Interquartile Range (IQR) on Calculator

The interquartile range (IQR) is one of the most useful measures of variability in statistics. It describes the spread of the middle 50% of observations in a dataset and is especially useful when a dataset contains outliers.

This guide explains what the IQR means, how to calculate it, how to use a TI-84 calculator to find quartiles, and how to interpret the result.

What Is the Interquartile Range?

The interquartile range is the difference between the third quartile (Q3) and the first quartile (Q1):

IQR = Q3 − Q1

Where:

• Q1 = first quartile, or 25th percentile
• Q2 = median, or 50th percentile
• Q3 = third quartile, or 75th percentile

The IQR therefore represents the range containing approximately the middle 50% of the data.

For example, if:

• Q1 = 12
• Q3 = 28

then:

IQR = 28 − 12 = 16

This means the middle 50% of observations span 16 units.

Why Is the IQR Important?

The IQR is particularly useful because it is less affected by extreme observations than measures such as the range or standard deviation.

For example, consider two datasets:

Dataset A: 10, 12, 13, 15, 16, 18, 20

Dataset B: 10, 12, 13, 15, 16, 18, 100

The extreme value of 100 substantially increases the range of Dataset B. However, the IQR focuses on the central portion of the data and is therefore much more resistant to that extreme value.

This makes the IQR useful when analyzing:

• Skewed distributions
• Salary and income data
• House prices
• Financial data
• Experimental measurements
• Survey responses
• Data containing potential outliers

IQR Formula

The basic formula is:

IQR = Q3 − Q1

For example, suppose a dataset has:

Q1 = 7

and

Q3 = 23

Then:

IQR = 23 − 7

IQR = 16

Therefore, the middle 50% of the observations have a spread of 16 units.

How to Calculate IQR Using a TI-84 Calculator

Let’s use the following dataset:

4, 6, 6, 7, 8, 12, 15, 17, 20, 21, 21, 23, 24, 27, 28

Step 1: Enter the Data

First, enter the observations into the TI-84 calculator.

1. Press STAT.
2. Select 1:Edit.
3. Enter the values into L1.

Your L1 list should contain:

4667812151720212123242728

Step 2: Open 1-Variable Statistics

After entering the data:

1. Press STAT.
2. Move to CALC.
3. Select 1-Var Stats.
4. Press ENTER.
5. Enter L1 if it is not automatically displayed.
6. Press ENTER.

The calculator will display descriptive statistics for the dataset.

Step 3: Find Q1 and Q3

Depending on your TI-84 model and software version, quartiles may be displayed in the statistical summary.

For this example:

• Q1 = 7
• Q3 = 23

If your calculator does not display Q1 and Q3 directly, you can determine the quartiles using the sorted data and the appropriate quartile method required by your course.

Step 4: Calculate the IQR

Once you have Q1 and Q3, calculate:

IQR = Q3 − Q1

Therefore:

IQR = 23 − 7

IQR = 16

So, the interquartile range is 16.

How to Interpret the IQR

An IQR of 16 means that the middle 50% of observations are spread across a range of 16 units.

It does not mean that every observation lies within a 16-unit interval centered around the mean.

Instead, it describes the distance between the 25th percentile and the 75th percentile.

For example:

• Q1 = 7
• Median = 17
• Q3 = 23

The middle 50% of the observations lie between 7 and 23, giving an IQR of 16.

IQR and Outliers

The IQR can also be used to identify potential outliers using the 1.5 × IQR rule.

First calculate:

Lower Fence = Q1 − 1.5(IQR)

Upper Fence = Q3 + 1.5(IQR)

Using Q1 = 7, Q3 = 23, and IQR = 16:

Lower Fence = 7 − 1.5(16)

Lower Fence = 7 − 24 = −17

And:

Upper Fence = 23 + 1.5(16)

Upper Fence = 23 + 24 = 47

Therefore, observations below −17 or above 47 would be considered potential outliers using this rule.

This approach is commonly used when constructing and interpreting box plots.

IQR vs. Range

The IQR and range both measure dispersion, but they use different parts of the dataset.

The range can be heavily influenced by a single extreme value, whereas the IQR is considerably more resistant to outliers.

IQR vs. Standard Deviation

The IQR and standard deviation answer slightly different questions.

IQR describes the spread of the middle 50% of observations and is particularly useful for skewed data or datasets containing outliers.

Standard deviation measures the typical distance of observations from the mean and is commonly used when the data are reasonably symmetric and other assumptions are appropriate.

For skewed distributions, reporting the median and IQR is often more informative than relying only on the mean and standard deviation.

Example: Interpreting IQR in Real-World Data

Suppose a company analyzes employee salaries and obtains:

• Q1 = ₹30,000
• Median = ₹45,000
• Q3 = ₹70,000

The IQR is:

IQR = ₹70,000 − ₹30,000

IQR = ₹40,000

This means that the middle 50% of employee salaries span a range of ₹40,000, from ₹30,000 to ₹70,000.

The IQR provides a useful summary even if a small number of employees have exceptionally high salaries.

Frequently Asked Questions

What does IQR stand for?

IQR stands for Interquartile Range. It measures the spread between the first and third quartiles.

What is the formula for IQR?

The formula is:

IQR = Q3 − Q1

What percentage of data does the IQR contain?

The IQR represents the middle 50% of the dataset, between the 25th and 75th percentiles.

Is a higher IQR better?

Not necessarily. A higher IQR means the middle 50% of observations are more spread out. Whether that is desirable depends on the context of the data.

Can IQR be used to identify outliers?

Yes. A common approach is to calculate the lower and upper fences using 1.5 × IQR.

Is IQR affected by outliers?

The IQR is relatively resistant to outliers because it is based on Q1 and Q3 rather than the minimum and maximum values.

Conclusion

The interquartile range (IQR) is an important measure of statistical dispersion that focuses on the middle 50% of a dataset. It is particularly useful for skewed datasets and data containing extreme observations.

On a TI-84 calculator, you can enter your dataset into L1 and use 1-Var Stats to examine the distribution and obtain the information needed to calculate Q1 and Q3.

For the example dataset:

4, 6, 6, 7, 8, 12, 15, 17, 20, 21, 21, 23, 24, 27, 28

with:

Q1 = 7

Q3 = 23

the IQR is:

IQR = 23 − 7 = 16

Understanding the IQR, along with the median and quartiles, provides a practical way to describe variability and identify potential outliers in real-world datasets.