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## Measurement & Data - Statistics & Probability 218-221

### Unit 1: Lesson 6

Interquartile range (IQR)

# Interquartile range review

## Interquartile Range (IQR)

Interquartile range is the amount of spread in the middle 50, percent of a dataset.
In other words, it is the distance between the first quartile left parenthesis, start text, Q, end text, start subscript, 1, end subscript, right parenthesis and the third quartile left parenthesis, start text, Q, end text, start subscript, 3, end subscript, right parenthesis.
start text, I, Q, R, end text, equals, start text, Q, end text, start subscript, 3, end subscript, minus, start text, Q, end text, start subscript, 1, end subscript
Here's how to find the IQR:
Step 1: Put the data in order from least to greatest.
Step 2: Find the median. If the number of data points is odd, the median is the middle data point. If the number of data points is even, the median is the average of the middle two data points.
Step 3: Find the first quartile left parenthesis, start text, Q, end text, start subscript, 1, end subscript, right parenthesis. The first quartile is the median of the data points to the left of the median in the ordered list.
Step 4: Find the third quartile left parenthesis, start text, Q, end text, start subscript, 3, end subscript, right parenthesis. The third quartile is the median of the data points to the right of the median in the ordered list.
Step 5: Calculate IQR by subtracting start text, Q, end text, start subscript, 3, end subscript, minus, start text, Q, end text, start subscript, 1, end subscript.

### Example

Essays in Ms. Fenchel's class are scored on a 6 point scale.
Find the IQR of these scores:
1, 3, 3, 3, 4, 4, 4, 6, 6
Step 1: The data is already in order.
Step 2: Find the median. There are 9 scores, so the median is the middle score.
1, 3, 3, 3, 4, 4, 4, 6, 6
The median is 4.
Step 3: Find start text, Q, end text, start subscript, 1, end subscript, which is the median of the data to the left of the median.
There is an even number of data points to the left of the median, so we need the average of the middle two data points.
1, 3, 3, 3
start text, Q, end text, start subscript, 1, end subscript, equals, start fraction, 3, plus, 3, divided by, 2, end fraction, equals, 3
The first quartile is 3.
Step 4: Find start text, Q, end text, start subscript, 3, end subscript, which is the median of the data to the right of the median.
There is an even number of data points to the right of the median, so we need the average of the middle two data points.
4, 4, 6, 6
start text, Q, end text, start subscript, 3, end subscript, equals, start fraction, 4, plus, 6, divided by, 2, end fraction, equals, 5
The third quartile is 5.
Step 5: Calculate the IQR.
\begin{aligned} \text{IQR} &= \text{Q}_3-\text{Q}_1 \\ \\ &= 5-3 \\ \\ &= 2 \end{aligned}
The IQR is 2 points.