How Do You Get The Lower Quartile

You're staring at a spreadsheet. In real terms, maybe it's customer wait times, or monthly revenue per store, or the number of steps your fitness tracker logged last week. So naturally, maybe it's test scores. Someone — your boss, your professor, that one LinkedIn thought-leader post — mentioned "the lower quartile" and now you need to find it.

This is the bit that actually matters in practice.

Here's the thing: it's not complicated. But it is one of those concepts where the details matter. Get the method wrong and your Q1 shifts. Sometimes by a lot.

Let's walk through it properly. Here's the thing — no jargon for jargon's sake. Just the steps, the traps, and the context you actually need.

What Is the Lower Quartile

The lower quartile — also called Q1 or the first quartile — is the value that sits at the 25th percentile of your dataset. One quarter of your data falls below it. Three quarters sits above it Worth knowing..

Simple definition. But here's where it gets interesting: there isn't just one way to calculate it.

Different software, different textbooks, different industries — they all use slightly different methods. R's default quantile() function uses type 7. Python's numpy.In practice, percentile() defaults to linear interpolation. On the flip side, QUARTILE. Your TI-84 calculator? And eXC gives you another. INCgives you one answer. So excel'sQUARTILE. It uses yet another method.

And they don't always agree Worth keeping that in mind..

The Core Idea Stays the Same

No matter the method, the goal is identical: split your ordered data into four roughly equal chunks. The lower quartile marks the boundary between the bottom chunk and the second chunk Surprisingly effective..

Think of it like this. Sort them smallest to largest. You've got 100 test scores. The lower quartile is the score where 25 students did worse and 75 did better (or tied).

With 100 clean data points, that's easy. Practically speaking, the 25th and 26th values straddle the line. Most methods will average them or pick one.

But what if you have 11 values? In practice, or 14? Or 7? That's where the methods diverge It's one of those things that adds up..

Why It Matters / Why People Care

You might wonder: does the method really change anything? In practice — yes, sometimes.

Outlier Detection Depends on It

The most common use of Q1? Building the interquartile range (IQR). Consider this: that's Q3 minus Q1. Then you flag outliers as anything below Q1 − 1.5×IQR or above Q3 + 1.5×IQR Took long enough..

If your Q1 shifts by even a little, your IQR shifts. Your outlier fences move. A data point that looked like an outlier under one method might look perfectly normal under another.

I've seen this cause real arguments in production dashboards. One analyst uses Excel. Another uses Python. They're looking at the same data but flagging different records as anomalies.

Box Plots Tell Different Stories

Box plots (box-and-whisker plots) anchor their box at Q1 and Q3. The whiskers extend to the furthest non-outlier points. Think about it: change the quartile method, and the whole visual shifts. Which means the box gets wider or narrower. Whiskers stretch or shrink Still holds up..

If you're comparing distributions across groups — say, customer satisfaction scores by region — inconsistent quartile methods make comparisons misleading The details matter here. But it adds up..

Grading Curves and Performance Benchmarks

Some grading schemes curve to quartiles. " "Top quartile gets an A."Bottom quartile gets a C." If you're the student right on the boundary, the calculation method literally changes your grade.

Same with sales quotas, call center handle-time targets, manufacturing defect rates. The quartile is the benchmark.

How to Calculate the Lower Quartile (Step by Step)

Let's get practical. Here are the three most common methods you'll encounter, plus how to execute each one.

Method 1: Inclusive (Excel QUARTILE.INC, Google Sheets QUARTILE)

This is the default in Excel and Google Sheets. It includes the median in both halves when splitting the data.

Steps:

1. Sort your data ascending
2. Find the median (Q2)
3. Split the data into a lower half and upper half — including the median in both halves if n is odd
4. Find the median of the lower half. That's your Q1.

Example: Data = [3, 5, 7, 8, 9, 11, 12] (n = 7)

• Median = 8 (4th value)
• Lower half including median = [3, 5, 7, 8]
• Median of lower half = (5 + 7) / 2 = 6

So Q1 = 6.

Method 2: Exclusive (Excel QUARTILE.EXC, TI-84 Calculator)

This method excludes the median when splitting. Used by TI-83/84 calculators and Excel's .EXC function.

Steps:

1. Sort your data ascending
2. Find the median
3. Split into lower and upper halves — excluding the median entirely
4. Find the median of the lower half.

Same example: [3, 5, 7, 8, 9, 11, 12]

• Median = 8
• Lower half excluding median = [3, 5, 7]
• Median of lower half = 5

Q1 = 5. Different from Method 1 That's the part that actually makes a difference..

Method 3: Linear Interpolation (R default, Python numpy default, Minitab, SPSS)

This is the most statistically rigorous approach. It treats the percentile as a continuous position and interpolates between values.

Formula: Position = (n + 1) × 0.25

If the position is an integer, take that value. If it's fractional, interpolate between the two surrounding values.

Same example: n = 7

• Position = (7 + 1) × 0.25 = 2
• 2nd value = 5

Q1 = 5.

But watch what happens with n = 8:

• Data = [3, 5, 7, 8, 9, 11, 12, 14]
• Position = (8 + 1) × 0.Practically speaking, 25 = 2. Day to day, 25
• Interpolate between 2nd value (5) and 3rd value (7): 5 + 0. 25×(7−5) = **5.

Quick Comparison Table

| Dataset | Inclusive (Excel .EXC) | Interpolation (R/Python) | |---------|------------------------|------------------------|--------------------------| | [3,5,7,8,9,11,12] | 6 | 5 | 5 | | [3,5,7,8,9,11,12,14] | 6 | 6 | 5.INC) | Exclusive (Excel .5 | | [10,20,30,40] | 15 | 15 | 17.

See the differences? Which means they're real. And they compound when you calculate IQR.

How to Do It in Your Tool of Choice

Excel / Google Sheets:

• =QUARTILE.INC(range, 1) — inclusive (default QUARTILE does this too)
• =QUARTILE.EXC(range, 1) — exclusive

R:

• quantile(x, 0.25) — default type 7 (interpolation)
• quantile(x, 0.25, type = 6) — matches Excel .INC
• `quantile(x, 0.25, type =

7)` — matches Excel.EXC

Python (NumPy/Pandas):

• numpy.percentile(data, 25) — Uses linear interpolation by default.
• pandas.Series.quantile(0.25) — Uses linear interpolation by default.

TI-84 Calculator:

• 1-Var Stats $\rightarrow$ Look for the value labeled $Q_1$ in the output list.

Summary: Which One Should You Use?

The "correct" method depends entirely on your context. There is no single mathematical truth; there is only the convention used by your software or your field of study No workaround needed..

1. Use Inclusive (.INC) when you are working in a standard business or administrative setting where you are using basic spreadsheet functions. It is the most "conservative" method because it tends to pull the quartiles closer to the center of the data.
2. Use Exclusive (.EXC) when you are using specialized statistical hardware (like a TI-84) or if you are following specific educational curricula that mandate excluding the median. This method often results in wider, more spread-out quartiles.
3. Use Interpolation (R/Python) when you are performing advanced statistical modeling, data science, or academic research. Because it treats the data as a continuous distribution rather than a discrete set of points, it is the most mathematically reliable for large datasets.

Conclusion

The discrepancy between these methods is a common source of frustration for students and data analysts alike. If you calculate the Interquartile Range (IQR) for a dataset and your answer doesn't match your textbook or a colleague's Python script, don't assume you made a calculation error. Instead, check the methodology.

Not the most exciting part, but easily the most useful And that's really what it comes down to..

Always document which method you are using when presenting data. In the world of statistics, knowing how you arrived at a number is just as important as the number itself.