Histograms and Cumulative Frequency for the TMUA

Understanding Histograms and Frequency Density

Histograms are the primary tool for representing continuous data or grouped discrete data where intervals are used. Unlike a standard bar chart, where the frequency is simply the height of the bar, a histogram represents frequency through the area of the bar. This principle is fundamental because it ensures that the visual 'weight' of a bar is proportional to the number of data points it contains, regardless of the width of the class interval.

To achieve this, we use Frequency Density on the vertical axis. Frequency density is defined as the frequency per unit of the class interval. The formula is:

$Frequency\ Density = \frac{Frequency}{Class\ Width}$

When we plot frequency density against the class intervals, the area of each bar is:

$Area = Class\ Width \times Frequency\ Density = Class\ Width \times \frac{Frequency}{Class\ Width} = Frequency$

Histograms with Unequal Class Intervals

In many data sets, class intervals are not equal. For example, a survey might group ages as 0 to 18, 18 to 30, 30 to 50, and 50 to 80. If we used frequency on the vertical axis, the 50 to 80 bar would appear disproportionately large even if its frequency were low. Frequency density corrects for this by thinning the bar height for wider classes.

Worked Example: Constructing a Histogram

A researcher records the mass ($m$) of 40 samples in grams:

1. 0 to 10g: Frequency 8. Class width 10. $Frequency\ Density = 8 / 10 = 0.8$.
2. 10 to 15g: Frequency 10. Class width 5. $Frequency\ Density = 10 / 5 = 2.0$.
3. 15 to 20g: Frequency 12. Class width 5. $Frequency\ Density = 12 / 5 = 2.4$.
4. 20 to 40g: Frequency 10. Class width 20. $Frequency\ Density = 10 / 20 = 0.5$.

When drawing this, the vertical axis must be labelled Frequency Density. The highest bar is the 15 to 20g class (height 2.4), and the lowest is the 20 to 40g class (height 0.5).

Cumulative Frequency Graphs

Cumulative frequency refers to the running total of the frequencies. It allows us to see how many data points fall below a certain value. These graphs are plotted with cumulative frequency on the vertical axis and the data values on the horizontal axis.

Construction and Plotting

To construct the graph, you must always plot the cumulative frequency at the upper class boundary of each interval. This is because, at the end of an interval, we know exactly how many items are less than or equal to that boundary value. We do not know the exact distribution within the interval, so we assume a steady increase and join the points with a smooth curve or a series of straight lines.

Worked Example: Estimating the Median

If we have a total frequency $n = 100$:

1. Calculate the cumulative frequencies for each class.
2. Plot the points (Upper Boundary, Cumulative Frequency).
3. To find the Median ($Q_2$), find the value $n / 2 = 50$ on the vertical axis. Draw a horizontal line to the curve, then a vertical line down to the horizontal axis. This value is the estimated median.
4. To find the Lower Quartile ($Q_1$), use the value $n / 4 = 25$ on the vertical axis.
5. To find the Upper Quartile ($Q_3$), use the value $3n / 4 = 75$ on the vertical axis.

The Interquartile Range (IQR) is $Q_3 - Q_1$ and provides a measure of the spread of the middle 50 percent of the data.

Appropriate Use of Diagrams

1. Bar Charts: Best for discrete, non grouped data where categories are distinct.
2. Histograms: Best for continuous data or grouped data where we want to observe the distribution or shape (skewness and modal class) of the data.
3. Cumulative Frequency Graphs: Best when the objective is to find the median, quartiles, percentiles, or to compare the spread of two different data sets using box plots derived from the quartiles.