Histograms

Histograms are a powerful tool for displaying the distribution of continuous data. When the data is grouped into intervals, histograms show the frequency of data within those intervals.

1. When are Histograms Appropriate?

Histograms are used when data is continuous and grouped into intervals.

2. Drawing a Histogram from a Grouped Frequency Table:

To draw a histogram, follow these steps:

1. Calculate the frequency density: Divide the frequency by the class width to get the frequency density.
2. Label the axes: The x-axis should represent the variable, and the y-axis should represent frequency.
3. Draw the bars: For each interval, draw a bar where the height is equal to the frequency density and the width corresponds to the class width.

3. Example: Drawing a Histogram from a Grouped Frequency Table

Consider the following grouped frequency table showing the number of students scoring within certain ranges on a test:

To draw the histogram:

• Label the x-axis with the score range: \( 0 - 50 \).
• Label the y-axis with the frequency density range: \( 0 - 1.2 \).
• For each interval, draw a bar where the height is equal to the frequency density and the width corresponds to the class width (10 in this case).

4. Estimating Frequencies Between Two Points:

Histograms can also be used to estimate the frequency between two points. The area of each bar represents the frequency, so by calculating the area of the relevant bars, you can estimate the frequency between the two points.

To estimate the frequency between two points:

• Identify the intervals that contain the points you're interested in.
• For each interval, calculate the frequency by multiplying the frequency by the class width (Area = Frequency × Class Width).
• If the range you're interested in doesn't align with a full interval, estimate the portion of the bar corresponding to the range of interest.

5. Example: Estimating Frequency Between Two Points

Consider the histogram based on the previous grouped frequency table. If we want to estimate the frequency of students scoring between 15 and 25, we look at the score ranges:

• The range \( 10 < x \leq 20 \) has a frequency of 8 and a class width of 10.
• The range \( 20 < x \leq 30 \) has a frequency of 12 and a class width of 10.

To estimate the frequency between 15 and 25:

• The range \( 10 < x \leq 20 \) spans scores from 10 to 20. The part between 15 and 20 is half of the interval, so we can estimate the area as half the total area of the bar:
• The range \( 20 < x \leq 30 \) spans scores from 20 to 30. The part between 20 and 25 is half of the interval, so we can estimate the area as half the total area of the bar:
• Therefore, the total estimated frequency between 15 and 25 is: \(4 + 6 = 10\)

6. Summary:

• Histograms are used to represent continuous data, with bars that are adjacent to each other and represent the frequency of data within specific intervals.
• To draw a histogram, plot the frequency of each interval and ensure the bars are adjacent with no gaps to show the continuity of the data.
• Estimating frequencies between two points involves calculating the area of the bars within the specified range, considering partial bars if necessary.

Histograms provide a clear visual representation of data distribution and can be used to estimate frequencies and analyse the spread of continuous data.