Is the lower quartile here $18.5$ or $27$?

Take the following list of data:

$$4\ 10\ 27\ 27\ 29\ 34\ 34\ 34\ 37$$

If I remove the median which is $$29$$ then the left hand side is:

$$4\ 10\ 27\ 27$$

The median of this side is $$\frac{10+27}{2}=18.5$$

But if I use the percentile formula: $$P_{25} =\frac{25}{100} 9= 2.25$$

$$2.25$$ is not a whole number so we take the next whole number which is $$3$$.

The $$P_{25}$$ is the third value which is $$27$$.

Note: Wikipedia defines the lower quartile as the middle number between the lowest number and the median. In this case it would be $$27$$.

Which of these is the lower quartile, $$18.5$$, $$27$$, or somethings else?

• R's quantile function offers 9 approaches to this question with answers ranging from $10$ to $27$. My favourite types (2 and 7) both suggest $27$, the former because if there are nine values then the third covers from $22.22\ldots\%$ of the distribution to $33.33\ldots\%$, including the $25\%$ point. May 27, 2020 at 23:07
• @Henry Interesting , I have been promising myself I would learn R for way too long now. May 27, 2020 at 23:41

The lower quartile of that data set is $$\frac{10+27}{2}=18.5$$. The lower quartile is essentially the median of the lower half of the data set. Calculating the lower quartile is no different when the lower half of the data set has an even number of data points than if it you were calculating the median: average the middle two.

• And what value would you get for $P_{25}$? May 27, 2020 at 19:09
• The 25th percentile ($P_{25}$) is equal to the first quartile. It is also $18.5$.
– Ty.
May 27, 2020 at 19:12
• How did you calculate $P_{25}$? Can you show all the details? If you did: $\frac{25}{100} 9=2.25$, why did you not take the third value? May 27, 2020 at 19:14
• I already calculated the first quartile in the original answer which was the average between the second and third data point in the lower quartile. The first quartile is equal to $P_{25}$. "A percentile is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations falls." If you take the third term for $P_{25}$ that would be the 33rd percentile, not the 25th percentile. Because there is no one data point where the 25th percentile falls on, you have to take the average of the 2nd data point and 3rd data point for this data set
– Ty.
May 27, 2020 at 19:22
• It seems that it depends on which definition of percentiles we are using. According to onlinestatbook.com/2/introduction/percentiles.html, $R=\frac{\text{Percentile}}{100} \cdot (N+1)=\frac{25}{100}\cdot(9+1)=2.5$. So the 25th percentile is $10+(0.25)(27-10)=14.25$, but it could be $10$ or $27$ as well. Thinking back, I think that the 25th percentile for this data set (I'm assuming its discrete) is 27. I'm sure someone else can provide a more definitive answer.
– Ty.
May 27, 2020 at 19:40