Lesson 4: Median and IQR

Property

The 5 number summary consists of five values: Minimum, Q1, Q2 (median), Q3, and Maximum. To find them:

1. Order the data from smallest to largest. The ends are the Minimum and Maximum.
2. Find the median of the entire data set (Q2).
3. Find the median of the lower half of the data (Q1).
4. Find the median of the upper half of the data (Q3).

The interquartile range (IQR) is the distance from Q1 to Q3, representing the middle 50% of the data. IQR = Q3 - Q1.

Examples

• For the data {2, 5, 6, 9, 11, 14, 17}, the 5 number summary is: Min=2, Q1=5, Q2=9, Q3=14, Max=17. The IQR is $14 - 5 = 9$.
• For the data {10, 20, 25, 35, 45, 50}, the median (Q2) is 30. The lower half is {10, 20, 25}, so Q1=20. The upper half is {35, 45, 50}, so Q3=45. The IQR is $45 - 20 = 25$.
• Given a 5 number summary of Min=5, Q1=12, Q2=18, Q3=22, Max=30, the range is $30-5=25$ and the interquartile range (IQR) is $22 - 12 = 10$.

Explanation

The 5 number summary provides a quick snapshot of your data's distribution. The IQR is a powerful measure of spread because it focuses on the middle half of the data, which means it isn't affected by unusually high or low outliers.

Property

Box plots provide a visual image of the 5 number summary. To create a box plot:

1. Draw a number line that covers the range of the data.
2. Draw a rectangle (the box) from Q1 to Q3.
3. Draw a vertical line segment inside the box at the median (Q2).
4. Draw horizontal line segments (the whiskers) from the box to the Minimum and Maximum values.

Examples

• For a 5 number summary of {10, 15, 19, 24, 30}, a box plot would have a box from 15 to 24, a line at 19, a left whisker to 10, and a right whisker to 30.
• If a box plot has a very long right whisker and the median line is to the left of the box's center, it suggests the data is skewed right, with some high values stretching the data out.
• Comparing two box plots, one with an IQR of 5 and another with an IQR of 20 shows that the middle 50% of the data in the second plot is much more spread out.

Explanation

A box plot turns the 5 Number Summary into a simple picture. The 'box' shows the spread of the middle 50% of data (IQR), and the 'whiskers' show the spread of the lowest and highest 25%. A wider section means the data in that part is more spread out.