Numerical & Statistical Skills

Calculating Central Tendency

• Mean: Sum all individual values in the dataset and divide by the total number of items ($n$). It is the most common average but is sensitive to extreme outliers.
• Median: Arrange data in ascending order and identify the middle value. If there is an even number of items, the median is the average of the two central values.
• Mode: Identify the value that appears with the highest frequency. A dataset can be unimodal, bimodal, or have no mode if all values are unique.

Measuring Dispersion

• Range: Calculate the difference between the maximum and minimum values ($Range = Max - Min$). This provides a simple measure of the total spread.
• Interquartile Range (IQR): Calculate the difference between the Upper Quartile ($75^{th}$ percentile) and the Lower Quartile ($25^{th}$ percentile). This measures the spread of the middle 50% of the data, ignoring outliers.

Proportional Analysis

• Percentage Change: Use the formula $\frac{Final - Original}{Original} \times 100$ to determine the relative increase or decrease between two points in time.

• Ratio vs. Proportion: A ratio compares two different groups (e.g., 3:1), while a proportion compares one group to the total population (e.g., 75%).
• Percentage vs. Percentage Change: A percentage represents a static portion of a whole, whereas percentage change measures the dynamic shift in a value over time.

• Check the Precision: Always verify if the question requires a specific number of decimal places or rounding to the nearest whole number. Rounding too early in a multi-step calculation can lead to inaccuracies.

• Show All Working: Examiners often award marks for the correct application of a formula even if the final arithmetic is incorrect. Clearly state the formula used and the values substituted into it.

• Units are Mandatory: Ensure that units (e.g., dollars, meters, %) are included in the final answer if specified. A numerical value without units may be considered incomplete.

• Sanity Check: Evaluate if the calculated answer is realistic. For example, a percentage decrease cannot exceed 100% if the value cannot be negative.

• Unordered Median: A common error is attempting to find the median without first sorting the data from smallest to largest. The median is a position-based measure, not just the middle of a random list.

• Denominator Error: In percentage change calculations, students often divide by the 'new' value instead of the 'original' value. Always divide by the starting point ($Original$).

• Negative Signs: Forgetting that a negative result in a percentage change formula indicates a decrease. It is often better to state 'a 15% decrease' rather than '-15% decrease' to avoid double negatives.

• Mean vs. Median Choice: Using the mean for datasets with extreme outliers can give a misleading 'average'. In such cases, the median is usually a more accurate representation of the typical value.