Range & Interquartile Range

Range

The range is the simplest measure of spread in a data set. It shows how spread out the values are by calculating the difference between the highest and lowest values.

To find the range:

• Identify the highest value in the data set
• Identify the lowest value in the data set
• Subtract the lowest value from the highest value

The formula is: $\text{Range} = \text{Highest value} - \text{Lowest value}$

The range is easy to calculate and gives a quick idea of the spread of the data. However, it only considers the two extreme values and ignores all other data points.

For instance, if the temperatures recorded over a week are $12^\circ\text{C}, 15^\circ\text{C}, 18^\circ\text{C}, 20^\circ\text{C}, 22^\circ\text{C}, 25^\circ\text{C}, 30^\circ\text{C}$, the range is:

$$\text{Range} = 30 - 12 = 18^\circ\text{C}$$

Interquartile Range (IQR)

The interquartile range (IQR) measures the spread of the middle 50% of the data. It is the difference between the upper quartile (Q3) and the lower quartile (Q1).

Unlike the range, the IQR is less affected by extreme values or outliers, making it a more reliable measure of spread when the data contains unusual values.

The formula is:

$$\text{IQR} = Q_3 - Q_1$$

Where:

• Q1 (lower quartile) is the median of the lower half of the data (below the median)
• Q3 (upper quartile) is the median of the upper half of the data (above the median)

The IQR shows the range within which the central half of the data lies, giving a better idea of typical values than the full range.

Calculating Quartiles

To calculate the quartiles:

1. Order the data set from smallest to largest
2. Find the median (Q2), which divides the data into two halves
3. Find the lower quartile (Q1) by finding the median of the lower half (values below Q2)
4. Find the upper quartile (Q3) by finding the median of the upper half (values above Q2)

If the number of data points is odd, exclude the median when finding Q1 and Q3.

For even numbers of data points, include all values in halves when finding Q1 and Q3.

For example, consider the data set:

3, 7, 8, 12, 13, 14, 18, 21, 23, 27, 30

Step 1: The data is already ordered.

Step 2: Find the median (Q2). There are 11 values, so the median is the 6th value:

$$\text{Median} = 14$$

Step 3: Lower half is 3, 7, 8, 12, 13 (values below 14). Median of these is 8 (3rd value), so:

$$Q_1 = 8$$

Step 4: Upper half is 18, 21, 23, 27, 30 (values above 14). Median of these is 23 (3rd value), so:

$$Q_3 = 23$$

Step 5: Calculate IQR:

$$\text{IQR} = 23 - 8 = 15$$