Calculate the third quartile (Q3/75th percentile) using exclusive, inclusive, and interpolated methods. Includes quartile number line, method comparison, derived measures, and percentile table.
The third quartile (Q3) calculator finds the 75th percentile of a dataset, the point below which roughly three quarters of the observations fall.
Q3 matters because it defines the upper edge of the interquartile range and feeds directly into box plots, Tukey fences, and other robust descriptive summaries. This tool shows several common quartile conventions side by side so you can see how software and textbooks may differ for the same dataset.
Along with Q3 itself, the page reports derived measures such as the upper fence, midhinge, trimean, and quartile-based spread metrics.
Q3 often gets used beyond a simple percentile lookup. It becomes the top of the box in a box plot, the starting point for upper-fence outlier checks, and a cutoff for describing the upper quarter of a distribution.
Seeing the alternative quartile methods and the downstream fence values together helps prevent silent method mismatches when you move between spreadsheets, textbooks, and code libraries.
Exclusive (Tukey): Q₃ = value at position 0.75(n+1) with linear interpolation. Inclusive: Q₃ = value at position 0.75(n−1) with linear interpolation. IQR = Q₃ − Q₁. Upper fence = Q₃ + 1.5×IQR.
Result: Q₃ = 89.00, Q₁ = 69.50, IQR = 19.50
With n=15 exam scores sorted in order, the exclusive method places Q₃ at position 0.75×16 = 12th position. The 12th value is 89. Q₁ falls at position 4 (interpolated between 68 and 71 = 69.50). The IQR = 89.00 − 69.50 = 19.50, and the upper fence = 89.00 + 1.5×19.50 = 118.25.
The box plot is built entirely around quartiles: Q₁ is the bottom of the box, Q₂ (median) is the line inside, Q₃ is the top. The whiskers extend to the most extreme non-outlier values, defined by the upper fence (Q₃ + 1.5×IQR) and lower fence (Q₁ − 1.5×IQR). When the Q₃–Max range is much larger than the Min–Q₁ range, the data shows positive skew — visible as a longer upper whisker.
Q₃ is especially important in income data because income distributions are typically right-skewed. The median income tells you the "typical" person, Q₃ tells you the boundary of the "upper-middle" income bracket, and Q₃ + 1.5×IQR identifies extreme high earners. Policy analysts and economists use Q₃ to define "upper quartile" income thresholds for tax brackets, benefits eligibility, and affordability metrics.
For odd sample sizes (e.g., n=15), the exclusive method excludes the median from both halves, giving Q₃ as the median of the upper 7 values. The inclusive method includes the median in both halves, giving Q₃ as the median of the upper 8 values. For even sample sizes (e.g., n=20), both methods split the data into equal halves of 10, but still differ in the interpolation formula. Standard software defaults: Excel QUARTILE.INC = inclusive, R quantile type 7 = interpolated, Python numpy.percentile = interpolated.
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The third quartile (Q₃) is the value that separates the top 25% from the bottom 75% of a sorted dataset. It's the 75th percentile. Q₃ is one of the five-number summary values (min, Q₁, median, Q₃, max) and forms the upper edge of the box in box-and-whisker plots.
Sort the data in ascending order. Find the median (Q₂), which splits the data into a lower half and upper half. Q₃ is the median of the upper half. If n is odd, whether you include the overall median in the upper half depends on the method: exclusive (Tukey) excludes it, inclusive includes it.
There's no single mathematical definition of quartiles for finite datasets — only for continuous distributions. Different statisticians proposed different conventions. Tukey's exclusive method excludes the median when splitting, Moore & McCabe's inclusive method includes it, and interpolation methods use weighted averages. For large datasets, they give nearly identical results.
The upper fence = Q₃ + 1.5×IQR is the threshold above which values are flagged as mild outliers. The upper outer fence = Q₃ + 3×IQR flags extreme outliers. These boundaries drive the whisker endpoints in box plots. Q₃ itself is the top of the box.
In a box plot, Q₃ is the top edge of the box. The box spans from Q₁ to Q₃ (the IQR). The upper whisker extends from Q₃ to the largest value within Q₃ + 1.5×IQR. Any points beyond the whisker are plotted as individual outlier dots.
They're the same concept by definition: Q₃ = P75 = the value below which 75% of observations fall. Some textbooks distinguish between "quartile" (which may use exclusive/inclusive splitting) and "percentile" (which may use interpolation), but in practice, the terms are interchangeable.