# Class Width – Explained with Examples

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The class width of a given frequency distribution refers to the length of the class intervals of the distribution. Therefore, class width is indeed the same as the class size. Another name for the class width is the class length. Class width is calculated by subtracting the lower class limits from the upper-class limits of the frequency distribution.

### How to Find the Class Width?

1. Identify the class intervals of the given distribution.
2. Write down the upper and lower limits of each class.
3. Subtract the upper limit from the lower in order to obtain the width for that class. This can be given by the formula below.

### Class Width Formula:

\text{Class width} = U - L where, U is the upper class limit and L is the lower class limit of the given class interval.

### Example 1:

Using the frequency distribution we can find the class width:

The size of the class 30-40 is the difference between 30 and 40 which is 10. Here we see that the size of each class is 10. Hence class width for each class in this example is 10.

In the above example, the class widths for each class were equal. It can also be the case that we have unequal class lengths for each class.

### Example 2:

Consider the following frequency distribution table:

The class width of the class 45-50 is 5 whereas the class width of the class 25-45 is 20. Therefore, the class widths are unequal in the above example. We compute all of the class widths are follows:

### Example 3: Class Width for Histogram

If we are given a histogram we can compute the class width by looking at the space that each bar occupies on the X-axis. Consider the histogram given below:

As we can see in the above histogram, two histograms together occupy a length of 10 units on the X-axis, therefore a single histogram has a width of 5 units. So based on the above histogram, the class width is 5.

#### Relationship between Class Width and Number of Intervals:

When constructing a frequency distribution, as the number of classes is decreased, the class width increases. The reason behind the increase in class length is that more and more values have to be accommodated inside a small number of classes. For example, if 100 was partitioned into 4 classes, we would have 4 classes each of length 25. But, if 100 was partitioned into 2 classes, we would have 2 classes each of length 50. The size of the class interval is inversely proportional to the number of classes (class intervals) in a given distribution.

#### Calculation Class Width using the Number of Classes:

We can use the given minimum and maximum data entries and the number of classes to find the class width. For example if we are given that the minimum entry is 20 and the maximum entry is 60 and the number of classes is 8 then we can calculate the class width using the formula, \begin{align*}\text{Class Width}&=\frac{\text{Maximum Entry}-\text{Minimum Entry}}{\text{Number of Classes}}\\ &=\frac{60-20}{8} = \frac{40}{8} = 5 \end{align*}

### CalculatingClass Width for Raw Data:

Using the above formula we know how to determine the class width of tabulated data. But what about raw untabulated data? In that case, we have the following method to calculate the class distribution:

1. Find the range of the data by subtracting the highest and the lowest number of values
2. Divide the result by the number of classes for the data

#### Example:

Suppose we are given the following raw data for marks out of 100 for a class of twenty students:

15, 45, 40, 53, 77, 76, 96, 86, 53, 12, 46, 75, 48, 36, 84, 15, 37, 47, 24, 29

Let us assume that we want to have 10 classes.

Now Range= Largest value-Smallest value=96-15=81

Therefore Class Width= 81/10=8.1

So we should take the class width to be approximately equal to 8.

### How to determine the number of classes for grouped frequency distribution?

In the above example, we took the number of classes to be 10. But in fact, there exists a formula called Sturges Rule or Sturges formula which states that the number of classes is given as,

\text{Number of classes} = 1+3.322*log_{10}N where, N is the total number of observations.

If we have N=100 observations then,

Number of classes=1+3.322*2=7.644, that is, approximately 8.

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