Quartiles and the interquartile range

Quartiles are the three values that split sorted data into four equal parts: Q1 (the 25th percentile), the median and Q3 (the 75th percentile), and the interquartile range (IQR = Q3 − Q1) is the spread of the middle half of the data.

Last updated: 07 Oct, 2026 · SciPy 1.18

Percentiles and percentile rank found the value at any percentile with the (n + 1)p position. Quartiles are three of those percentiles, and the distance between two of them gives a measure of spread that one outlier cannot inflate. The same two numbers set the fences that flag outliers.

Setting a lower fence and an upper fence

The board's dataset has 19 values, and one of them looks out of place:

1, 2, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 6, 6, 7, 8, 8, 9, 27

To decide which values are outliers, the data gets a lower fence and an upper fence. Values above the upper fence or below the lower fence are treated as outliers. A value of −50 in this data would be one too, on the low side. The fences are built from the quartiles and the IQR:

Finding Q1 and Q3 with the (n + 1)p position

Q1 is the 25th percentile and Q3 the 75th. With n = 19 the positions are 25/100 × (19 + 1) = 5 and 75/100 × 20 = 15, both whole numbers, so no interpolation is needed. The 5th sorted value is 3, so Q1 = 3. The 15th is 7, so Q3 = 7. The interquartile range is IQR = 7 − 3 = 4.

Computing the fences for the 19 values

• Lower fence = 3 − 1.5 × 4 = 3 − 6 = −3.
• Upper fence = 7 + 1.5 × 4 = 7 + 6 = 13.

Anything greater than 13 or less than −3 is treated as an outlier. Only 27 is outside the range from −3 to 13, so 27 is flagged. The 1.5 × IQR rule is Tukey's convention for flagging candidates: whether a flagged value is a mistake or a real extreme value is a separate question (Outlier detection with IQR and z-score).

Comparing quantile methods

The board's quartiles use the (n + 1)p position. NumPy, pandas, matplotlib and seaborn use the 'linear' method by default, and on these 19 values it gives a different Q3:

Every method flags 27 as the only outlier, so the outlier verdict does not change. The quartiles themselves do, which matters when you compare a hand calculation with software, or one library with another.

Computing quartiles in Python

np.percentile with an explicit method

import numpy as np

q1, q3 = np.percentile(data, [25, 75], method="weibull")   # the (n + 1)p method
iqr = q3 - q1
lower_fence, upper_fence = q1 - 1.5 * iqr, q3 + 1.5 * iqr

pandas quantile and scipy.stats.iqr

Both use the linear method by default. scipy.stats.iqr takes the same method names through its interpolation argument.

import pandas as pd
from scipy import stats

pd.Series(data).quantile([0.25, 0.75])    # Q1 and Q3, linear method
stats.iqr(data)                           # Q3 - Q1, linear method

Flagging 27 with the fences

import numpy as np
import matplotlib.pyplot as plt

data = [1, 2, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 6, 6, 7, 8, 8, 9, 27]
q1, q3 = np.percentile(data, [25, 75], method="weibull")    # the (n + 1)p method
iqr = q3 - q1
lower_fence, upper_fence = q1 - 1.5 * iqr, q3 + 1.5 * iqr
print("Q1:", q1, " Q3:", q3, " IQR:", iqr)
print("fences:", lower_fence, upper_fence)
print("outside the fences:", [x for x in data if x < lower_fence or x > upper_fence])

fig, ax = plt.subplots(figsize=(9, 2.6))
seen = {}
for v in data:
    seen[v] = seen.get(v, 0) + 1
    ax.scatter(v, seen[v], color="red" if v > upper_fence else "grey", s=45, zorder=3)
ax.axvspan(q1, q3, color="lightblue", label="Q1 to Q3 (IQR = 4)")
ax.axvline(lower_fence, color="black", linestyle="--", label="fences -3 and 13")
ax.axvline(upper_fence, color="black", linestyle="--")
ax.set_xticks([-3, 0, 3, 5, 7, 10, 13, 20, 27])
ax.set_yticks([])
ax.set_ylim(0, 5.5)
ax.set_title("The 19 values with the lower and upper fence")
ax.legend(loc="upper right")
plt.show()

Q1: 3.0  Q3: 7.0  IQR: 4.0
fences: -3.0 13.0
outside the fences: [27]

What the fences show

• Q1 = 3.0, Q3 = 7.0, IQR = 4.0, the board's values, because method='weibull' is the (n + 1)p method.
• The fences are −3.0 and 13.0, and the only value outside them is 27.

Running every quantile method on the 19 values

import numpy as np
import pandas as pd

data = [1, 2, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 6, 6, 7, 8, 8, 9, 27]
for method in ["weibull", "linear", "hazen", "median_unbiased"]:
    q1, q3 = np.percentile(data, [25, 75], method=method)
    iqr = q3 - q1
    print(f"{method:16} Q1={q1:.3f}  Q3={q3:.3f}  IQR={iqr:.3f}"
          f"  fences={q1 - 1.5 * iqr:.3f}, {q3 + 1.5 * iqr:.3f}")
print("pandas quantile: ", pd.Series(data).quantile([0.25, 0.75]).tolist())

s = sorted(data)
print("Tukey hinges:    ", np.median(s[:9]), np.median(s[10:]))   # medians of the two halves

without = data[:-1]                                              # the 18 values without 27
q1w, q3w = np.percentile(without, [25, 75], method="weibull")
print("IQR without 27:", q3w - q1w)
print("sample SD with 27:", round(np.std(data, ddof=1), 2), " without:", round(np.std(without, ddof=1), 2))

weibull          Q1=3.000  Q3=7.000  IQR=4.000  fences=-3.000, 13.000
linear           Q1=3.000  Q3=6.500  IQR=3.500  fences=-2.250, 11.750
hazen            Q1=3.000  Q3=6.750  IQR=3.750  fences=-2.625, 12.375
median_unbiased  Q1=3.000  Q3=6.833  IQR=3.833  fences=-2.750, 12.583
pandas quantile:  [3.0, 6.5]
Tukey hinges:     3.0 7.0
IQR without 27: 3.5
sample SD with 27: 5.56  without: 2.35

What the methods show

• Q1 is 3 for every method: the values around the 25th percentile are tied.
• Q3 ranges from 6.5 to 7: 7.0 with weibull, 6.5 with linear, 6.75 with hazen and 6.833 with median_unbiased.
• pandas agrees with NumPy's default, [3.0, 6.5].
• Tukey's hinges, the medians of the nine values on either side of the median, give 3.0 and 7.0, the same as the board.
• Without 27 the IQR moves only from 4 to 3.5, while the sample standard deviation drops from 5.56 to 2.35.

IQR vs standard deviation

Where you use quartiles and the IQR

• Describing skewed data such as salaries or house prices: the median with Q1 and Q3 says more than the mean and the standard deviation.
• Flagging outliers with the 1.5 × IQR fences before training a model.
• Scaling features: scikit-learn's RobustScaler subtracts the median and divides by the IQR, so outliers do not squash the other values.

Related

• Previous: Percentiles and percentile rank
• Next: Five-number summary and box plot
• See also: Range, MAD and coefficient of variation