8, f5 = 5, and f6 = 3. The sum of all frequencies is Σfi = 40.

## 5.4 Number of Classes and Ranges

Sturges’ rule can help determine the number of classes: i = 1 + 3.3 * log(n), where ‘n’ is the total number of data points. However, this is a guideline, and the final decision depends on the specific data.

In our example (n = 40), Sturges’ rule suggests i = 6 classes. With a sample range of 23 cm, an interval of 4 cm for each class is reasonable.

## 5.5 Types of Frequencies

### 5.5.1 Simple or Absolute Frequency (fi)

This is the actual number of data points in each class (as shown in Table 5.4).

### 5.5.2 Relative Frequency (fri)

This is the ratio of the simple frequency of a class to the total frequency: fri = fi / Σfi.

### 5.5.3 Cumulative Frequency (Fi)

This is the sum of the frequencies of all classes up to a given class.

### 5.5.4 Relative Cumulative Frequency (Fri)

This is the ratio of the cumulative frequency of a class to the total frequency: Fri = Fi / Σfi.

## Exercises and Examples

The provided exercises and examples demonstrate calculations for relative frequency, cumulative frequency, and relative cumulative frequency. They also include questions about interpreting frequency distributions.