Interpreting Cumulative Frequency Diagrams

• The curve represents the distribution of the population. A steep section of the curve indicates a high frequency of data points within that specific range, while a flat section indicates fewer data points.

• The total frequency ($n$) is the maximum value on the y-axis. All positional statistics are calculated as fractions of this total value, allowing for comparisons between datasets of different sizes.

• The Interquartile Range (IQR) measures the spread of the middle 50% of the data, calculated as $IQR = Q_3 - Q_1$. This provides a measure of variability that is less sensitive to outliers than the total range.

Finding Positional Statistics

• The Median ($Q_2$): Locate the $\frac{n}{2}$ position on the y-axis, move horizontally to the curve, and then vertically down to read the value on the x-axis.
• Lower Quartile ($Q_1$): Locate the $\frac{n}{4}$ (or 25%) position on the y-axis and map it to the x-axis via the curve.
• Upper Quartile ($Q_3$): Locate the $\frac{3n}{4}$ (or 75%) position on the y-axis and map it to the x-axis via the curve.

Estimating Frequencies

• To find the number of items below a value: Locate the value on the x-axis, move up to the curve, and then across to the y-axis to read the cumulative frequency.
• To find the number of items above a value: Find the cumulative frequency for that value and subtract it from the total frequency ($n$).

Percentile Formula: To find the $p$-th percentile, locate the value at $y = \frac{p}{100} \times n$ and map it to the x-axis.

• Check the Total Frequency: Always identify the maximum value on the y-axis ($n$) before performing any calculations. Do not assume the y-axis ends exactly at $n$.

• Precision with Rulers: When mapping values between axes in an exam, use a ruler to draw dashed lines. Small errors in reading the graph can lead to significant errors in the final answer.

• Sanity Check: Ensure that $Q_1 < Q_2 < Q_3$. If your median is smaller than your lower quartile, you have likely miscalculated the positions on the y-axis.

• 'More Than' Questions: A common exam trap is asking for the number of items greater than a certain value. Remember to subtract the y-value from the total frequency: $\text{Count} = n - \text{CF}$.

• Using the wrong axis: Students often confuse the data value (x-axis) with the frequency (y-axis) when asked to find a percentile.

• Misinterpreting the slope: A steep slope means a high density of data, not necessarily 'higher' values. A flat line means there is no data in that interval.

• Incorrect Quartile Positions: Using $n+1$ instead of $n$ for large datasets. In cumulative frequency diagrams, using $\frac{n}{4}$ and $\frac{3n}{4}$ is the standard convention for continuous data.