If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

## Statistics and probability

### Course: Statistics and probability>Unit 3

Lesson 3: Interquartile range (IQR)

# Comparing range and interquartile range (IQR)

Range and interquartile range (IQR) both measure the "spread" in a data set. Looking at spread lets us see how much data varies. Range is a quick way to get an idea of spread. It takes longer to find the IQR, but it sometimes gives us more useful information about spread.

## Part 1: The range

Ron recorded the daily high temperatures for two different cities in a recent week in degree Celsius. The temperatures for each city are shown below.
Kansas City, MO: 23, comma, 25, comma, 28, comma, 28, comma, 32, comma, 33, comma, 35
Paradise, MI: 16, comma, 24, comma, 26, comma, 26, comma, 26, comma, 27, comma, 28
problem a
Calculate the range of the temperatures in Kansas City, MO.
range =

problem b
Calculate the range of the temperatures in Paradise, MI.
range =

problem c
What is the best interpretation of the range?

problem d
According to the ranges, which city's temperatures were more varied in that week?

## Part 2: The interquartile range (IQR)

problem a
Calculate the IQR of the temperatures in Kansas City, MO.
As a reminder, here are the temperatures: 23, comma, 25, comma, 28, comma, 28, comma, 32, comma, 33, comma, 35
IQR =

problem b
Calculate the IQR of the temperatures in Paradise, MI.
As a reminder, here are the temperatures: 16, comma, 24, comma, 26, comma, 26, comma, 26, comma, 27, comma, 28
IQR =

problem c
What is the best interpretation of the IQR?

problem d
According to the IQRs, which city's temperatures were more varied in that week?

## Part 3: Comparing range and IQR

Ron made a dot plot for the temperatures in each city.
A double dot plot with the upper half modeling the Kansas City, Missouri and the lower half models the Paradise, Michigan. The number line is labeled temperature in degrees celsius. The Kansas City, Missouri dots range from 21 to 35. The Paradise, Michigan dots range from 16 to 28, but there is a cluster of dots from 26 to 28 with only one dot at 16 and a gap from 17 to 23.
problem a
Looking at the dot plots, which city's temperatures seem to be more varied from day to day?

problem b
Why might IQR be the preferred measure of spread in this context?

## Want to join the conversation?

• IQR is used to find the dispersion between the quartiles means of Q1 to Q3?
• Not quite. It's the difference between Q1 (the boundary between the first and second quartile groups) and Q3 (the boundary between the third and fourth quartile groups). They're not means; they're just points.

So, consider the following data points:
2 4 5 6 9 10 11
Q1 is the boundary between the the first and second quartile groups, which is 4. Q3 is the boundary between the third and fourth quartile groups, which is 10. That means that the IQR is their difference, Q3-Q1, which is 6.

IQR is an easy way to measure how spread out numbers are, without worrying about outliers. If I look at the page counts of the Harry Potter books, I see: 223, 251, 317, 636, 766, 607, 607. The IQR is 385. On the other hand, if I look at the page counts for the Mercy Thompson books, I see 298, 306, 321, 305, 351, 337, 343, 353, 350. The IQR is only 45. From this, I can see that Mercy Thompson books tend to be more or less the same size, while the lengths of Harry Potter books tend to vary a lot.
• How would we use IQR in real-life situations?
• I'll try an example. You work for the regional manager of some kind of chain business -- restaurant, hair salon, whatever. Your boss wants to know, roughly how many employees does the average location have? Is it, like, about 15? Or is it about 50? Or is it something like, between 15 and 30? So, you know that there are some locations with only a handful of employees; another location in a big city has over 100. But your boss doesn't want to worry about such details, and just wants a "ballpark estimate".

The mean or the median might be thrown off by the one or two big locations, or maybe by the handful of really small locations. (Maybe they balance each other out; or maybe not.) In that case, the IQR could usefully predict how much weekly dry cleaning of uniforms (or whatever) needs to be done at the "average" location.
• is there a Q4? if not why is it called IQR?
(1 vote)
• There is no Q4. The Quartiles split the data up into 4 equal portions. To do so, we need just three values - Q1, Q2, and Q3. For comparison, think of cutting a string (or anything) in half. You wind up with 2 pieces, but made only 1 cut. This is because you start with "one" piece, and each cut gives you one more. So 3 cuts gives you 4 pieces. The Quartiles are the "cuts".

The range is the distance from the highest value to the lowest value. The Inter-Quartile Range is quite literally just the range of the quartiles: the distance from the largest quartile to the smallest quartile, which is IQR=Q3-Q1.

Sometimes people will group the minimum and the maximum along with the Quartiles in what is called the "5 Number Summary". This is because the Min and Max are kind of like the 0th and 4th Quartiles, if those things existed. Going back to the string example, the Min and the Max would be the original ends of the string, and the Quartiles would be the places that you cut the string.
• What is the meaning of outlier and why it's used?
• If you were to make a graph, the outlier wouldn't be where most of the other numbers were. So, let's say the data is 10, 11, 9, 10, 12, and 20. The outlier would be 20 because it is farther away from the other numbers.
I hope this helps.
• if you have a normally distributed bell curve and a known mean, but no known standard deviation, how do you find the interquartile range?
• It's not possible to do this without other information. If only the mean of a normal distribution is known, then clearly the larger the standard deviation, the larger the interquartile range.
• what do you mean by variability
• emm.. - Variability is the extent to which data points in a statistical distribution or data set diverge from the average, or mean, value as well as the extent to which these data points differ from each other. There are four commonly used measures of variability: range, mean, variance and standard deviation-from https://www.investopedia.com › Insights › Markets & Economy
.
Hope it is helpful. Thank you.
• I wonder whether my understandings of IQR as follows is right:
Q1 is equal to the digit which is located in 25% of all values in the sample from least to greatest( the number is at 1/4 of the sample);
Q2 is also known as the "median";
Q3 is equal to the digit located in 75% of all values in the sample from least to greatest(the number is at 3/4 of the sample);
IQR=Q3-Q1.
Thank you.