This is a very important statistics lesson that is almost guaranteed to be in the exam. It covers drawing three types of graphs (Histogram, Frequency Polygon, Cumulative Frequency Curve) and using them to analyze data, especially finding quartiles.

1. Class Limits vs. Class Boundaries (Crucial First Step!)

Before drawing a histogram, you must understand the difference between limits and boundaries.

• Class Limits: The numbers written in the table (e.g., 30-34, 35-39). Notice the gap between 34 and 35.

• Class Boundaries: The values that close the gaps. They are found by taking the midpoint of the upper limit of one class and the lower limit of the next. For the classes 30-34 and 35-39, the boundary is 34.5.

  • Rule of Thumb: Subtract 0.5 from the lower limit and add 0.5 to the upper limit to find the boundaries (e.g., 30-34 becomes 29.5 - 34.5).

Exam Tip: Histograms must be drawn using class boundaries on the x-axis so that the bars touch each other. If the classes are already continuous (e.g., 10-20, 20-30), then the limits are also the boundaries.

2. The Histogram

A histogram is a bar chart that represents a frequency distribution.

Case 1: Equal Class Intervals

1. Mark the class boundaries on the horizontal axis (x-axis).

2. Mark the frequency on the vertical axis (y-axis).

3. Draw a rectangular bar for each class. The width of the bar is the class interval, and the height of the bar is the frequency.

Case 2: Unequal Class Intervals (Very Common in Exams)

The key principle here is that the AREA of the bar represents the frequency, not the height.

1. Identify the smallest class width in the table (e.g., if you have intervals 0-10, 10-20, 20-40, the smallest width is 10).

2. For each class interval, calculate the new height of the bar using this formula: Height = Frequency / (Class Width / Smallest Class Width)

   • For the 0-10 class (width 10): Height = Frequency / (10/10) = Frequency

   • For the 20-40 class (width 20): Height = Frequency / (20/10) = Frequency / 2

Common Mistake: Forgetting to adjust the heights for unequal intervals. If you see a class interval that is wider than the others, you must adjust its height downwards.

3. The Frequency Polygon

This is a line graph that shows the shape of the distribution.

1. Find the mid-value of each class interval. Mid-value = (Lower Limit + Upper Limit) / 2.

2. Plot the points: (mid-value, frequency).

3. Connect the points with straight lines.

4. Anchor the polygon: Anchor the line to the horizontal axis by including the mid-value of the class before the first class and the class after the last class (their frequency is 0).

4. The Cumulative Frequency Curve (Ogive)

This S-shaped curve is used to find medians and quartiles.

Steps:

1. Create a Cumulative Frequency Table: Add a new column to your table and calculate the "running total" of the frequencies.

2. Plot the Points: Plot the points using the (UPPER class boundary, cumulative frequency).

3. Anchor the Curve: Start the curve from the lower boundary of the first class, at a frequency of 0.

4. Draw a smooth S-shaped curve through the points. Do not use a ruler.

5. Finding Quartiles from the Cumulative Frequency Curve

This is the main application of the Ogive. Let N be the total frequency (the final value in your cumulative frequency column).

• First Quartile (Q1): Find the position (1/4) * N on the vertical (cumulative frequency) axis. Draw a horizontal line to the curve, then a vertical line down to the horizontal axis. The value you read is Q1.

• Median (Second Quartile, Q2): Find the position (1/2) * N on the vertical axis. Repeat the process to find the median.

• Third Quartile (Q3): Find the position (3/4) * N on the vertical axis. Repeat the process to find Q3.

• Interquartile Range (IQR): This is a measure of spread. Calculate it using the formula: IQR = Q3 - Q1.

Exam Tip: When asked for the "interquartile range," don't just state the values of Q1 and Q3. You must do the final subtraction to get full marks.