Measures of Spread and Outliers

The Impact of Outliers

An outlier is an extreme value that does not fit the general pattern of the data. The range is highly sensitive to outliers because it relies exclusively on the highest and lowest values.

When a data set contains extreme outliers, the range "distorts" the view of the data by stretching the distance without reflecting the rest of the set. The IQR is much more robust because it ignores the extreme top and bottom 25%.

• Data Set A: $2, 4, 5, 6, 8, 10, 12$
  • $\text{Range} = 12 - 2 = 10$
  • $\text{IQR} = 10 - 4 = 6$
• Data Set B (with an outlier): $2, 4, 5, 6, 8, 10, 80$
  • $\text{Range} = 80 - 2 = 78$
  • $\text{IQR} = 10 - 4 = 6$

Calculating Quartiles from Discrete Data

Quartiles divide an ordered data set into four equal parts. The Lower Quartile ($Q_1$) is the value one-quarter of the way through the data, and the Upper Quartile ($Q_3$) is three-quarters of the way through.

For discrete lists of data, Edexcel accepts finding the median of the lower and upper halves of the data, or using the position formulas:

• Position of $Q_1$: $\frac{n+1}{4}\text{th term}$
• Position of $Q_3$: $\frac{3(n+1)}{4}\text{th term}$

Worked Example: Discrete Data

Find the range, $Q_1$, $Q_3$, and the IQR for the following data set: $3, 5, 8, 12, 15, 18, 20$.

Step 1: Calculate the Range.

• $\text{Range} = \text{Maximum} - \text{Minimum}$
• $\text{Range} = 20 - 3 = 17$

Step 2: Find the median to split the data into halves.

• There are 7 values ($n=7$).
• The median is the middle value: $12$.

Step 3: Find $Q_1$ and $Q_3$ using the lower and upper halves.

• Lower half is $3, 5, 8$. The median of this half ($Q_1$) is $5$.
• Upper half is $15, 18, 20$. The median of this half ($Q_3$) is $18$.

Step 4: Calculate the IQR.

• $\text{IQR} = Q_3 - Q_1$
• $\text{IQR} = 18 - 5 = 13$

Estimating Spread from Grouped Data

When data is grouped into classes, the exact individual values are lost. We can only calculate an Estimated Range by finding the difference between the upper boundary of the highest class and the lower boundary of the lowest class.

Worked Example: Estimating Range from a Table

Find the estimated range for the following masses:

Mass ($m$ kg)	Frequency
$0 < m \leq 10$	5
$10 < m \leq 20$	12
$20 < m \leq 30$	3

Step 1: Identify the lowest and highest boundaries.

• Lower boundary of the lowest class $= 0$
• Upper boundary of the highest class $= 30$

Step 2: Calculate the Estimated Range.

• $\text{Estimated Range} = 30 - 0 = 30\text{ kg}$

Estimating Quartiles from a Cumulative Frequency Graph

To find quartiles for grouped data, you must use a Cumulative Frequency Graph.

• Find the $Q_1$ position at $\frac{n}{4}$ on the frequency ($y$) axis.
• Find the $Q_3$ position at $\frac{3n}{4}$ on the frequency ($y$) axis.
• Read across to the curve and down to the $x$-axis to find the actual values.

Worked Example: IQR from a Cumulative Frequency Graph

A cumulative frequency graph shows the test marks of 80 students. Estimate the IQR if the curve crosses the $\frac{n}{4}$ frequency line at 32 marks and the $\frac{3n}{4}$ frequency line at 58 marks.

Step 1: Identify the positions on the $y$-axis.

• $n = 80$
• $Q_1$ position $= \frac{80}{4} = 20\text{th value}$
• $Q_3$ position $= \frac{3 \times 80}{4} = 60\text{th value}$

Step 2: Read the corresponding values from the $x$-axis.

• Reading across from 20 on the $y$-axis gives $Q_1 = 32\text{ marks}$.
• Reading across from 60 on the $y$-axis gives $Q_3 = 58\text{ marks}$.

Step 3: Calculate the IQR.

• $\text{IQR} = Q_3 - Q_1$
• $\text{IQR} = 58 - 32 = 26\text{ marks}$

Identifying Outliers Mathematically (Higher Tier)

At Higher Tier, you must use a specific mathematical rule to provide an objective boundary, determining exactly which values are outliers.

An extreme value is an outlier if it falls outside the Lower Boundary or the Upper Boundary:

$$\text{Lower Boundary} = Q_1 - (1.5 \times \text{IQR})$$ $$\text{Upper Boundary} = Q_3 + (1.5 \times \text{IQR})$$

If asked to draw a box plot, any calculated outliers must be plotted as an 'x' or an asterisk (*). The whiskers should only extend to the smallest and largest values that are not outliers.

Worked Example: Identifying Outliers

A data set has $Q_1 = 14$ and $Q_3 = 17$. The maximum value in the data set is $22$. Determine mathematically if $22$ is an outlier.

Step 1: Calculate the IQR.

• $\text{IQR} = 17 - 14 = 3$

Step 2: Calculate the upper boundary.

• $\text{Upper Boundary} = Q_3 + (1.5 \times \text{IQR})$
• $\text{Upper Boundary} = 17 + (1.5 \times 3)$
• $\text{Upper Boundary} = 17 + 4.5 = 21.5$

Step 3: Compare the value to the boundary.

• $22 > 21.5$
• Therefore, $22$ is an outlier.