12.1.B. Histogram

A histogram is a form of representation like the bar graph, but it is used for continuous class intervals. For instance, consider the frequency distribution Table 12.2, representing the weights of 36 students of a class:

A histogram is similar to a bar graph but is specifically designed for continuous data, where the data falls into ranges or intervals. It visually displays how many data points fall within each range (or class interval). In the example provided, we will be looking at the weights of 36 students.

Imagine measuring the heights of children at a school. Instead of knowing the exact height of each child, we might be interested in how many children fall into certain height ranges, like 4-5 feet, 5-6 feet, etc. A histogram helps us quickly see how many children fall into each height range.

(i) We represent the weights on the horizontal axis on a suitable scale. We can choose the scale as 1 cm = 5 kg. Also, since the first class interval is starting from 30.5 and not zero, we show it on the graph by marking a kink or a break on the axis.

To create a histogram, we start by plotting our intervals along the horizontal axis (x-axis). We choose a scale that allows us to represent our data clearly. If our first class interval starts at 30.5 kg, we need to indicate that this is not starting from zero by adding a break in the axis. This ensures our visualization accurately reflects the data.

Think about a ramp that starts at the ground level (zero). If our first data point starts a little above the ground, we need to create a clear break showing that the ramp starts higher and doesn't touch the ground.

(ii) We represent the number of students (frequency) on the vertical axis on a suitable scale. Since the maximum frequency is 15, we need to choose the scale to accommodate this maximum frequency.

Next, we plot the frequencies of our intervals along the vertical axis (y-axis). We need to ensure that our scale allows us to visualize the highest grouping clearly. If the highest number of students in any interval is 15, our y-axis scale must be enough to represent at least this number.

It's like measuring how many students scored above a certain grade. If the highest score is 100 in a test and you decided to plot it, you should ensure your y-axis reaches higher than 100 so everyone can see the scores clearly.

(iii) We now draw rectangles (or rectangular bars) of width equal to the class-size and lengths according to the frequencies of the corresponding class intervals.

Now it’s time to draw the histogram. Each interval will be represented by a rectangle where the width corresponds to the class size (e.g., from 30.5 to 35.5 is one interval), and the height will reflect the number of students (frequency) in that range. Therefore, if we have 9 students in the 30.5-35.5 kg interval, we draw a rectangle above that bar extending to 9 on the vertical scale.

Consider building a wall of different heights to represent how many people fit into different height groups. If 9 people are between 30.5 and 35.5 kg, you would build a wall that rises 9 bricks high at that position.

Observe that since there are no gaps in between consecutive rectangles, the resultant graph appears like a solid figure. This is called a histogram, which is a graphical representation of a grouped frequency distribution with continuous classes.

The key feature of a histogram is that there are no gaps between adjacent rectangles. This solid appearance signifies that the data are continuous, meaning they flow into each other without interruption. Each rectangle represents a range of data points, and the height is proportional to the frequency of those points.

Think of a water slide at an amusement park. The slide is continuous; you go from one section to another without any breaks or gaps in between. Just like that, a histogram allows for a smooth transition between values, showing how data flows through different intervals.

Also, unlike a bar graph, the width of the bar plays a significant role in its construction. Here, in fact, areas of the rectangles erected are proportional to the corresponding frequencies.

In a histogram, the width of each rectangle can impact how the frequency is represented, especially when intervals vary in size. It's important to ensure that the area of each rectangle is proportional to the frequency of the data it represents. In simpler terms, wider intervals should have corresponding heights adjusted accordingly to ensure accurate representation.

Imagine a garden with flower beds of various sizes. If one bed is much wider, but has fewer flowers than a narrower bed, just counting the number of flowers may give a misleading picture. The size of each bed (or bar) matters to understand how many flowers are truly there in comparison to each other.

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