Determine quartiles and interquartile range

For an ordered data set, determines the first quartile (Q1), median (Q2), and third quartile (Q3) by finding the medians of the lower and upper halves, omitting the overall median from those halves when the number of values is odd. Calculates and interprets the interquartile range, $Q3-Q1$ , as the spread of the middle 50% of the data and connects these values to box plots; alternate quartile conventions, percentile interpolation, and grouped-data methods are not included.

Detailed Explanation: Determine quartiles and interquartile range

Quartiles divide an ordered data set into four parts:

$(Q1)$ : the median of the lower half
$(Q2)$ : the overall median
$(Q3)$ : the median of the upper half
Interquartile range: $(Q3-Q1)$ , which shows the spread of the middle $(50\%)$ of the data

Example

Find $(Q1)$ , $(Q2)$ , $(Q3)$ , and the interquartile range for:

3,\ 5,\ 6,\ 8,\ 9,\ 12,\ 14,\ 15,\ 18,\ 20,\ 24

The data is already in order.

Step 1: Find the median, $(Q2)$ .

There are $(11)$ values, so the middle value is the $6$ th value:

Q2=12

Because there is an odd number of values, leave the overall median out of both halves.

Step 2: Find $(Q1)$ .

The lower half is:

3,\ 5,\ 6,\ 8,\ 9

The middle value is $6$ , so:

Q1=6

Step 3: Find $(Q3)$ .

The upper half is:

14,\ 15,\ 18,\ 20,\ 24

The middle value is $(18)$ , so:

Q3=18

Step 4: Find the interquartile range.

Subtract $(Q1)$ from $(Q3)$ :

\text{IQR}=Q3-Q1=18-6=12

So the results are:

\boxed{Q1=6,\quad Q2=12,\quad Q3=18,\quad \text{IQR}=12}

The interquartile range of $(12)$ means that the middle $(50\%)$ of the data stretches from $6$ to $(18)$ , covering a range of $(12)$ units.

These values are used in a box plot. The box extends from $(Q1)$ to $(Q3)$ , the line inside the box marks $(Q2)$ , and the IQR is the length of the box.

Learn by doing: Determine quartiles and interquartile range

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Statistics - Quartiles - Data Set (No Outliers) to Interquartile Range

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