Measures of Position

Quartiles divide a sorted data set into four equal parts, each containing approximately 25% of the observations.

• First Quartile ($Q_1$): The 25th percentile; 25% of data is $\le Q_1$.

• Second Quartile ($Q_2$): The 50th percentile, better known as the Median.

• Third Quartile ($Q_3$): The 75th percentile; 75% of data is $\le Q_3$.

Calculating Quartiles

• Step 1: Sort the data in ascending order. This is the most critical step; calculations on unsorted data will be incorrect.
• Step 2: Find the median ($Q_2$) of the entire data set. This splits the data into a lower half and an upper half.
• Step 3: Find the median of the lower half to determine $Q_1$. If the original data set has an odd number of values, the median itself is excluded from both halves.
• Step 4: Find the median of the upper half to determine $Q_3$.

Calculating Percentiles

• To find the position $i$ of the $k$-th percentile in a data set of $n$ values, use the formula: $i = \frac{k}{100} \cdot n$
• If $i$ is not a whole number, round up to the next integer to find the position. If $i$ is a whole number, the percentile is the average of the value at position $i$ and the value at position $i+1$.

It is important to distinguish between the value of a measure and its position in the list.

• Always Sort First: Examiners often provide data in random order. Students frequently lose marks by calculating the median or quartiles before sorting.

• Check the 'n': When finding the median of halves for quartiles, be careful with odd vs. even counts. For an odd $n$, the median is a specific data point and should be 'covered up' or ignored when looking at the lower and upper halves.

• Interpretation: If asked to interpret a percentile, always use the phrase 'less than or equal to'. For example, 'A score at the 90th percentile means 90% of students scored the same as or lower than this score.'

• Sanity Check: $Q_1$ must always be less than or equal to the Median, and the Median must be less than or equal to $Q_3$.