How to Calculate the Upper Quartile Step-by-Step

Calculating the upper quartile of a data set is a key task in statistics that allows you to understand the distribution and spread of your data. But for many of us non-statisticians, quartiles can seem complicated at first. Not to worry! In this article, I’ll walk you through how to calculate the upper quartile in an easy, step-by-step manner.

What is a Quartile?

Before jumping into the calculation, it helps to understand exactly what a quartile is. Quartiles divide a set of data into four equal groups, or quarters There are three quartile values

• The first quartile (Q1) is the 25th percentile It splits off the lowest 25% of data from the highest 75%

• The second quartile (Q2) is the 50th percentile, or the median. It splits the data set in half.

• The third quartile (Q3) is the 75th percentile, or the upper quartile. It splits off the highest 25% of data from the lowest 75%.

The upper quartile, specifically, tells us the value that separates the top 25% of data points from the bottom 75%. It gives us insight into the spread and skewness of the data.

When to Calculate the Upper Quartile

Some examples of when calculating the upper quartile is useful:

• When constructing a box plot diagram. The upper quartile forms the top line of the box.

• When describing the distribution of data. Knowing the quartiles helps us see how tightly or broadly distributed the data is.

• When identifying outliers. Data points above the upper quartile may be outliers that skew the rest of the data.

• When comparing data sets. Comparing upper quartiles helps us see how one set of data differs from another.

• When summarizing a scoring data set. The upper quartile tells us the score that 25% of people scored above.

How to Calculate the Upper Quartile by Hand

Calculating the upper quartile by hand just takes a few steps:

1. Order the Data from Lowest to Highest

The first step is to arrange all of the observations in your data set from smallest to largest. For example:

Original data: {1, 5, 7, 3, 11, 9, 4, 8, 6}
Ordered data: {1, 3, 4, 5, 6, 7, 8, 9, 11}

This puts the data in ascending order ready for the quartile calculation.

2. Find the Total Number of Data Points

Next, count the total number of data points you have.

For our example, there are 9 data points.

3. Plug the Values into the Formula

The formula to find the upper quartile is:

Q3 = (n + 1) * (3/4)

Where:

• Q3 is the upper quartile
• n is the total number of data points

For our example with 9 data points, this would be:

Q3 = (9 + 1) * (3/4)

Q3 = 10 * (3/4)

Q3 = 7.5

4. Determine the Upper Quartile Value

The formula gives us the upper quartile position, but not the actual value yet.

If it results in a whole number, we simply take the data point at that position. For example, if the result was 5, the upper quartile would be the 5th data point.

For decimals or fractions, take the two data points on either side of the quartile position. Find their mean (average) to get the upper quartile value.

In our case, the position is 7.5. The two data points on either side are 7 and 8.

(7 + 8) / 2 = 7.5

Therefore, the upper quartile value of our data set is 7.5.

And that’s it – we’ve calculated the upper quartile by hand in just 4 steps!

Using Excel to Find the Upper Quartile

While the manual method works for small datasets, computer programs are better suited for larger amounts of data. Excel has a handy quartile function that does the heavy lifting for us.

Here are the steps to find the upper quartile in Excel:

1. Enter the Data into a Column

Type or paste your data set into one column in Excel. For example in cells A1 to A9.

2. Click an Empty Cell and Type “=QUARTILE”

Click another cell where you want the quartile result to appear. Type “=” to begin a function then start typing “QUARTILE”. Select the QUARTILE function from the dropdown.

3. Select the Data Range

After typing the function, Excel will prompt you to select the range of data cells. Highlight the column of data you want to analyze.

4. Type a Comma Followed by “3” for the Third Quartile

Excel’s quartile function takes two inputs: the data range, followed by the quartile number. For the upper quartile, we want the third quartile, so type a comma after selecting the data range, then type “3”.

5. Hit Enter to Reveal the Result

The complete function will look something like:

=QUARTILE(A1:A9,3)

Hit enter and Excel calculates the upper quartile for you! The quartile function does all the ordering and math automatically.

Comparing Methods for Different Data Sets

To help demonstrate, let’s run through a couple examples comparing the manual and Excel methods:

Data Set 1: {1, 5, 8, 9, 10, 11, 13, 15, 17}

Manual Method:

• n = 9 data points
• Plug into formula: Q3 = (9 + 1) * (3/4) = 7.5
• The upper quartile position is 7.5, between data points 7 and 8
• Mean is (7 + 8) / 2 = 7.5
• Upper quartile = 7.5

Excel Method:

• Input data into A1 through A9
• Enter formula =QUARTILE(A1:A9,3)
• Result is 7.5
• Upper quartile = 7.5

Data Set 2: {10, 13, 14, 15, 16, 17, 20, 26, 42, 510}

Manual Method:

• n = 10 data points
• Q3 = (10 + 1) * (3/4) = 8.25
• Position 8.25, between data points 17 and 20
• Mean is (17 + 20) / 2 = 18.5
• Upper quartile = 18.5

Excel Method:

• Enter data into A1 through A10
• Formula is =QUARTILE(A1:A10,3)
• Result is 18.5
• Upper quartile = 18.5

As you can see, both methods arrive at the same upper quartile result. The manual approach is great for small datasets and understanding what’s going on behind the scenes. But for large datasets, Excel provides convenient automation to handle the calculation quickly.

Interpreting and Using the Upper Quartile Value

Once you’ve found the upper quartile, what can you do with it? Here are some ways to apply it:

• Use it to construct a box plot diagram, marking the upper hinge.

• Assess the data spread – a small IQR (Q3 – Q1 difference) means the data is clustered, while a large IQR means it’s more dispersed.

• Look for outliers – data points above Q3 + 1.5(IQR) are potential outliers.

• Compare groups – Differences in Q3 values can show distribution shifts between groups.

• Summarize a score dataset – We can say “25% of test takers scored above the upper quartile.”

• Set benchmarks or goals – For example, “We aim to have 25% of sales above $500k per month.”

The upper quartile builds understanding of our data and aids decision making. Combined with the other quartiles, we gain valuable insight into the shape and spread.

Finding Other Quartiles

While we focused on the upper quartile here, you can use the exact same methods to find the other quartiles:

Lower Quartile (Q1):

Manual: Q1 = (n + 1) * (1/4)

Excel: =QUARTILE(range,1)

Second Quartile or Median (Q2):

Manual: Q2 = (n + 1) * (2/4)

Excel: =QUARTILE(range,2)

Q1, Q2 and Q3 each have their own handy interpretations and use cases for understanding your data.

Key Takeaways for Calculating Quartiles

The key points on finding the upper and other quartiles:

• Quartiles split data into four equal quarters – the upper quartile separates the top 25%.

• Man